![Page 1: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/1.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Fermions, Bosons and their Statistics
Peter Hertel
University of Osnabruck, Germany
Lecture presented at APS, Nankai University, China
http://www.home.uni-osnabrueck.de/phertel
March/April 2011
![Page 2: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/2.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Structure fermions
charge -1 -2/3 -1/3 0 +1/3 +2/3 +1
generation 1 e− u d νe , νe d u e+
generation 2 µ− c s νµ, νµ s c µ+
generation 3 τ− t b ντ , ντ b t τ+
• all structure particles have spin 1/2
• proton p=(uud) and neutron n=(udd)
• both have spin 1/2
• excited states ∆++=(uuu)∗, ∆+=(uud)∗ etc.
• they have spin 3/2
• p is stable, n → p+e−+ν is allowed
![Page 3: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/3.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Structure fermions
charge -1 -2/3 -1/3 0 +1/3 +2/3 +1
generation 1 e− u d νe , νe d u e+
generation 2 µ− c s νµ, νµ s c µ+
generation 3 τ− t b ντ , ντ b t τ+
• all structure particles have spin 1/2
• proton p=(uud) and neutron n=(udd)
• both have spin 1/2
• excited states ∆++=(uuu)∗, ∆+=(uud)∗ etc.
• they have spin 3/2
• p is stable, n → p+e−+ν is allowed
![Page 4: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/4.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Structure fermions
charge -1 -2/3 -1/3 0 +1/3 +2/3 +1
generation 1 e− u d νe , νe d u e+
generation 2 µ− c s νµ, νµ s c µ+
generation 3 τ− t b ντ , ντ b t τ+
• all structure particles have spin 1/2
• proton p=(uud) and neutron n=(udd)
• both have spin 1/2
• excited states ∆++=(uuu)∗, ∆+=(uud)∗ etc.
• they have spin 3/2
• p is stable, n → p+e−+ν is allowed
![Page 5: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/5.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Structure fermions
charge -1 -2/3 -1/3 0 +1/3 +2/3 +1
generation 1 e− u d νe , νe d u e+
generation 2 µ− c s νµ, νµ s c µ+
generation 3 τ− t b ντ , ντ b t τ+
• all structure particles have spin 1/2
• proton p=(uud) and neutron n=(udd)
• both have spin 1/2
• excited states ∆++=(uuu)∗, ∆+=(uud)∗ etc.
• they have spin 3/2
• p is stable, n → p+e−+ν is allowed
![Page 6: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/6.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Structure fermions
charge -1 -2/3 -1/3 0 +1/3 +2/3 +1
generation 1 e− u d νe , νe d u e+
generation 2 µ− c s νµ, νµ s c µ+
generation 3 τ− t b ντ , ντ b t τ+
• all structure particles have spin 1/2
• proton p=(uud) and neutron n=(udd)
• both have spin 1/2
• excited states ∆++=(uuu)∗, ∆+=(uud)∗ etc.
• they have spin 3/2
• p is stable, n → p+e−+ν is allowed
![Page 7: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/7.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Structure fermions
charge -1 -2/3 -1/3 0 +1/3 +2/3 +1
generation 1 e− u d νe , νe d u e+
generation 2 µ− c s νµ, νµ s c µ+
generation 3 τ− t b ντ , ντ b t τ+
• all structure particles have spin 1/2
• proton p=(uud) and neutron n=(udd)
• both have spin 1/2
• excited states ∆++=(uuu)∗, ∆+=(uud)∗ etc.
• they have spin 3/2
• p is stable, n → p+e−+ν is allowed
![Page 8: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/8.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Exchange bosons
W− νee− νµµ
− νττ− ud cs tb
γ,Z 0 e+e− µ+µ− τ+τ− (νeνe νµνµ ντντ ) dd uu ss cc bb tt
W+ e+νe µ+νµ τ
+ντ du sc bt
• W+, γ, Z0, W− mediate electro-weak interaction
• the photon couples to charged particles only
• there are also gluons which mediate strong interactions
• e.g. n=(udd) → (uud)+W− → p+νe+e−
• is there also a graviton G ?
• is there also a Higgs boson H ?
![Page 9: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/9.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Exchange bosons
W− νee− νµµ
− νττ− ud cs tb
γ,Z 0 e+e− µ+µ− τ+τ− (νeνe νµνµ ντντ ) dd uu ss cc bb tt
W+ e+νe µ+νµ τ
+ντ du sc bt
• W+, γ, Z0, W− mediate electro-weak interaction
• the photon couples to charged particles only
• there are also gluons which mediate strong interactions
• e.g. n=(udd) → (uud)+W− → p+νe+e−
• is there also a graviton G ?
• is there also a Higgs boson H ?
![Page 10: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/10.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Exchange bosons
W− νee− νµµ
− νττ− ud cs tb
γ,Z 0 e+e− µ+µ− τ+τ− (νeνe νµνµ ντντ ) dd uu ss cc bb tt
W+ e+νe µ+νµ τ
+ντ du sc bt
• W+, γ, Z0, W− mediate electro-weak interaction
• the photon couples to charged particles only
• there are also gluons which mediate strong interactions
• e.g. n=(udd) → (uud)+W− → p+νe+e−
• is there also a graviton G ?
• is there also a Higgs boson H ?
![Page 11: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/11.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Exchange bosons
W− νee− νµµ
− νττ− ud cs tb
γ,Z 0 e+e− µ+µ− τ+τ− (νeνe νµνµ ντντ ) dd uu ss cc bb tt
W+ e+νe µ+νµ τ
+ντ du sc bt
• W+, γ, Z0, W− mediate electro-weak interaction
• the photon couples to charged particles only
• there are also gluons which mediate strong interactions
• e.g. n=(udd) → (uud)+W− → p+νe+e−
• is there also a graviton G ?
• is there also a Higgs boson H ?
![Page 12: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/12.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Exchange bosons
W− νee− νµµ
− νττ− ud cs tb
γ,Z 0 e+e− µ+µ− τ+τ− (νeνe νµνµ ντντ ) dd uu ss cc bb tt
W+ e+νe µ+νµ τ
+ντ du sc bt
• W+, γ, Z0, W− mediate electro-weak interaction
• the photon couples to charged particles only
• there are also gluons which mediate strong interactions
• e.g. n=(udd) → (uud)+W− → p+νe+e−
• is there also a graviton G ?
• is there also a Higgs boson H ?
![Page 13: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/13.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Exchange bosons
W− νee− νµµ
− νττ− ud cs tb
γ,Z 0 e+e− µ+µ− τ+τ− (νeνe νµνµ ντντ ) dd uu ss cc bb tt
W+ e+νe µ+νµ τ
+ντ du sc bt
• W+, γ, Z0, W− mediate electro-weak interaction
• the photon couples to charged particles only
• there are also gluons which mediate strong interactions
• e.g. n=(udd) → (uud)+W− → p+νe+e−
• is there also a graviton G ?
• is there also a Higgs boson H ?
![Page 14: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/14.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation The Aleph detector (opened) at Cern
![Page 15: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/15.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Spin 1/2 fermions
• massive structure fermion (quarks, electrons) have spin1/2
• or helicity 1/2 and -1/2
• massless structure fermions (neutrinos) have either helicity1/2 or -1/2
•
• this explains parity violation
• spin and statistics are related
![Page 16: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/16.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Spin 1/2 fermions
• massive structure fermion (quarks, electrons) have spin1/2
• or helicity 1/2 and -1/2
• massless structure fermions (neutrinos) have either helicity1/2 or -1/2
•
• this explains parity violation
• spin and statistics are related
![Page 17: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/17.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Spin 1/2 fermions
• massive structure fermion (quarks, electrons) have spin1/2
• or helicity 1/2 and -1/2
• massless structure fermions (neutrinos) have either helicity1/2 or -1/2
•
• this explains parity violation
• spin and statistics are related
![Page 18: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/18.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Spin 1/2 fermions
• massive structure fermion (quarks, electrons) have spin1/2
• or helicity 1/2 and -1/2
• massless structure fermions (neutrinos) have either helicity1/2 or -1/2
•
• this explains parity violation
• spin and statistics are related
![Page 19: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/19.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Spin 1/2 fermions
• massive structure fermion (quarks, electrons) have spin1/2
• or helicity 1/2 and -1/2
• massless structure fermions (neutrinos) have either helicity1/2 or -1/2
•
• this explains parity violation
• spin and statistics are related
![Page 20: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/20.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Spin 1/2 fermions
• massive structure fermion (quarks, electrons) have spin1/2
• or helicity 1/2 and -1/2
• massless structure fermions (neutrinos) have either helicity1/2 or -1/2
•
• this explains parity violation
• spin and statistics are related
![Page 21: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/21.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Spin and statisics
• rotation of a particle with spin s is described by
ψR = ei~α · Sψ
• a 360 rotation yields
ψ = e2πis
ψ = φRψ
• φR = +1 for integer spin (bosons), and -1 for half-integerspin (fermions)
• exchanging two identicle particles must not change |ψ|2
ψ(σ1x1, σ2x2) = φXψ(σ2x2, σ1x1)
• spin-statistics theorem says φR = φX
• for fermions: ψ(σ1x1, σ2x2) = −ψ(σ2x2, σ1x1)
• Pauli exclusion principle
![Page 22: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/22.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Spin and statisics
• rotation of a particle with spin s is described by
ψR = ei~α · Sψ
• a 360 rotation yields
ψ = e2πis
ψ = φRψ
• φR = +1 for integer spin (bosons), and -1 for half-integerspin (fermions)
• exchanging two identicle particles must not change |ψ|2
ψ(σ1x1, σ2x2) = φXψ(σ2x2, σ1x1)
• spin-statistics theorem says φR = φX
• for fermions: ψ(σ1x1, σ2x2) = −ψ(σ2x2, σ1x1)
• Pauli exclusion principle
![Page 23: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/23.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Spin and statisics
• rotation of a particle with spin s is described by
ψR = ei~α · Sψ
• a 360 rotation yields
ψ = e2πis
ψ = φRψ
• φR = +1 for integer spin (bosons), and -1 for half-integerspin (fermions)
• exchanging two identicle particles must not change |ψ|2
ψ(σ1x1, σ2x2) = φXψ(σ2x2, σ1x1)
• spin-statistics theorem says φR = φX
• for fermions: ψ(σ1x1, σ2x2) = −ψ(σ2x2, σ1x1)
• Pauli exclusion principle
![Page 24: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/24.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Spin and statisics
• rotation of a particle with spin s is described by
ψR = ei~α · Sψ
• a 360 rotation yields
ψ = e2πis
ψ = φRψ
• φR = +1 for integer spin (bosons), and -1 for half-integerspin (fermions)
• exchanging two identicle particles must not change |ψ|2
ψ(σ1x1, σ2x2) = φXψ(σ2x2, σ1x1)
• spin-statistics theorem says φR = φX
• for fermions: ψ(σ1x1, σ2x2) = −ψ(σ2x2, σ1x1)
• Pauli exclusion principle
![Page 25: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/25.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Spin and statisics
• rotation of a particle with spin s is described by
ψR = ei~α · Sψ
• a 360 rotation yields
ψ = e2πis
ψ = φRψ
• φR = +1 for integer spin (bosons), and -1 for half-integerspin (fermions)
• exchanging two identicle particles must not change |ψ|2
ψ(σ1x1, σ2x2) = φXψ(σ2x2, σ1x1)
• spin-statistics theorem says φR = φX
• for fermions: ψ(σ1x1, σ2x2) = −ψ(σ2x2, σ1x1)
• Pauli exclusion principle
![Page 26: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/26.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Spin and statisics
• rotation of a particle with spin s is described by
ψR = ei~α · Sψ
• a 360 rotation yields
ψ = e2πis
ψ = φRψ
• φR = +1 for integer spin (bosons), and -1 for half-integerspin (fermions)
• exchanging two identicle particles must not change |ψ|2
ψ(σ1x1, σ2x2) = φXψ(σ2x2, σ1x1)
• spin-statistics theorem says φR = φX
• for fermions: ψ(σ1x1, σ2x2) = −ψ(σ2x2, σ1x1)
• Pauli exclusion principle
![Page 27: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/27.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Spin and statisics
• rotation of a particle with spin s is described by
ψR = ei~α · Sψ
• a 360 rotation yields
ψ = e2πis
ψ = φRψ
• φR = +1 for integer spin (bosons), and -1 for half-integerspin (fermions)
• exchanging two identicle particles must not change |ψ|2
ψ(σ1x1, σ2x2) = φXψ(σ2x2, σ1x1)
• spin-statistics theorem says φR = φX
• for fermions: ψ(σ1x1, σ2x2) = −ψ(σ2x2, σ1x1)
• Pauli exclusion principle
![Page 28: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/28.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Wolfgang Pauli, Austrian/Swiss Physicist, Nobel prize 1945
![Page 29: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/29.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Arbitrary number of particles
• A∗ creates a particle, A = (A∗)† annihilates a particle
• (A,A) = 0, (A∗,A∗) = 0, (A,A∗) = I
• bosons: (X ,Y ) = [X ,Y ] = XY − YX , the commutator
• fermions: (X ,Y ) = X ,Y = XY + YX , theanti-commutator
• N = A∗A is number operator
• Ω is vacuum
• create n particles
ψn =1√n!
(A∗)n Ω
• Nψn = nψn
• bosons: n = 0, 1, 2, . . .
• fermions: n = 0, 1
![Page 30: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/30.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Arbitrary number of particles
• A∗ creates a particle, A = (A∗)† annihilates a particle
• (A,A) = 0, (A∗,A∗) = 0, (A,A∗) = I
• bosons: (X ,Y ) = [X ,Y ] = XY − YX , the commutator
• fermions: (X ,Y ) = X ,Y = XY + YX , theanti-commutator
• N = A∗A is number operator
• Ω is vacuum
• create n particles
ψn =1√n!
(A∗)n Ω
• Nψn = nψn
• bosons: n = 0, 1, 2, . . .
• fermions: n = 0, 1
![Page 31: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/31.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Arbitrary number of particles
• A∗ creates a particle, A = (A∗)† annihilates a particle
• (A,A) = 0, (A∗,A∗) = 0, (A,A∗) = I
• bosons: (X ,Y ) = [X ,Y ] = XY − YX , the commutator
• fermions: (X ,Y ) = X ,Y = XY + YX , theanti-commutator
• N = A∗A is number operator
• Ω is vacuum
• create n particles
ψn =1√n!
(A∗)n Ω
• Nψn = nψn
• bosons: n = 0, 1, 2, . . .
• fermions: n = 0, 1
![Page 32: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/32.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Arbitrary number of particles
• A∗ creates a particle, A = (A∗)† annihilates a particle
• (A,A) = 0, (A∗,A∗) = 0, (A,A∗) = I
• bosons: (X ,Y ) = [X ,Y ] = XY − YX , the commutator
• fermions: (X ,Y ) = X ,Y = XY + YX , theanti-commutator
• N = A∗A is number operator
• Ω is vacuum
• create n particles
ψn =1√n!
(A∗)n Ω
• Nψn = nψn
• bosons: n = 0, 1, 2, . . .
• fermions: n = 0, 1
![Page 33: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/33.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Arbitrary number of particles
• A∗ creates a particle, A = (A∗)† annihilates a particle
• (A,A) = 0, (A∗,A∗) = 0, (A,A∗) = I
• bosons: (X ,Y ) = [X ,Y ] = XY − YX , the commutator
• fermions: (X ,Y ) = X ,Y = XY + YX , theanti-commutator
• N = A∗A is number operator
• Ω is vacuum
• create n particles
ψn =1√n!
(A∗)n Ω
• Nψn = nψn
• bosons: n = 0, 1, 2, . . .
• fermions: n = 0, 1
![Page 34: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/34.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Arbitrary number of particles
• A∗ creates a particle, A = (A∗)† annihilates a particle
• (A,A) = 0, (A∗,A∗) = 0, (A,A∗) = I
• bosons: (X ,Y ) = [X ,Y ] = XY − YX , the commutator
• fermions: (X ,Y ) = X ,Y = XY + YX , theanti-commutator
• N = A∗A is number operator
• Ω is vacuum
• create n particles
ψn =1√n!
(A∗)n Ω
• Nψn = nψn
• bosons: n = 0, 1, 2, . . .
• fermions: n = 0, 1
![Page 35: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/35.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Arbitrary number of particles
• A∗ creates a particle, A = (A∗)† annihilates a particle
• (A,A) = 0, (A∗,A∗) = 0, (A,A∗) = I
• bosons: (X ,Y ) = [X ,Y ] = XY − YX , the commutator
• fermions: (X ,Y ) = X ,Y = XY + YX , theanti-commutator
• N = A∗A is number operator
• Ω is vacuum
• create n particles
ψn =1√n!
(A∗)n Ω
• Nψn = nψn
• bosons: n = 0, 1, 2, . . .
• fermions: n = 0, 1
![Page 36: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/36.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Arbitrary number of particles
• A∗ creates a particle, A = (A∗)† annihilates a particle
• (A,A) = 0, (A∗,A∗) = 0, (A,A∗) = I
• bosons: (X ,Y ) = [X ,Y ] = XY − YX , the commutator
• fermions: (X ,Y ) = X ,Y = XY + YX , theanti-commutator
• N = A∗A is number operator
• Ω is vacuum
• create n particles
ψn =1√n!
(A∗)n Ω
• Nψn = nψn
• bosons: n = 0, 1, 2, . . .
• fermions: n = 0, 1
![Page 37: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/37.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Arbitrary number of particles
• A∗ creates a particle, A = (A∗)† annihilates a particle
• (A,A) = 0, (A∗,A∗) = 0, (A,A∗) = I
• bosons: (X ,Y ) = [X ,Y ] = XY − YX , the commutator
• fermions: (X ,Y ) = X ,Y = XY + YX , theanti-commutator
• N = A∗A is number operator
• Ω is vacuum
• create n particles
ψn =1√n!
(A∗)n Ω
• Nψn = nψn
• bosons: n = 0, 1, 2, . . .
• fermions: n = 0, 1
![Page 38: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/38.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Arbitrary number of particles
• A∗ creates a particle, A = (A∗)† annihilates a particle
• (A,A) = 0, (A∗,A∗) = 0, (A,A∗) = I
• bosons: (X ,Y ) = [X ,Y ] = XY − YX , the commutator
• fermions: (X ,Y ) = X ,Y = XY + YX , theanti-commutator
• N = A∗A is number operator
• Ω is vacuum
• create n particles
ψn =1√n!
(A∗)n Ω
• Nψn = nψn
• bosons: n = 0, 1, 2, . . .
• fermions: n = 0, 1
![Page 39: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/39.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Many states
• j labels one-particle states
• creators A∗j and annihilators Aj
• (Aj ,Ak) = (A∗j ,A∗k) = 0 and (Aj ,A
∗k) = δjk
• for fermions:
• A∗j A∗kΩ = −A∗kA∗j Ω
• Nj = A∗j Aj has eigenvalues 0 and 1, i.e.
• a state is occupied at most once
• note that [Nj ,Nk ] = 0
![Page 40: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/40.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Many states
• j labels one-particle states
• creators A∗j and annihilators Aj
• (Aj ,Ak) = (A∗j ,A∗k) = 0 and (Aj ,A
∗k) = δjk
• for fermions:
• A∗j A∗kΩ = −A∗kA∗j Ω
• Nj = A∗j Aj has eigenvalues 0 and 1, i.e.
• a state is occupied at most once
• note that [Nj ,Nk ] = 0
![Page 41: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/41.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Many states
• j labels one-particle states
• creators A∗j and annihilators Aj
• (Aj ,Ak) = (A∗j ,A∗k) = 0 and (Aj ,A
∗k) = δjk
• for fermions:
• A∗j A∗kΩ = −A∗kA∗j Ω
• Nj = A∗j Aj has eigenvalues 0 and 1, i.e.
• a state is occupied at most once
• note that [Nj ,Nk ] = 0
![Page 42: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/42.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Many states
• j labels one-particle states
• creators A∗j and annihilators Aj
• (Aj ,Ak) = (A∗j ,A∗k) = 0 and (Aj ,A
∗k) = δjk
• for fermions:
• A∗j A∗kΩ = −A∗kA∗j Ω
• Nj = A∗j Aj has eigenvalues 0 and 1, i.e.
• a state is occupied at most once
• note that [Nj ,Nk ] = 0
![Page 43: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/43.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Many states
• j labels one-particle states
• creators A∗j and annihilators Aj
• (Aj ,Ak) = (A∗j ,A∗k) = 0 and (Aj ,A
∗k) = δjk
• for fermions:
• A∗j A∗kΩ = −A∗kA∗j Ω
• Nj = A∗j Aj has eigenvalues 0 and 1, i.e.
• a state is occupied at most once
• note that [Nj ,Nk ] = 0
![Page 44: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/44.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Many states
• j labels one-particle states
• creators A∗j and annihilators Aj
• (Aj ,Ak) = (A∗j ,A∗k) = 0 and (Aj ,A
∗k) = δjk
• for fermions:
• A∗j A∗kΩ = −A∗kA∗j Ω
• Nj = A∗j Aj has eigenvalues 0 and 1, i.e.
• a state is occupied at most once
• note that [Nj ,Nk ] = 0
![Page 45: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/45.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Many states
• j labels one-particle states
• creators A∗j and annihilators Aj
• (Aj ,Ak) = (A∗j ,A∗k) = 0 and (Aj ,A
∗k) = δjk
• for fermions:
• A∗j A∗kΩ = −A∗kA∗j Ω
• Nj = A∗j Aj has eigenvalues 0 and 1, i.e.
• a state is occupied at most once
• note that [Nj ,Nk ] = 0
![Page 46: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/46.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Many states
• j labels one-particle states
• creators A∗j and annihilators Aj
• (Aj ,Ak) = (A∗j ,A∗k) = 0 and (Aj ,A
∗k) = δjk
• for fermions:
• A∗j A∗kΩ = −A∗kA∗j Ω
• Nj = A∗j Aj has eigenvalues 0 and 1, i.e.
• a state is occupied at most once
• note that [Nj ,Nk ] = 0
![Page 47: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/47.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Energy
• H =∑
j εj A∗j Aj +
∑jklm Vjklm A∗j A
∗kAlAm
• if interaction term can be neglected
• H =∑
j εj Nj
• examples are quasi-free electrons (hopping model) orphonons
• N =∑
j Nj is the particle number operator
• note that the number of particles in a system is not fixed(open system)
![Page 48: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/48.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Energy
• H =∑
j εj A∗j Aj +
∑jklm Vjklm A∗j A
∗kAlAm
• if interaction term can be neglected
• H =∑
j εj Nj
• examples are quasi-free electrons (hopping model) orphonons
• N =∑
j Nj is the particle number operator
• note that the number of particles in a system is not fixed(open system)
![Page 49: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/49.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Energy
• H =∑
j εj A∗j Aj +
∑jklm Vjklm A∗j A
∗kAlAm
• if interaction term can be neglected
• H =∑
j εj Nj
• examples are quasi-free electrons (hopping model) orphonons
• N =∑
j Nj is the particle number operator
• note that the number of particles in a system is not fixed(open system)
![Page 50: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/50.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Energy
• H =∑
j εj A∗j Aj +
∑jklm Vjklm A∗j A
∗kAlAm
• if interaction term can be neglected
• H =∑
j εj Nj
• examples are quasi-free electrons (hopping model) orphonons
• N =∑
j Nj is the particle number operator
• note that the number of particles in a system is not fixed(open system)
![Page 51: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/51.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Energy
• H =∑
j εj A∗j Aj +
∑jklm Vjklm A∗j A
∗kAlAm
• if interaction term can be neglected
• H =∑
j εj Nj
• examples are quasi-free electrons (hopping model) orphonons
• N =∑
j Nj is the particle number operator
• note that the number of particles in a system is not fixed(open system)
![Page 52: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/52.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Energy
• H =∑
j εj A∗j Aj +
∑jklm Vjklm A∗j A
∗kAlAm
• if interaction term can be neglected
• H =∑
j εj Nj
• examples are quasi-free electrons (hopping model) orphonons
• N =∑
j Nj is the particle number operator
• note that the number of particles in a system is not fixed(open system)
![Page 53: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/53.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Free energy
• The equilibrium (Gibbs) state of an open system is
G = e(F − H + µN)/kBT
• free energy F defined by trG = 1
• which gives
F = −kBT ln tr e(µN − H)/kBT
• temperature T defined by trGH = U = H
• chemical potential µ defined by trGN = N
• generalization to more than one species of particles isobvious
![Page 54: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/54.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Free energy
• The equilibrium (Gibbs) state of an open system is
G = e(F − H + µN)/kBT
• free energy F defined by trG = 1
• which gives
F = −kBT ln tr e(µN − H)/kBT
• temperature T defined by trGH = U = H
• chemical potential µ defined by trGN = N
• generalization to more than one species of particles isobvious
![Page 55: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/55.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Free energy
• The equilibrium (Gibbs) state of an open system is
G = e(F − H + µN)/kBT
• free energy F defined by trG = 1
• which gives
F = −kBT ln tr e(µN − H)/kBT
• temperature T defined by trGH = U = H
• chemical potential µ defined by trGN = N
• generalization to more than one species of particles isobvious
![Page 56: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/56.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Free energy
• The equilibrium (Gibbs) state of an open system is
G = e(F − H + µN)/kBT
• free energy F defined by trG = 1
• which gives
F = −kBT ln tr e(µN − H)/kBT
• temperature T defined by trGH = U = H
• chemical potential µ defined by trGN = N
• generalization to more than one species of particles isobvious
![Page 57: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/57.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Free energy
• The equilibrium (Gibbs) state of an open system is
G = e(F − H + µN)/kBT
• free energy F defined by trG = 1
• which gives
F = −kBT ln tr e(µN − H)/kBT
• temperature T defined by trGH = U = H
• chemical potential µ defined by trGN = N
• generalization to more than one species of particles isobvious
![Page 58: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/58.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Free energy
• The equilibrium (Gibbs) state of an open system is
G = e(F − H + µN)/kBT
• free energy F defined by trG = 1
• which gives
F = −kBT ln tr e(µN − H)/kBT
• temperature T defined by trGH = U = H
• chemical potential µ defined by trGN = N
• generalization to more than one species of particles isobvious
![Page 59: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/59.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Josiah Willard Gibbs, US American physicist, 1839-1903
![Page 60: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/60.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Fermi-Dirac statistics
• because particle number operators commute
e(µN − H)/kBT =
∏j
e(µ− εj)Nj/kBT
• work out the trace, i. e. sum over eigenvalues n = 0, 1
tr e(µN − H)/kBT =
∏j
(1 + e
(µ− εj)/kBT)
• free energy
F = −kBT∑j
ln
(1 + e
(µ− εj)/kBT)
• in [ε, ε+ dε] there are V z(ε)dε single particle states
• one may write
F = −kBTV
∫dε z(ε) ln
(1 + e
(µ− ε)/kBT)
![Page 61: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/61.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Fermi-Dirac statistics
• because particle number operators commute
e(µN − H)/kBT =
∏j
e(µ− εj)Nj/kBT
• work out the trace, i. e. sum over eigenvalues n = 0, 1
tr e(µN − H)/kBT =
∏j
(1 + e
(µ− εj)/kBT)
• free energy
F = −kBT∑j
ln
(1 + e
(µ− εj)/kBT)
• in [ε, ε+ dε] there are V z(ε)dε single particle states
• one may write
F = −kBTV
∫dε z(ε) ln
(1 + e
(µ− ε)/kBT)
![Page 62: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/62.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Fermi-Dirac statistics
• because particle number operators commute
e(µN − H)/kBT =
∏j
e(µ− εj)Nj/kBT
• work out the trace, i. e. sum over eigenvalues n = 0, 1
tr e(µN − H)/kBT =
∏j
(1 + e
(µ− εj)/kBT)
• free energy
F = −kBT∑j
ln
(1 + e
(µ− εj)/kBT)
• in [ε, ε+ dε] there are V z(ε)dε single particle states
• one may write
F = −kBTV
∫dε z(ε) ln
(1 + e
(µ− ε)/kBT)
![Page 63: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/63.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Fermi-Dirac statistics
• because particle number operators commute
e(µN − H)/kBT =
∏j
e(µ− εj)Nj/kBT
• work out the trace, i. e. sum over eigenvalues n = 0, 1
tr e(µN − H)/kBT =
∏j
(1 + e
(µ− εj)/kBT)
• free energy
F = −kBT∑j
ln
(1 + e
(µ− εj)/kBT)
• in [ε, ε+ dε] there are V z(ε) dε single particle states
• one may write
F = −kBTV
∫dε z(ε) ln
(1 + e
(µ− ε)/kBT)
![Page 64: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/64.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Fermi-Dirac statistics
• because particle number operators commute
e(µN − H)/kBT =
∏j
e(µ− εj)Nj/kBT
• work out the trace, i. e. sum over eigenvalues n = 0, 1
tr e(µN − H)/kBT =
∏j
(1 + e
(µ− εj)/kBT)
• free energy
F = −kBT∑j
ln
(1 + e
(µ− εj)/kBT)
• in [ε, ε+ dε] there are V z(ε) dε single particle states
• one may write
F = −kBTV
∫dε z(ε) ln
(1 + e
(µ− ε)/kBT)
![Page 65: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/65.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Enrico Fermi, Italian/USA physicist, 1901-1954
![Page 66: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/66.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Paul Dirac, Britisch physicist, 1902-1984
![Page 67: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/67.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Non-relativistic fermions
• sum over states (factor 2 for spin)∑j
= 2
∫d3x d3p
(2π~)3
• ε = p2/2m
• gives∑j
= V
∫dε z(ε)
= V1
π2
(2m
~2
)3/2 ∫dε√ε
![Page 68: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/68.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Non-relativistic fermions
• sum over states (factor 2 for spin)∑j
= 2
∫d3x d3p
(2π~)3
• ε = p2/2m
• gives∑j
= V
∫dε z(ε)
= V1
π2
(2m
~2
)3/2 ∫dε√ε
![Page 69: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/69.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Non-relativistic fermions
• sum over states (factor 2 for spin)∑j
= 2
∫d3x d3p
(2π~)3
• ε = p2/2m
• gives∑j
= V
∫dε z(ε)
= V1
π2
(2m
~2
)3/2 ∫dε√ε
![Page 70: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/70.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Particle density and pressure
• recall
F = −kBTV
∫dε z(ε) ln
(1 + e
(µ− ε)/kBT)
• particle density is
n(T , µ) = − 1
V
∂F
∂µ
=
∫dε z(ε)
1
e(ε− µ)/kBT + 1
• pressure is
p = − ∂F∂V
= kBT
∫dε z(ε) ln
(1 + e
(µ− ε)/kBT)
=2
3
∫dε z(ε)
ε
e(ε− µ)/kBT + 1
![Page 71: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/71.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Particle density and pressure
• recall
F = −kBTV
∫dε z(ε) ln
(1 + e
(µ− ε)/kBT)
• particle density is
n(T , µ) = − 1
V
∂F
∂µ
=
∫dε z(ε)
1
e(ε− µ)/kBT + 1
• pressure is
p = − ∂F∂V
= kBT
∫dε z(ε) ln
(1 + e
(µ− ε)/kBT)
=2
3
∫dε z(ε)
ε
e(ε− µ)/kBT + 1
![Page 72: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/72.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Particle density and pressure
• recall
F = −kBTV
∫dε z(ε) ln
(1 + e
(µ− ε)/kBT)
• particle density is
n(T , µ) = − 1
V
∂F
∂µ
=
∫dε z(ε)
1
e(ε− µ)/kBT + 1
• pressure is
p = − ∂F∂V
= kBT
∫dε z(ε) ln
(1 + e
(µ− ε)/kBT)
=2
3
∫dε z(ε)
ε
e(ε− µ)/kBT + 1
![Page 73: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/73.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Zero temperature
• if T → 01
e(ε− µ)/kBT + 1
= θ(µ− ε)
• particle density
n(0, µ) =
∫ µ
−∞dε z(ε)
• pressure
p(0, µ) =2
3
∫ µ
−∞dε z(ε) ε
• if particle density n is given, then
n = n(0, εF) =
∫ εF
−∞dε z(ε)
defines Fermi energy
![Page 74: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/74.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Zero temperature
• if T → 01
e(ε− µ)/kBT + 1
= θ(µ− ε)
• particle density
n(0, µ) =
∫ µ
−∞dε z(ε)
• pressure
p(0, µ) =2
3
∫ µ
−∞dε z(ε) ε
• if particle density n is given, then
n = n(0, εF) =
∫ εF
−∞dε z(ε)
defines Fermi energy
![Page 75: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/75.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Zero temperature
• if T → 01
e(ε− µ)/kBT + 1
= θ(µ− ε)
• particle density
n(0, µ) =
∫ µ
−∞dε z(ε)
• pressure
p(0, µ) =2
3
∫ µ
−∞dε z(ε) ε
• if particle density n is given, then
n = n(0, εF) =
∫ εF
−∞dε z(ε)
defines Fermi energy
![Page 76: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/76.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Zero temperature
• if T → 01
e(ε− µ)/kBT + 1
= θ(µ− ε)
• particle density
n(0, µ) =
∫ µ
−∞dε z(ε)
• pressure
p(0, µ) =2
3
∫ µ
−∞dε z(ε) ε
• if particle density n is given, then
n = n(0, εF) =
∫ εF
−∞dε z(ε)
defines Fermi energy
![Page 77: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/77.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Atoms and so forth
• Periodic system
• from small to huge molecules
• dielectrics, conductors, semiconductors, ferromagnets
• stability of normal matter
• stars can be stable even at T = 0
![Page 78: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/78.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Atoms and so forth
• Periodic system
• from small to huge molecules
• dielectrics, conductors, semiconductors, ferromagnets
• stability of normal matter
• stars can be stable even at T = 0
![Page 79: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/79.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Atoms and so forth
• Periodic system
• from small to huge molecules
• dielectrics, conductors, semiconductors, ferromagnets
• stability of normal matter
• stars can be stable even at T = 0
![Page 80: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/80.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Atoms and so forth
• Periodic system
• from small to huge molecules
• dielectrics, conductors, semiconductors, ferromagnets
• stability of normal matter
• stars can be stable even at T = 0
![Page 81: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/81.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Atoms and so forth
• Periodic system
• from small to huge molecules
• dielectrics, conductors, semiconductors, ferromagnets
• stability of normal matter
• stars can be stable even at T = 0
![Page 82: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/82.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Hydrodynamic equilibrium
• The mass within a sphere of radius R is
M(r) = 4π
∫ r
0ds s2ρ(s)
• gravitational force per unit volume
f (r) = −G ρ(r)M(r)
r2
• pressure gradient and force must balance
p ′(r) = f (r)
• relate pressure with mass density
![Page 83: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/83.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Hydrodynamic equilibrium
• The mass within a sphere of radius R is
M(r) = 4π
∫ r
0ds s2ρ(s)
• gravitational force per unit volume
f (r) = −G ρ(r)M(r)
r2
• pressure gradient and force must balance
p ′(r) = f (r)
• relate pressure with mass density
![Page 84: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/84.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Hydrodynamic equilibrium
• The mass within a sphere of radius R is
M(r) = 4π
∫ r
0ds s2ρ(s)
• gravitational force per unit volume
f (r) = −G ρ(r)M(r)
r2
• pressure gradient and force must balance
p ′(r) = f (r)
• relate pressure with mass density
![Page 85: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/85.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Hydrodynamic equilibrium
• The mass within a sphere of radius R is
M(r) = 4π
∫ r
0ds s2ρ(s)
• gravitational force per unit volume
f (r) = −G ρ(r)M(r)
r2
• pressure gradient and force must balance
p ′(r) = f (r)
• relate pressure with mass density
![Page 86: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/86.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Equation of state
• electron gas at T = 0
• recall
z(ε) =1
π2
2m
~2
3/2√ε
• particle density
n(0, µ) =2
3π2
2m
~2
3/2
µ3/2
• pressure
p(0, µ) =4
15π2
2m
~2
3/2
µ5/2
• eliminate chemical potential
p = a~2
mn5/3 where a = 1.2058
• since one electron is related with two nucleons
p = a~2
me
ρ
2mp
5/3
![Page 87: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/87.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Equation of state
• electron gas at T = 0
• recall
z(ε) =1
π2
2m
~2
3/2√ε
• particle density
n(0, µ) =2
3π2
2m
~2
3/2
µ3/2
• pressure
p(0, µ) =4
15π2
2m
~2
3/2
µ5/2
• eliminate chemical potential
p = a~2
mn5/3 where a = 1.2058
• since one electron is related with two nucleons
p = a~2
me
ρ
2mp
5/3
![Page 88: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/88.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Equation of state
• electron gas at T = 0
• recall
z(ε) =1
π2
2m
~2
3/2√ε
• particle density
n(0, µ) =2
3π2
2m
~2
3/2
µ3/2
• pressure
p(0, µ) =4
15π2
2m
~2
3/2
µ5/2
• eliminate chemical potential
p = a~2
mn5/3 where a = 1.2058
• since one electron is related with two nucleons
p = a~2
me
ρ
2mp
5/3
![Page 89: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/89.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Equation of state
• electron gas at T = 0
• recall
z(ε) =1
π2
2m
~2
3/2√ε
• particle density
n(0, µ) =2
3π2
2m
~2
3/2
µ3/2
• pressure
p(0, µ) =4
15π2
2m
~2
3/2
µ5/2
• eliminate chemical potential
p = a~2
mn5/3 where a = 1.2058
• since one electron is related with two nucleons
p = a~2
me
ρ
2mp
5/3
![Page 90: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/90.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Equation of state
• electron gas at T = 0
• recall
z(ε) =1
π2
2m
~2
3/2√ε
• particle density
n(0, µ) =2
3π2
2m
~2
3/2
µ3/2
• pressure
p(0, µ) =4
15π2
2m
~2
3/2
µ5/2
• eliminate chemical potential
p = a~2
mn5/3 where a = 1.2058
• since one electron is related with two nucleons
p = a~2
me
ρ
2mp
5/3
![Page 91: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/91.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Equation of state
• electron gas at T = 0
• recall
z(ε) =1
π2
2m
~2
3/2√ε
• particle density
n(0, µ) =2
3π2
2m
~2
3/2
µ3/2
• pressure
p(0, µ) =4
15π2
2m
~2
3/2
µ5/2
• eliminate chemical potential
p = a~2
mn5/3 where a = 1.2058
• since one electron is related with two nucleons
p = a~2
me
ρ
2mp
5/3
![Page 92: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/92.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Pressure p (decreasing) and mass M (increasing) of a white dwarf vs. distance r
from the center in natural units. We have assumed zero temperature, two
nucleons per electron which are treated non-relativistically. r = 1 corresponds to
6500 km, M = 1 to 0.85 sun masses.
![Page 93: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/93.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Crab nebula, the cloud of debris of the 1054 supernova
![Page 94: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/94.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
There is a neutron star (pulsar) at its center.
![Page 95: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/95.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Jagadris Chandra Bose, Indian physicist, 1858-1937
![Page 96: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/96.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Bosons
• bosons have integer spin or integer helicity
• such as the photon
• such as (qq) states - mesons
• or (ee) - Cooper pairs
• or (ppnn), the helium nucleus or helium atom
• or phonons, the quanta of lattice vibrations
![Page 97: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/97.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Bosons
• bosons have integer spin or integer helicity
• such as the photon
• such as (qq) states - mesons
• or (ee) - Cooper pairs
• or (ppnn), the helium nucleus or helium atom
• or phonons, the quanta of lattice vibrations
![Page 98: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/98.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Bosons
• bosons have integer spin or integer helicity
• such as the photon
• such as (qq) states - mesons
• or (ee) - Cooper pairs
• or (ppnn), the helium nucleus or helium atom
• or phonons, the quanta of lattice vibrations
![Page 99: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/99.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Bosons
• bosons have integer spin or integer helicity
• such as the photon
• such as (qq) states - mesons
• or (ee) - Cooper pairs
• or (ppnn), the helium nucleus or helium atom
• or phonons, the quanta of lattice vibrations
![Page 100: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/100.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Bosons
• bosons have integer spin or integer helicity
• such as the photon
• such as (qq) states - mesons
• or (ee) - Cooper pairs
• or (ppnn), the helium nucleus or helium atom
• or phonons, the quanta of lattice vibrations
![Page 101: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/101.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Bosons
• bosons have integer spin or integer helicity
• such as the photon
• such as (qq) states - mesons
• or (ee) - Cooper pairs
• or (ppnn), the helium nucleus or helium atom
• or phonons, the quanta of lattice vibrations
![Page 102: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/102.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Many bosons
• recall the spin/statistics theorem
• [Aj ,Ak ] = [A∗j ,A∗k ] = 0 - commutators!
• [Aj ,A∗k ] = δjk I
• Nj = A∗j ,Aj - number of particles in state j
• eigenvalues are n = 0, 1, 2, . . .
• work out the trace, i. e. sum over eigenvaluesn = 0, 1, 2, . . .
• N =∑
j Nj - total number of indistinguishable particles
![Page 103: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/103.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Many bosons
• recall the spin/statistics theorem
• [Aj ,Ak ] = [A∗j ,A∗k ] = 0 - commutators!
• [Aj ,A∗k ] = δjk I
• Nj = A∗j ,Aj - number of particles in state j
• eigenvalues are n = 0, 1, 2, . . .
• work out the trace, i. e. sum over eigenvaluesn = 0, 1, 2, . . .
• N =∑
j Nj - total number of indistinguishable particles
![Page 104: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/104.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Many bosons
• recall the spin/statistics theorem
• [Aj ,Ak ] = [A∗j ,A∗k ] = 0 - commutators!
• [Aj ,A∗k ] = δjk I
• Nj = A∗j ,Aj - number of particles in state j
• eigenvalues are n = 0, 1, 2, . . .
• work out the trace, i. e. sum over eigenvaluesn = 0, 1, 2, . . .
• N =∑
j Nj - total number of indistinguishable particles
![Page 105: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/105.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Many bosons
• recall the spin/statistics theorem
• [Aj ,Ak ] = [A∗j ,A∗k ] = 0 - commutators!
• [Aj ,A∗k ] = δjk I
• Nj = A∗j ,Aj - number of particles in state j
• eigenvalues are n = 0, 1, 2, . . .
• work out the trace, i. e. sum over eigenvaluesn = 0, 1, 2, . . .
• N =∑
j Nj - total number of indistinguishable particles
![Page 106: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/106.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Many bosons
• recall the spin/statistics theorem
• [Aj ,Ak ] = [A∗j ,A∗k ] = 0 - commutators!
• [Aj ,A∗k ] = δjk I
• Nj = A∗j ,Aj - number of particles in state j
• eigenvalues are n = 0, 1, 2, . . .
• work out the trace, i. e. sum over eigenvaluesn = 0, 1, 2, . . .
• N =∑
j Nj - total number of indistinguishable particles
![Page 107: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/107.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Many bosons
• recall the spin/statistics theorem
• [Aj ,Ak ] = [A∗j ,A∗k ] = 0 - commutators!
• [Aj ,A∗k ] = δjk I
• Nj = A∗j ,Aj - number of particles in state j
• eigenvalues are n = 0, 1, 2, . . .
• work out the trace, i. e. sum over eigenvaluesn = 0, 1, 2, . . .
• N =∑
j Nj - total number of indistinguishable particles
![Page 108: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/108.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Many bosons
• recall the spin/statistics theorem
• [Aj ,Ak ] = [A∗j ,A∗k ] = 0 - commutators!
• [Aj ,A∗k ] = δjk I
• Nj = A∗j ,Aj - number of particles in state j
• eigenvalues are n = 0, 1, 2, . . .
• work out the trace, i. e. sum over eigenvaluesn = 0, 1, 2, . . .
• N =∑
j Nj - total number of indistinguishable particles
![Page 109: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/109.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Free energy
• energy is H =∑
j εj A∗j Aj + . . .
• free energy is
F = −kBT ln tr e(µN − H)/kBT
• Nj commute, therefore
e(µN − H)/kBT =
∏j
e(µ− εj)Nj/kBT
• work out the trace, i. e. sum over eigenvaluesn = 0, 1, 2, . . .
tr e(µ− εj)Nj/kBT =
∞∑n=0
e
(µ− εj)/kBTn
=1
1− e(µ− εj)/kBT
![Page 110: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/110.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Free energy
• energy is H =∑
j εj A∗j Aj + . . .
• free energy is
F = −kBT ln tr e(µN − H)/kBT
• Nj commute, therefore
e(µN − H)/kBT =
∏j
e(µ− εj)Nj/kBT
• work out the trace, i. e. sum over eigenvaluesn = 0, 1, 2, . . .
tr e(µ− εj)Nj/kBT =
∞∑n=0
e
(µ− εj)/kBTn
=1
1− e(µ− εj)/kBT
![Page 111: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/111.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Free energy
• energy is H =∑
j εj A∗j Aj + . . .
• free energy is
F = −kBT ln tr e(µN − H)/kBT
• Nj commute, therefore
e(µN − H)/kBT =
∏j
e(µ− εj)Nj/kBT
• work out the trace, i. e. sum over eigenvaluesn = 0, 1, 2, . . .
tr e(µ− εj)Nj/kBT =
∞∑n=0
e
(µ− εj)/kBTn
=1
1− e(µ− εj)/kBT
![Page 112: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/112.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Free energy
• energy is H =∑
j εj A∗j Aj + . . .
• free energy is
F = −kBT ln tr e(µN − H)/kBT
• Nj commute, therefore
e(µN − H)/kBT =
∏j
e(µ− εj)Nj/kBT
• work out the trace, i. e. sum over eigenvaluesn = 0, 1, 2, . . .
tr e(µ− εj)Nj/kBT =
∞∑n=0
e
(µ− εj)/kBTn
=1
1− e(µ− εj)/kBT
![Page 113: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/113.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Free energy ctd.
• the free energy of a boson gas is
F = kBT∑j
ln
(1− e
(µ− εj)/kBT)
= kBT V
∫dε z(ε) ln
(1− e
(µ− ε)/kBT)
• compare with the free energy of a fermi gas
F = −kBT V
∫dε z(ε) ln
(1 + e
(µ− ε)/kBT)
• recall spin/statistics theorem: again the +/− difference
![Page 114: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/114.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Free energy ctd.
• the free energy of a boson gas is
F = kBT∑j
ln
(1− e
(µ− εj)/kBT)
= kBT V
∫dε z(ε) ln
(1− e
(µ− ε)/kBT)
• compare with the free energy of a fermi gas
F = −kBT V
∫dε z(ε) ln
(1 + e
(µ− ε)/kBT)
• recall spin/statistics theorem: again the +/− difference
![Page 115: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/115.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Free energy ctd.
• the free energy of a boson gas is
F = kBT∑j
ln
(1− e
(µ− εj)/kBT)
= kBT V
∫dε z(ε) ln
(1− e
(µ− ε)/kBT)
• compare with the free energy of a fermi gas
F = −kBT V
∫dε z(ε) ln
(1 + e
(µ− ε)/kBT)
• recall spin/statistics theorem: again the +/− difference
![Page 116: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/116.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Black body radiation
• photon energy is ~ω = pc
• density of states can be read off from
2
∫d3x d3p
(2π~)3= V
∫dω ω2
π2c3
• µ = 0
• black body free energy is
F = kBT V
∫ ∞0
dω ω2
π2c3ln
(1− e
−~ω/kBT)
• internal energy U
• U = F + TS = F − T∂F/∂T
• Planck’s formula
U = V
∫ ∞0
dω ω2
π2c3~ω
e~ω/kBT − 1
![Page 117: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/117.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Black body radiation
• photon energy is ~ω = pc
• density of states can be read off from
2
∫d3x d3p
(2π~)3= V
∫dω ω2
π2c3
• µ = 0
• black body free energy is
F = kBT V
∫ ∞0
dω ω2
π2c3ln
(1− e
−~ω/kBT)
• internal energy U
• U = F + TS = F − T∂F/∂T
• Planck’s formula
U = V
∫ ∞0
dω ω2
π2c3~ω
e~ω/kBT − 1
![Page 118: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/118.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Black body radiation
• photon energy is ~ω = pc
• density of states can be read off from
2
∫d3x d3p
(2π~)3= V
∫dω ω2
π2c3
• µ = 0
• black body free energy is
F = kBT V
∫ ∞0
dω ω2
π2c3ln
(1− e
−~ω/kBT)
• internal energy U
• U = F + TS = F − T∂F/∂T
• Planck’s formula
U = V
∫ ∞0
dω ω2
π2c3~ω
e~ω/kBT − 1
![Page 119: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/119.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Black body radiation
• photon energy is ~ω = pc
• density of states can be read off from
2
∫d3x d3p
(2π~)3= V
∫dω ω2
π2c3
• µ = 0
• black body free energy is
F = kBT V
∫ ∞0
dω ω2
π2c3ln
(1− e
−~ω/kBT)
• internal energy U
• U = F + TS = F − T∂F/∂T
• Planck’s formula
U = V
∫ ∞0
dω ω2
π2c3~ω
e~ω/kBT − 1
![Page 120: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/120.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Black body radiation
• photon energy is ~ω = pc
• density of states can be read off from
2
∫d3x d3p
(2π~)3= V
∫dω ω2
π2c3
• µ = 0
• black body free energy is
F = kBT V
∫ ∞0
dω ω2
π2c3ln
(1− e
−~ω/kBT)
• internal energy U
• U = F + TS = F − T∂F/∂T
• Planck’s formula
U = V
∫ ∞0
dω ω2
π2c3~ω
e~ω/kBT − 1
![Page 121: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/121.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Black body radiation
• photon energy is ~ω = pc
• density of states can be read off from
2
∫d3x d3p
(2π~)3= V
∫dω ω2
π2c3
• µ = 0
• black body free energy is
F = kBT V
∫ ∞0
dω ω2
π2c3ln
(1− e
−~ω/kBT)
• internal energy U
• U = F + TS = F − T∂F/∂T
• Planck’s formula
U = V
∫ ∞0
dω ω2
π2c3~ω
e~ω/kBT − 1
![Page 122: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/122.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Max Planck, German physicist, 1858-1947
![Page 123: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/123.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Particle density
• recall∞∑n=0
en(µ− εj)/kBT =
1
1− e(µ− εj)/kBT
• valid only if µ < min εj = 0
• −∂F/∂µ = N is average particle number
• the average particle density n is therefore
n(T , µ) =
∫ ∞0
dε z(ε)1
e(ε− µ)/kBT − 1
• n(T , 0) is maximal density:
nmax(T ) =
∫ ∞0
dε z(ε)1
eε/kBT − 1
![Page 124: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/124.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Particle density
• recall∞∑n=0
en(µ− εj)/kBT =
1
1− e(µ− εj)/kBT
• valid only if µ < min εj = 0
• −∂F/∂µ = N is average particle number
• the average particle density n is therefore
n(T , µ) =
∫ ∞0
dε z(ε)1
e(ε− µ)/kBT − 1
• n(T , 0) is maximal density:
nmax(T ) =
∫ ∞0
dε z(ε)1
eε/kBT − 1
![Page 125: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/125.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Particle density
• recall∞∑n=0
en(µ− εj)/kBT =
1
1− e(µ− εj)/kBT
• valid only if µ < min εj = 0
• −∂F/∂µ = N is average particle number
• the average particle density n is therefore
n(T , µ) =
∫ ∞0
dε z(ε)1
e(ε− µ)/kBT − 1
• n(T , 0) is maximal density:
nmax(T ) =
∫ ∞0
dε z(ε)1
eε/kBT − 1
![Page 126: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/126.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Particle density
• recall∞∑n=0
en(µ− εj)/kBT =
1
1− e(µ− εj)/kBT
• valid only if µ < min εj = 0
• −∂F/∂µ = N is average particle number
• the average particle density n is therefore
n(T , µ) =
∫ ∞0
dε z(ε)1
e(ε− µ)/kBT − 1
• n(T , 0) is maximal density:
nmax(T ) =
∫ ∞0
dε z(ε)1
eε/kBT − 1
![Page 127: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/127.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Particle density
• recall∞∑n=0
en(µ− εj)/kBT =
1
1− e(µ− εj)/kBT
• valid only if µ < min εj = 0
• −∂F/∂µ = N is average particle number
• the average particle density n is therefore
n(T , µ) =
∫ ∞0
dε z(ε)1
e(ε− µ)/kBT − 1
• n(T , 0) is maximal density:
nmax(T ) =
∫ ∞0
dε z(ε)1
eε/kBT − 1
![Page 128: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/128.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Bose-Einstein condensation
• If T decreases, so does nmax(T )
• if n is given, there is a temperature Tc such that
nmax(Tc) = n
• for even lower temperature, only a fraction of the particlesare in thermal equilibrium
• the rest is in the multiply occupied ground state
• supra-fluidity
• superconductivity
• light
![Page 129: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/129.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Bose-Einstein condensation
• If T decreases, so does nmax(T )
• if n is given, there is a temperature Tc such that
nmax(Tc) = n
• for even lower temperature, only a fraction of the particlesare in thermal equilibrium
• the rest is in the multiply occupied ground state
• supra-fluidity
• superconductivity
• light
![Page 130: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/130.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Bose-Einstein condensation
• If T decreases, so does nmax(T )
• if n is given, there is a temperature Tc such that
nmax(Tc) = n
• for even lower temperature, only a fraction of the particlesare in thermal equilibrium
• the rest is in the multiply occupied ground state
• supra-fluidity
• superconductivity
• light
![Page 131: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/131.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Bose-Einstein condensation
• If T decreases, so does nmax(T )
• if n is given, there is a temperature Tc such that
nmax(Tc) = n
• for even lower temperature, only a fraction of the particlesare in thermal equilibrium
• the rest is in the multiply occupied ground state
• supra-fluidity
• superconductivity
• light
![Page 132: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/132.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Bose-Einstein condensation
• If T decreases, so does nmax(T )
• if n is given, there is a temperature Tc such that
nmax(Tc) = n
• for even lower temperature, only a fraction of the particlesare in thermal equilibrium
• the rest is in the multiply occupied ground state
• supra-fluidity
• superconductivity
• light
![Page 133: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/133.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Bose-Einstein condensation
• If T decreases, so does nmax(T )
• if n is given, there is a temperature Tc such that
nmax(Tc) = n
• for even lower temperature, only a fraction of the particlesare in thermal equilibrium
• the rest is in the multiply occupied ground state
• supra-fluidity
• superconductivity
• light
![Page 134: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/134.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Bose-Einstein condensation
• If T decreases, so does nmax(T )
• if n is given, there is a temperature Tc such that
nmax(Tc) = n
• for even lower temperature, only a fraction of the particlesare in thermal equilibrium
• the rest is in the multiply occupied ground state
• supra-fluidity
• superconductivity
• light
![Page 135: Fermions, Bosons and their Statistics - uni-osnabrueck.deFermions, Bosons and their Statistics Peter Hertel Fundamental particles Spin and statistics Many particles Gibbs state Fermions](https://reader034.vdocuments.us/reader034/viewer/2022051310/5ff268e4883ec91fec74dd5a/html5/thumbnails/135.jpg)
Fermions,Bosons and
theirStatistics
Peter Hertel
Fundamentalparticles
Spin andstatistics
Many particles
Gibbs state
Fermions
Zerotemperature
White dwarfsand neutronstars
Bosons
Black bodyradiation
Bose-Einsteincondensation
Albert Einstein, German physicist, 1879-1955