1

Chapter 9 Electron Spin and Pauli Principle

§9.1 Electron Spin: Experimental evidences

2p

1s 1s

2p

a b c

3p-3s transition Na

Double lines detected in experiments are in conflict with the theory of atomic spectroscopy.

O.Stern and W. Gerlach, Z. Physik, 1922, 110, 9, 349.

Existence of electron spin

2

§9.2 Electron Spin: Theory

1928, Dirac, Relativistic Quantum Mechanics

1925 Uhlenbeck and Goudsmit

Electron spin

Electron spin Theorem

Theorem 1: Electron has an intrinsic angular momentum, the relationship between the corresponding operators is similar with that of angular momentum operators.

2 2 22x y zS S S S

yxzxzyzyx SiSSSiSSSiSS ],[,],[,],[

0],[],[],[ 222 zyx SSSSSS

3

Theorem 2: For single electron, Sz is connected with two eigenfunctions only, which correspond to eigenvalues of 1/2 ћ and -1/2 ћ, respectively. We denote the eigenfunction with eigenvalue of ½ ћ as a, and the eigenfunction with eigenvalue of -1/2 ћ as b.

2

1zS

2

1zS

22 )12

1(

2

1S 22 )1

2

1(

2

1S

Obviously, a and b are not eigenfunctions of Sx and Sy

4

0, 0S S

5

S

S

图 10.1 　电子自旋向量对 z 轴的两个可能取向

( ) , ( )s sm m

6

2| ( , , ) | 1x y z dxdydz

1/ 22

1/ 2

| ( ) | 1s

sm

m

1/ 2

2

1/ 2

| ( ) | 1s

sm

m

1/ 22

1/ 2

| ( , , , ) | 1s

sm

x y z m dxdydz

12

12

*( ) ( ) 0s

s sm

m m

12

,( )

ss m

m 12

,( )

ss m

m

7

Theorem 3: Spinning electron can be taken as a tiny magnet, with magnetic momentum m

Sge

s 0

Orbital motion Current I

r

ev

vr

eI

2/2

Theory of electric-magnetic field: A current around an area of A can be treated as a magnet with magnetic momentum IA/c

mc

erp

rc

rev

c

IA

22

2

L

mc

e

rc

rev

c

IA

22

2

Classic Quantum

Sge

s 0

8

The Hamiltonian does not involve spin

( , , ) ( )sx y z g m

2 22n n

, , ( ) ( ) ( , , )s sH x y z g m g m H x y z

( , , ) ( )sE x y z g m

§ 9.3 Spin and the Hydrogen Atom§ 9.3 Spin and the Hydrogen Atom

degeneracy

For H atom 𝐸 [𝜓 (𝑥 , 𝑦 ,𝑧 )𝛼] 𝐸 [𝜓 (𝑥 , 𝑦 ,𝑧 ) 𝛽 ]

9

1 (1)2 (2)s s

§ 9.4 The Pauli Princeple§ 9.4 The Pauli Princeple

Identical particles:

In classic mechanics: distinguishable

In quantum chemistry: indistinguishable

),,,(:),(1111121 smzyxqqq

),,,(22222 smzyxq

),(),(),,,( 21212211 qqEqqqpqpH

),(),( 122112 qqqqP Permutation operator:

12 1 (1) (1)3 (2) (2) 1 (2) (2)3 (1) (1)P s s s s

10

),(ˆ),(),,,(ˆ211221221112 qqEPqqqpqpHP

12

222211

1)]2()1([

2

1),,,(

rqpqpH

Hamiltonian is symmetric with respect to the coordinates qs

0)],,,(,[ 221112 qpqpHP

),(ˆ),(ˆ),,,( 211221122211 qqPEqqPqpqpH

),(ˆ2112 qqP is also an eigenfunction of H with eigenvalue of E.

),(),(),(ˆ21122112 qqcqqqqP

),(),(),(ˆ),(ˆˆ21

2211212211212 qqcqqqqPqqPP

112 cc

ricantisymmetqqqqP

symmetricqqqqP

),(),(ˆ

),(),(ˆ

212112

212112

11

2 1c

1c

symmetric

antisymmetric

12

1c

2ijP f f

ijP f cf

Since the particles are indistinguishable,

the eigenfunctions of Pij

symmetric or antisymmetric

Both wavefunctions correspond to the same state of the system.

13

1 2q q

2 0

Pauli principle: The wave function of a system of electrons must beantisymmetric with respect to interchange of any two electrons.

Half-integral spin: antisymmetric FermionsIntegral spin: symmetric Bosons

Pauli repulsion (not a real physical force)

),,,(),,,( 1111 nn qqqqqq

0),,,( 11 nqqq

14

( ) ( )sr g m

1 (1)1 (2)s s

§ 9.5 Ground State of the Helium Atom§ 9.5 Ground State of the Helium Atom

the ground state

(0) 1 (1)1 (2) (1) (2) (1) (2) 2s s Ground state

triplet

singlet

a(1)a(2)b(1)b(2)

b(1) b(2)

a(1) a(2)

a(1) b(2)

a(2) b(1)

Sym

Sym

None

a(1) a(2)

b(1) b(2)

a(1) b(2)+ a(2) b(1)

a(1) b(2)- a(2) b(1)

Sym

Sym

Sym

A-Sym

15

§ 9.6 First excited state of the Helium Atom§ 9.6 First excited state of the Helium Atom

SSStateExcitedFirst 21:__

1S(1)2S(2) and 2S(1)1S(2)

1/ 21/2[1S(1)2S(2) + 2S(1)1S(2)]

1/ 21/2[1S(1)2S(2) - 2S(1)1S(2)]

Sym

A-sym

))1()2()2()1())(1(2)2(1)2(2)1(1(2

1

))1()2()2()1())(1(2)2(1)2(2)1(1(2

1

))2()1())(1(2)2(1)2(2)1(1(2

1

))2()1())(1(2)2(1)2(2)1(1(2

1

SSSS

SSSS

SSSS

SSSS

Triplet

Singlet

16

§ 9.7 The Pauli Exclusion Principle§ 9.7 The Pauli Exclusion Principle

(0) 1 (1)1 (2)1 (3)s s s Li

2 2(0)

2 2 20

1 1 1 '27(13.6)eV 367.4eV

1 1 1 2

Z eE

a

2(1) 2 2 2

112

22 2 2

223

22 2 2

313

|1 (1) | |1 (2) | |1 (3) |

|1 (1) | |1 (2) | |1 (3) |

|1 (1) | |1 (2) | |1 (3) |

eE s s s d

r

es s s d

r

es s s d

r

17

(0) (1) 214.3eVE E

(5.39 75.64 122.45)eV= 203.5eV

1 (1)1 (2)1 (3)s s s

(1) (2) (3)

2(1) 2 2 2

1 2 312

3 |1 (1) | |1 (2) | |1 (3) |e

E s s d d s dr

2

(1)

0

53 153.1eV

4 2

Z eE

a

sym

antisym. impossible

Li experimental:

18

How to construct antisymmetric wavefunction with three functions?

f, g, h: Orth-normalized functions

f(1)g(2)h(3)

P12

P13

P23

f(2)g(1)h(3)

f(3)g(2)h(1)

f(1)g(3)h(2)

P12f(3)g(1)h(2)

P12 f(2)g(3)h(1)

Anti-symmetric wavefunction can be described as a linear combination of the functions above.

)1()3()2()2()1()3(

)2()3()1()1()2()3(

)3()1()2()3()2()1(

65

43

21

hgfchgfc

hgfchgfc

hgfchgfc

19

)1()3()2()2()1()3(

)2()3()1()1()2()3(

)3()1()2()3()2()1(

65

43

21

hgfchgfc

hgfchgfc

hgfchgfc

Anti-symmetric requirement leads to:

432651 cccccc

)]1()3()2()2()1()3()2()3()1(

)1()2()3()3()1()2()3()2()1([1

hgfhgfhgf

hgfhgfhgfc

Normalization requirement leads to:

6

11 c

)3()3()3(

)2()2()2(

)1()1()1(

6

1

hgf

hgf

hgf

Slater Determine

20

(1) (1) (1)1

(2) (2) (2)6

(3) (3) (3)

f g h

f g h

f g h

(0)

1 (1) (1) 1 (1) (1) 2 (1) (1)

1 (2) (2) 1 (2) (2) 2 (2) (2)

1 (3) (3) 1 (3) (3) 2 (3) (3)

s s s

s s s

s s s

1 (1)1 (2)2 (3)s s s

How to construct antisymmetric wave function?

Slater Det.

1 (1) 1 (1) 2 (1)1

1 (2) 1 (2) 2 (2)6

1 (3) 1 (3) 2 (3)

s s s

s s s

s s s

Pauli exclusion priciple:Each spin-orbital can haveonly one electron.

Spin-orbital

21

xS求 的矩阵表示

§ 9.8 Pauli Matrix§ 9.8 Pauli Matrix

22

02

12

10

2221

1211

||)(,||)(

||)(,||)(

xxxx

xxxx

SSSS

SSSS

01

10

2

1xS

23

同理可求得其它表示矩阵

0

0

2

1

i

iS y

10

01

2

1zS

10

01

4

3 22 S

00

10S

01

00S

24

Pauli 算符与 Pauli 矩阵

S2

10

012

10

01z

01

10x

0

0

i

iy