Postulates

Postulate 1: A physical state is represented by a wavefunction . The probablility to find the particle at within is .

Postulate 2: Physical quantities are represented by Hermitian operators acting on wavefunctions.

Postulate 3: The evolution of a wavefunction is given by the Schrödinger equation .

Postulate 4: The measurement of a quantity (operator A) can only give an eigenvalue an of A.

Postulate 5: The probability to get an is . After the measurement, the wavefunction collapes to (corresponding eigenfunction).

Postulate 6: N identical particles. The wavefunctions are either symmentrical (bosons) or antisymmetrical (fermions).

),( tr

r rd

rdtr 2|),(|

Ht

i

2|),(|| trn

n

Quantum mechanics

If H is time-independent

Time-independent Schrödinger equation: H

te-iEt

ntiE

nnn

nn

nectc )()0(

A, B, C, ... Commutating Hermitian operators

There exists a common set of orthormal egenfunctions

Orbital angular momentum

prL

ip

zyx LiLL ],[ + circ. perm. Commutation relations

Eigenfunctions common to L2, Lz Spherical harmonics

),()1(),( 22 mm YYL

),(),( mmz YmYL

m,0 m

integers

12/1)]1()1([ mm YmmYL yx iLLL

Raising, lowering operators

Orthonormality''*

''0

2

0),(),(sin mmmm YYdd

One particle in a spherically symmetric potential

H, L2, Lz commute

),()(

)( mE Yr

rur

)()()(2

)1(

2 2

2

2

22

rEururVmrdr

d

m EE

Eigenfunctions common to H, L2, Lz

)(2

1

2 2

22

2

2

rVmr

L

rr

rrmH

Centrifugal potential

Degeneracy 12

Wavefunctions parity: )1(

0,)( 1 rrruE

Angular momentum

zyx JiJJ ],[ + circ. perm. Commutation relations

Eigenfunctions common to J2, Jz

jmjm jjJ 22 )1( mj,

0j jmj Integers or half-integers

jmjmz mJ

Addition of two angular momenta:

SLJ

sjs ||

Angular momentum

Triangle rule

L2, Lz ,S2, Sz commute

L2, S2, J2, Jz commute

ssmmY

s

s

j smmmm

jssjm Ysjmsmm

,

|

Clebsch-Gordan coefficients

One particle in a spherically symmetric potential

Eigenfunctions common to H, L2, Lz , S2, Sz

ss smmnsmmn YrRr ),()(),(

Eigenvalues2)1( m 2)1( ss smnE

Eigenfunctions common to H, L2, S2, J2, Jz

),,()(),(

sjmnsjmn jj

rRr

Eigenvalues2)1( 2)1( ss jmnE 2)1( jj

Also eigenfunctions to the spin-orbit interaction SLr

.)(

Time-independent perturbation theory

HHH 0 kkk EH 0 known

kkkH E ? Approximation ?

...

...)1()0(

)1()0(

kkk

kkk

EE

kk EE )0(

Non-degenerate level

')1(

)0(

kkk

kk

HE

km mk

mkk

kmm

mk

mkk

EE

HE

EE

H

2')2(

')1(

||

Degenerate level (s times)

First diagonalize H´ in the subspace corresponding to the degeneracy

)1()0( ,ss kk E

Time-dependent perturbation theory)(0 tHHH

kkk EH 0 known

ktiE

kk

ketc /)( System in a at t=0Probability to be in b at time t?

Constant perturbation switched on at t=0

ab

ba

EE

2)1(

0

1)1(

|)(|)(

)()()(

tctP

tdetHitc

bba

t tibab

ba

),(||2

)( 22 bababa tFHtP

2

2 )2/(sin2),(

t

tF

Continuum of final states with an energy distribution b(E), width

ba EEEbbaba EHt

tP |)(||2

)( 2

For 2

t

Fermi’s Golden rule

One particle in an electromagnetic field (I)

.42 0

22

Am

ier

e

mH

cceAA trki ).(

0 ˆ

),(||.ˆ||2

)( 2.2

20

2

baarki

bba tFem

AetP

2

2 )2/(sin2),(

t

tF

Plane wave

a

b

ba

Absorption

tWgem

tAetP babaa

rkibba )(||.ˆ||

2)( 2.

2

20

2

abba WW Stimulated emission

Line broadening

baM

One particle in an electromagnetic field (II)

a

b

ba

AbsorptionDipole approximation

1. rkie

)(||.ˆ|| 2.2

02

2

baarki

bba

ba Igecm

eW

baM

)(|.ˆ| 2

02

2

bababa Igrc

eW

qmnmn

qqba Ir

'''

1,0,1

*.ˆ

)ˆˆ(2

1;ˆ);ˆˆ(

2

1101 yxzyx ii

''|10'|1001'2

12)()(''0

3''' mmqrRrRrI nn

qmnmn

Selection rules

)1,0(1,0

1

0

qm

ms

2||3

2ba

baba r

mf

Oscillator strength

One particle in a magnetic field

zzzB BSLSLr

r

e

mH )2(.)(

42 0

22

Paschen-Back effect

Anomal Zeeman effect

Zeeman effect )2( sBsmm mmBE

jjBsjm mBgEj

snsBssmmn mmmmBE )2(

0H SOH BH

0SOH

BSO HHH 0

SOB HHH 0

)1(2

)1()1()1(1

jj

ssjjg j

zeEr

e

mH z

0

22

42

Quadratic Stark effect (ground state)

One particle in an electric field

Tunnel ionisation

Linear Stark effect

110

1

10010100

)1(100

1 1

21001022)2(

100

||

||||

nn

n

n

n n

n

EE

zeE

EE

zEeE

0

1,1

0

m

m

m

2n

)( 21020021

)( 21020021

m21

Many-electron atom

ji ij

P

i ii rr

ZH

1

2

1

1

EH

P identical particles

),(0],[ jiPH ij

antisymmetrical or symmetrical /permutation of two electrons

antisymmetrical

Postulate 6: N identical particles. The wavefunctions are either symmetrical (bosons) or antisymmetrical (fermions).

Many-electron atom

ji ij

P

iieff

i

P

iieffi r

rVr

ZrVH

1)()(

2

1

11

Hc central field H1 perturbation

cccc EH

P

inc ii

EE1

)()(

)()(

!

111

PP

c

ququ

ququ

P

Slater determinant Pauli principle

Wavefunctions common to Hc, L2, Lz, S2, Sz 2S+1L

Electron configuration, periodic system etc..

terms

uEuh nc ssmmn

Many-electron atom

ji ij

N

iieff

i

N

iieffi r

rVr

ZrVH

1)()(

2

1

11

Hc central field H1 perturbation

cccc EH

N

inc ii

EE1

)()(

)()(

!

111

NN

c

ququ

ququ

N

Slater determinant Pauli principle

Wavefunctions common to Hc, L2, Lz, S2, Sz 2S+1L

Beyond the central field approximation:

N

iiii SLrH

12 .)(

21 HHH c

12 HHH c

LS coupling

jj coupling

c antisymmetrical/ permutation of two electrons

Electon configuration, periodic system etc..

terms