Physics 451
Quantum mechanics I
Fall 2012
Nov 26, 2012
Karine Chesnel
Quantum mechanics
Test 3:
Tuesday Nov 27 – Fri Nov 30
Homework
• HW 22 Thursday Nov 29
• HW 23 Monday Dec 3
• HW 24 Wednesday Dec 5
Quantum mechanics
Two-particles systems
 (r1 , r2 , t )
H 
2
2m1
 
2
1
2
 2  V (r1 , r2 , t )
2
2m2
Pb 5.1, 5.2
Time-independent potential
 (r1 , r2 , t )   (r1 , r2 )e
 iEt /
Quantum mechanics
Two-particles systems
If we could distinguish two identical particles
 (r1 , r2 )   a (r1 ) b (r2 )
a
b
 a (r1 ) b (r2 )
a
b
 b (r1 ) a (r2 )
Quantum mechanics
Two-particles systems
In QM: we can not distinguish two identical particles
a
b
  (r1 , r2 )   a (r1 ) b (r2 )  b (r1 ) a (r2 )
Normalization Pb 5.4
Quantum mechanics
Bosons and fermions
Bosons:
S = integer
  (r1 , r2 )  A  a  r1  b  r2   b  r1  a  r2  
Fermions:
S = half-integer
  (r1 , r2 )  A  a  r1  b  r2   b  r1  a  r2  
Pauli exclusion principle:
Two identical fermions can not occupy the same state
Quantum mechanics
Two-particles system
Symmetrization requirement:
Symmetric:
Antisymmetric:
  (r1 , r2 )    (r2 , r1 )
  (r1 , r2 )    (r2 , r1 )
Quantum mechanics
Two particles system
• For distinguishable particles
 ( x1 , x2 )   a ( x1 ) b ( x2 )
• For indistinguishable (identical) particles
  (r1 , r2 )   a (r1 ) b (r2 )  b (r1 ) a (r2 )
symmetrical
  (r1 , r2 )   a (r1 ) b (r2 )  b (r1 ) a (r2 )
antisymmetrical
Example: 2 particles in infinite square well
Quantum mechanics
Separation distance
• For distinguishable particles
 ( x1 , x2 )   a ( x1 ) b ( x2 )
 x 
2

d
 x1  x2 
2
 x2
d
a
 x2
b
2 x
x
a
• For indistinguishable particles
  (r1 , r2 )   a (r1 ) b (r2 )  b (r1 ) a (r2 )
 x 
2

 x
2
a
 x
2
b
2 x
a
x
b
2 x
2
ab
b
Quantum mechanics
Exchange forces
 x 
2


 x 
2
2 x
d
2
ab
Bosons are closer than if they were distinguishable
 x 
2


 x 
2
2 x
d
2
ab
Fermions are farther apart than if they were distinguishable
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 29
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