Physics 451
Quantum mechanics I
Fall 2012
Nov 28, 2012
Karine Chesnel
Quantum mechanics
Test 3
Tuesday Nov 27 – Friday Nov 30
Homework
• HW 22 Thur Nov 29, 7pm
• HW 23 Tuesday Dec 4 , 7pm
• HW 24 Thursday Dec 6, 7pm
Quantum mechanics
Exchange forces
Attraction force
Symmetrical state:
 x 
2


 x 
2
2 x
d
2
ab
Covalent bound
Repulsion force
Antisymmetrical state
 x 
2


 x 
2
2 x
d
2
ab
Quantum mechanics
Two electrons
Total state antisymmetrical
 tot    r 1 , r 2    ,  
Spin state: singulet
antisymmetrical
Spatial state symmetrical
 x 
2


 x 
2
2 x
d
Attraction force
2
ab
Covalent bound
Pb 5.6
Quantum mechanics
Quiz 29a
If two electrons would occupy a triplet state (S=1)
what can we say about their spatial wave function?
A. It is antisymmetric (antibounding)
B. It is symmetric (bounding)
C. It could be both
Quantum mechanics
Homework
Pb 5.1:
Pb 5.2:
Pb 5.6:
Reduced coordinates
Reduced
coordinates
 x 
H 
2
2M

E   me

E
me
 x 
2
d
R2 

2
2

r 2  V


 x 
2
f
2
b
Quantum mechanics
Atoms
e-
ee-
Z=1
Z=2
Hydrogen
Helium
Quantum mechanics
e-
Helium
r2
er1
 (r1 , r2 , t )
Z=2
H 
2
2m1
 
2
1
2
 2  V (r1 , r2 )
2
2m2
2
2
2
2
2




1
2
e
1
2
e
1
e
2
H  
12 



 

2 
4 0 r1   2m2
4 0 r2  4 0 r1  r2
 2m1
electron 1
in the nucleus potential
electron 2
in the nucleus potential
Interaction
between
electrons 1 & 2
Quantum mechanics
Helium
To a first order:
2
2
2
2




1
2
e
1
2
e
2
2
H  
1 
2 
  

4 0 r1   2m2
4 0 r2 
 2m1
electron 1
in the nucleus potential
electron 2
in the nucleus potential
 (r1 , r2 )   nlm (r1 ) n 'l ' m ' (r2 )
Etot  Z 2 En  Z 2 En '
Etot  4  En  En ' 
Quantum mechanics
Helium
Ground state:
 (r1 , r2 )   100 (r1 ) 100 (r2 )
Etot  4  E1  E1   8E1
In this approximation:
Etot  8  (13.6eV )
Experimentally:
Etot
79eV
109eV
Difference comes from
the missing term
In the potential
Quantum mechanics
Helium
E
Excited states:
 (r1 , r2 )   nlm (r1 ) 100 (r2 )
1 

Etot  4  En  E1   4 E1 1  2 
 n 
4 E1
 17 
4 E1  
 16 
 10 
4 E1  
 9 
5
4 E1  
4
8 E1
Quantum mechanics
Helium
Symmetrization requirements
Total state antisymmetric (electrons)
 tot    r 1 , r 2    s1 , s2 
Spin state
Spatial wave function
Parahelium
Orthohelium
Symmetric
Antisymmetric


Singlet : antisymmetric
Triplet : symmetric
Quantum mechanics
Helium
Energy levels
Quantum mechanics
Quiz 29b
Which of these states corresponds to the energy
Level 3P in 3P Orthohelium ?
A.  200 100  100 200
B.  310 100  100 310
C.  210 100  100 210
D.  310 100  100 310
E.
 311 100  100 311