1 chapter 9 electron spin and pauli principle §9.1 electron spin: experimental evidences double...
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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
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