5.formulation of quantum statistics
DESCRIPTION
5.Formulation of Quantum Statistics. Quantum Mechanical Ensemble Theory: The Density Matrix Statistics of the Various Ensembles Examples Systems Composed of Indistinguishable Particles The Density Matrix & the Partition Function of a System of Free Particles. - PowerPoint PPT PresentationTRANSCRIPT
5. Formulation of Quantum Statistics
1. Quantum Mechanical Ensemble Theory: The Density Matrix
2. Statistics of the Various Ensembles
3. Examples
4. Systems Composed of Indistinguishable Particles
5. The Density Matrix & the Partition Function of a System of
Free Particles
Statistics Particle type Math Object
Classical Distinguishable Phase space density
Quantum Indistinguishable Density matrix
Advantage of using density matrix :
Quantum & ensemble averaging are
combined into one averaging.
Classical Statistical Mechanics
(Probability) density function ( p,q,t ) :
d ff
d
3 3N Nd d p d q
Liouville’s theorem : , Ht
Microcanonical ensemble :0
const E H E
otherwise
d
Canonical ensemble : He 3
1
!N NZ d
N h
Grand canonical ensemble : NH NN e 3
1
! N NNN
dN h
Z
Caution:Some authors, e.g., Landau-Lifshitz, use a normalized version of .
Quantum Statistical Mechanics (To be Proved)
k kk
d ff p f
d
Classical mechanics :
k kk
k
f p f Quantum mechanics :
Tr fk k
kk
p
Ensemble = phase space
Ensemble = Hilbert space
1E P
NMicrocanonical :
1 HeZ
Canonical :
1 H Ne Z
Grand canonical :
PE = projection operator onto the N-D subspace of states with energy E.
HZ Tr e
H NTr e Z
Pure State Density Operator
Expectation value of f :
f f
,n m
n n f m m
n
n n I
Orthonormal basis { | n } is complete :
Density operator for | : nm n m
n
Tr f n f n ,n m
n m m f n
,n m
n m m f n f
1
n
n n
nn
n a*
,n nm m
n m
a f a
Tr f 1Tr
r-Representation
f f ,n m
n n f m m n
n n I
d I r r r f d d f r r r r r r
*d d f r r r r r r
f is a 1-particle operator f f r r r r r
*f d f r r r r
Mixed State Density Operator
Averaged value of f :
k kk
k
f p f
, ,
k kk
k n m
p n n f m m n
n n I
Orthonormal basis { | n } is complete :
Density operator :k k
kk
p k knm k
k
p n m
n
Tr f n f n ,n m
n m m f n
, ,
k kk
k n m
p n m m f n f
Skip to ensembles Ex: Derive the quantum Liouville eq.
1 1kk
p Tr
5.1. Quantum Mechanical Ensemble Theory: The Density Matrix
, ,i ii t H tt
r r
Consider ensemble of N identical systems labelled by k = 1, 2,..., N.
Each system is described by i = 1,2,..., N
Let ,k kit t r be the wave function of the kth system in the ensemble.
Let n n i r be a set of complete orthonormal basis that spans the Hilbert space of H & satisfies the relevant B.C.s.
,k kit t r k k
n n i n nn n
a t a t r
with *k kn na t d t 3
ii
d d r
k runs through all independent solutions of this Schrodinger eq.
*k k kn na t d t n t
*k
knn t
d ai d i t
d t * k
nd H t
k kn n
n
t a t
*km n m
m
a t d H
k
knnm m
m
d ai H a t
d t where
*nm n mH d H
21kd t * *
,
1k km n m n
m n
a t a t d
*m n mnd m n 2
1kn
n
a t k
n H m
H can be t-dep
Density Operator
Density operator : *
1
k kk
k
p t t
N
1
ˆ k kk
k
p t t
N
Matrix elements : mn t m n
*
1
k kk m n
k
p a t a t
N
*m n ens
a t a t
2
nn nens
a t 21nn n
n n ens
a t
n or d ~ quantum averaging ens ~ k ~ ensemble averaging
1
k kk
k
p m t t n
N
k kn n
n
t a t
pk = weighting (or probability) factor with 1kk
p
*
1
k kmn k m n
k
p a t a t
N
k
knnm m
m
d ai H a t
d t
*
*
1
k kmn k km n
k n mk
d d a d ai p i a t a t i
d t d t d t
N
* *
1
k k k kk ml l n m l n l
k l l
p H a t a t a t H a t
N
*mn nmH H
ml l n m l l nl
H t t H
mnH H
_,d
i Hd t
,A B AB BA where
m H H n
H can be t-dep
Equilibrium Ensemble
System in equilibrium ensemble stationary :
0d
d t
_,d
i Hd t
i.e. _, 0H H and 0tH
Energy representation : n n nH E
*mn nm
mn m mnH E
H mn m mn
In a general basis , is hermitian
k
knnm m
m
d ai H a t
d t k
n nE a t expkn n n
ia t a E t
*
1
k kmn k m n
k
p a t a t
N
*
1
expk m n m nk
ip a a E E t
N
System in equilibrium 2
m mna
detailed balance
Expectation Values
*
*1
k k
k k kk
d GG p
d
N
Expectation value of a physical quantity G :
( Quantum + ensemble av. )
k kn n
n
t a t *
1 ,
k kk m n mn
k m n
G p a a G
N
*mn m nG d G
,nm mn
m n
G G *
1
k kmn k m n
k
p a t a t
N
G Tr Gi.e.
nnn
Tr 2
1
kk n
k n
p a t
N
1k
k
p
N
21k
nn
a t k normalized :1
*
1
k kk
k
G p d G
N
1k k k Assuming k normalized, i.e.,
5.2. Statistics of the Various
Ensembles
Microcanonical ensemble : Fixed N, V, E or ,2 2
E E E E
( N, V, E; ) = # of accessible microstates
Equal a priori probabilities postulate 1
0
k
k
is accessiblep
otherwise
i.e.
lnS k ( quantum statistics: no Gibbs’ paradox )
1
k kk
k
p
N
k kn n
n
a
Energy representation:
k
kn k na
n n nH E
1
0
nn
E E
otherwise
mn m mn H
k kn n
n
t a t 1
k k kk
p
N
Pure State
1 0S Only 1 state p is accessible 3rd law
k k pp
2ml l nmn
l
m p l p l p n pl
Thus m p n p mn
i.e. 2 idempotent
In another representation with basis { m } so that
mn m p n p
pl l
l
t a
p pmn m n
, normalized
( is a projector )
Energy representation :k
k
p pmn m n *
m na a
2 * *m l l nmn
l
a a a a *m na a mn 2
1ll
a
p p
Mixed State
Multiple states are accessible, i.e. > 1.
Any representation :
1 k kmn m n
k
A
1
k kk
k
p
N
1
0
k
k
is accessiblep
otherwise
Let K be the subspace spanned by the accessible k ’s.
Consider any orthonormal basis {n } such that 1, ,n
n
n
n
K
K
Since { k } is a basis of K, its completeness means
1
,
0
mnmn
m n
otherwise
= set of accessible state indices
( is diagonal w.r.t. {n } )
k k
k
I
KA
k kn n
n
a
1
1 k kmn m n
k
N
N
1,
0
mn m n
otherwise
*
1
1 k km n
k
a a
N
Nk = ensemble member index
2
1
k km ni
mnk
ae
N
N
k
nikna a e
So that 1mn
2
m nia e
Postulate of a priori random phases
Let
Canonical Ensemble 1
k kk
k
p
N
E-representation : n n nH E i.e.
kkn k na k k
n nn
t a t
Canonical ensemble : Fixed N, V, T. 1
kEkp e
Z
1kE
k kk
eZ
1 H
k kk
eZ
kE
k
Z e
1 HeZ
HTr e kE
k
Z e H
k
k e k H
k kk
e
NQ
0 !
jjH
j
e Hj
By definition
G Tr G 1 HTr e GZ
H
H
Tr e G
Tr e
Tr AB Tr BA
Grand Canonical Ensemble1
k kk
k
p
N
Grand canonical ensemble : Fixed , V, T
,
,
r s sE N
r s
e Z 1 H Ne
Z
G Tr G 1 H NTr e G Z
H NTr e Q
0
rEN
N r
e e
Z 0
N H
N
z Tr e
0
, ,N
N
z Z N V T
0
NN
N
z Q
0
rH N E NNr r
N r
Tr e G e e E N G E N
0
, ,N
NN
z Z N V T G
0
0
, ,
, ,
N
NN
N
N
z Z N V T G
Gz Z N V T
0
0
NN N
N
NN
N
z Q G
z Q
Er, s = Er (Ns )
= E of r th state of Ns p’cle sys
5.3. Examples
An Electron in a Magnetic Field
Single e with spin1
2σ & magnetic moment Bμ σ
BH σ B
2B
e
mc
B zB ˆBB z
Pauli matrices :
0 1
1 0x
0
0y
i
i
1 0
0 1z
HZ Tr e
nni j i ii j
AAA diagonal i iA
i ji je eA
B zBHe e 0
0
B
B
B
B
e
e
B BB Be e
01
0
B
B
B
B
e
Z e
0
0
B
B
B
z B
e
e
1
z zTrZ
tanh B B
agrees with § 3.9-10
signed
A Free Particle in a Box
Free particle of mass m in a cubical box of sides L.
2 22
2 2H
m m
p 2 2 2 2
2 2 22m x y z
H E
Periodic B.C : , , , , , , , ,x y z x L y z x y L z x y z L
3/2
1 iE e
L k rr with
2 2
2
kE
m
2
L
k n with , ,x y zn n nn 0, 1, 2,in
3/2
1 iE e E
L k rr r ( r - representation )
H H
E
e e E E r r r r E
E
e E E r r *EE E
E
e r r
2
2H
m
p
2 2
2
kE
m 3/2
1 iE e
L k rr with
2 2
3
1exp
2H k
e iL m
k
r r k r r
3
3
1
2V
d k
V
k
3 2 2
3 exp22
d k ki
m
k r r
3/22
2 2exp
2 2
m m
r r ( see next page )
3 2 2
3 exp22
d k kI i
m
k r r
2
2m
2
20
1
2
i k i kkd k k e e ei
r r r r
r r
12 2
20 1
1cos exp cos
2d kk d k i k
r r
22 2
1
41
2k ik yi
d k k e e d y y e
r rr r r r
2 20
0
k ik k ikd k k e d k k e
r r r r
2
2
1
2
k ikd k k ei
r r
r r
2
iy k
r r
21
43/22
ie
r r
r r
21
42 3/2
1
22I e
r r 22
3/2
222
mm
e
r r
3/22
2 2exp
2 2H m m
e
r r r r
HZ Tr e 3 Hd r e r r3/2
22
mV
2
2
1exp
2
m
V
r r
1 HeZ
r r r r r r
is symmetric
1
V r r rParticle density at r : Location
uncertainty :th
mkT
H Tr H 31 Hd r H eZ
r r
ln Z
3
2kT
1 Z
Z
31 Hd r eZ
r r
31 HH d r H eZ
r rAlternatively
, ,f fi
r r p r r
3 31 Hd r d r H eZ
r r r r
2
3 3 21
2HH d r d r e
Z m
r r r r
2
23 3 22
1exp
2 2
md r d r
V m
r r r r
2 22 22
1r re r er r r
232
12 rr e
r r
22 3 2 rr e
2 23 32 2
13 exp
2 2
m mH d r d r
V
r r r r r r
3
2kT
Uising & integrate by parts twice :
r r r r
A Simple Harmonic Oscillator
22 21
2 2
pH m q
m
1
2nE n
n = 0,1,2,...n n nH E
2
1/4
/21
2 !n nn
mq H e
n
1/4m
q
2 2n
n
n n
dH e e
d
Hermite polynomials : Rodrigues’ formula
2
1/4
/21
2 !n nn
mq H e
n
0
nEH
n
q e q q e n n q
0
nEn n
n
e q q
is real
2 2
1/2/2 1/2
0
1
2 !n
n nnn
me e H H
n
1/2
2 21 1exp tanh coth
2 sinh 4 2 2
m mq q q q
Kubo, “Stat Mech.”, p.175Mathematica
HZ d q q e q
1/22 1
exp tanh2 sinh 2
m m qd q
1/2
12 sinh tanh2
m
m
11
2sinh2
/2
1
e
e
1/2
2 21 1exp tanh coth
2 sinh 4 2 2H m m
q e q q q q q
Probability density :
1 Hq q q q e qZ
1/2 21 1tanh exp tanh
2 2
m m q
11
2sinh2
Z
q is a Gaussian with dispersion ( r.m.s. deviation ) :
2rmsq q q
2
2
2
x x
eg
1/21
coth2 2m
Classical limit :(purely thermal)
1
tanh
x
x x
2 2
2exp2 2
m mq q
kT kT
2rms
kTq
m
Quantum limit : (non-thermal)
1
tanh 1
x
x
2
expm m q
q
1/2 21 1
tanh exp tanh2 2
m m qq
2rmsqm
2
0 q = Probability density of ground state
11
2sinh2
Z
ln Z
H Tr H1 HeZ
1 1coth
2 2
1/2 21 1
tanh exp tanh2 2
m m qq
2 2
2exp2 2
m mq q
H H
2rms
Hq
m
2 21. .
2P E m q 2 21
2 rmsm q 0q 1
2H
. . . .K E H P E 1
2H
5.4. Systems Composed of Indistinguishable Particles
N non-interacting particles subject to the same 1-particle hamiltonian h.
1
, ,N
i ii
H h q p
q p h u u
E EH E q q
1
i
N
E ii
u q
q1
i
N
i
E
i = label of the eigenstate assumed by the i th particle.
Let n = # of particles occupying the th eigenstate.
N n
E n
,1
n
E L jj
u q
q
L( , j ) = label of the j th particle that occupies the th eigenstate.
,1
n
E L jj
u q
q Note: [ ... ] = 1 if n = 0.
Let P denote a permutation of the particle labels :
,1
n
E E P L jj
P P u q
q q
i P iq P q q q
1, 2, 3, 1 , 2 , 3 , ...,i N P i P P P P N
Indistinguishable particles : 1W n
Distinguishable particles :
permutations within the same counted as the same.
permutations across different ’s counted as distinct.
# of distinct microstates is !
!
NW n
n
,1
n
E L jj
u q
q Boltzmannian ( distinguishable p’cles)
Boltz q
Indistinguishable Particles
Particles indistinguishable Physical properties unchanged under particle exchange
2 2P P
symmetricP
anti symmetric
i.e. P P
2P
Anti-symmetric : P
A A BoltzP
C P q q detA iC u q
1 1 1
2 2 2
1 2
1 2
1 2N N N
N
NA
N
u q u q u q
u q u q u qC
u q u q u q
Pauli’s exclusiion principle 0A qi j
0,1n
i.e.
Symmetric : S S BoltzP
C P q q perS iC u q
21
0FD
if n NW n
otherwise
Fermi-Dirac statistics
1BEW n Bose-Einstein statistics
5.5. The Density Matrix & the Partition Function of a System of Free Particles
N non-interacting, indistinguishable particles :
1 1 1 1
1, , , , , , , ,
,H
N N N NeZ N
r r r r r r r r
31 1, , , , ,H N H
N NZ N Tr e d r e r r r r
Let i stands for ri , & i for ri .
e.g.,
1
1, , 1 , , 1, , 1 , ,,
HN N N e NZ N
Goal: To write , 1,N
Z N Z 1N
NQ Q or
1, , 1 , , 1, , 1 , ,H H
E
N e N N e E E N
1
!
1, ,!
j
j
j
NP
P j
n
N u P jN
kk
K k
*1, , 1 , ,HE E
E
e N N
Non-interacting particles
1/3
2
V
k n
2 2
2E
m
K 22
12
N
iim
k
Periodic B.C. 0, 1, 2,in 1 iu eV
k rk r
BosonsFermions
1
!
!
j
j
P j
NP
P j
n
u jN
k
k
k
1 1 1
2 2 2
1 2
1 2
1 2N N N
u u u N
u u u N
u u u N
k k k
k k k
k k k
1 2
1 2
1 2
1 1 1
2 2 2N
N
N
u u u
u u u
u N u N u N
k k k
k k k
k k k
Mathematica
2 2 / 2 *1, , 1 , , 1, , 1 , ,
mHN e N e N N
K
K KK
1
!
1, ,!
j
j
j
NP
P j
n
N u P jN
kk
K k
2 2 22
12 2
N
ii
Em m
Kk
Consider the N ! permutations among { ki } associated with a given K.
E is unchanged
1
1, ,P ji i
NP
P j
P N P u j
K kk k
1
i
P j
NP P
P j
u j
k
k
iP P P k
1, ,iP
N k
K
1
!
!
j
j
P j
NP
P j
n
u jN
k
k
k
22
12 *1
1, , 1 , , 1, , 1 , ,!
N
ii
i i
i
mHN e N e N NN
k
k kk
22
12 *
2, ,
1
!
N
ii
j P m
i
m P P
P P j m
e u P j u mN
k
k kk
nk > 1 cases neglected(measure 0)
1
!
1, ,!
j
j
j
NP
P j
n
N u P jN
kk
K k
1
!
!
j
j
P j
NP
P j
n
u jN
k
k
k
22
12 *
2, , 1
11, , 1 , ,
!
N
ii
j P m
i
Nm P PH
P P j m
N e N e u P j u mN
k
k kk
22
12 *
, 1
1
!
N
ii
j P m
i
Nm P P
P j m
e u P j u mN
k
k kk
22
12 *
1
1
!
N
ii
j j
i
Nm P
P j
e u P j u jN
k
k kk
2
2*2
1
1
!
j
j j
j
NP m
P j
e u P j u jN
k
k kk
1
1
!
NP h
jP jP j
eN
r r
arbitrary P
P = I
2
2h
m
p2-p'cle
1
11, , 1 , ,
!
NPH h
jP jP j
N e N eN
r r
3/22
2 21
1exp
! 2 2
NP
jP jP j
m m
N
r r
from § 5.3
3 /22
2 2
1exp
! 2 2
NP
jP jP j
m m
N
r r
2
3 2
1exp
!P
jP jNP jN
r r
22 2
m m k T
= thermal ( de Broglie ) wavelength
2
3 2
11, , 1 , , exp
!PH
jP jNP j
N e NN
r r
3
11, , 1 , , ,
!PH
NP j
N e N f P j jN
2
2, exp i jf i j
r r 2
2, exp 1i if i i
r rLet with
3
11, , 1, , 1 , , , , ,
!H
Ni j i j k
N e N f i j f j i f i j f j k f k iN
mean inter-particle distance =1/3
1/3Vn
N
n = particle density
1/3n , i jf i j 3
11, , 1, ,
!H
NN e N
N
3, , 1, , 1, ,N HZ N T V d r N e N 3!
N
N
V
N 1
1, ,!
NZ T V
N
Mathematica
3, ,
!
N
N
VZ N T V
N 1
1, ,!
NZ T V
N
Resolution of problems in classical statistics:
1.Gibbs correction factor ( 1 / N! ).
2. Phase space volume per state 0 h 3 33
1 N NN
d d q d ph
1/3n Classical limit :
Non-classical systems are said to be degenerate.
n 3 = degeneracy discriminant
1
1, , 1, , 1, , 1, ,, ,
HN N N e NZ N T V
3
3
! 1
!
N
N N
N
V N
1NV
Classical limitN r r ( no spatial
correlation )
1/3n
Exchange Correlation
Let N = 2 :
6
11, 2 1,2 1 1,2 2,1
2He f f
2
2, exp i jf i j
r r
2
1 26 2
1 21 exp
2
r r
23 31 2 1 26 2
1 22, , 1 exp
2Z T V d r d r
r r
3 26 2
1 21 exp
2V d r r
2 26 2
0
1 24 exp
2V V d r r r
3
11, , 1, , 1 , , , , ,
!H
Ni j i j k
N e N f i j f j i f i j f j k f k iN
2 26 2
0
1 22, , 4 exp
2Z T V V V d r r r
3/22 2
3
1 4 1 31
2 2 2 2
V
V
2 1 / 2
0
1 11
2 2nn xd x x e n
2 3
3 3/2
1 11
2 2
V
V
3 1 1
2 2 2 2
2
3
1
2
V
Classical limit
1
1, 2 1,2 1, 2 1,22, ,
HeZ T V
2
1 22 2
1 21 exp
V
r r 2
1 2
21
0
Bosons
FermionsV
r r
1, 2 1,2 1,2 1,2k k k
k
p
2
1 2,k k
k
p r r
Statistical Potential
22
21 expsv re r
2
2
2ln 1 expsv r kT r
Mathematica
2 1, 2 1,2V