Quantum statistics of free particles
Identical particles
Two particles are said to be identical if all their intrinsic properties
(e.g. mass, electrical charge, spin, color, . . . ) are exactly the same.
Imagine: 2 identical classical objects
We can label them because we can keep track of the trajectories
1
2
Heisenberg’s uncertainty principle prevents us from keeping track in qm
identical quantum particles are indistinguishable
Implications from indistinguishability
Consider Hamilton operator H  h1  h2 ,e.g., H  
2
2m
 
2
1
2
2m
2
2
With corresponding Schroedinger eq.
2
 2 2
2




 2m 1 2m 2  ( r 1 , r 2 )  E ( r 1 , r 2 )


For the interaction free situation considered here
spin variable
look at solutions of
hˆ i  r ,     i i  r ,  
i labels set of quantum numbers of particular single particle
eigenfunction (do not confuse with particle label)
for basis functions appropriate to build up  ( r 11 , r 2 2 )
Why is a simple product ansatz ( r 1 1 , r 2 2 )  i  r1 ,  1  i  r 2 ,  2  not appropriate?
1
2
If we conduct an experiment with indistinguishable particles a correct quantum
description cannot allow anything which distinguishes between them.
 ( r 1 1 , r 2 2 )  i  r1 ,  1  i  r 2 ,  2  artificially distinguishes between the 2 particles
1
2
because indistinguishability requires
however in general
2
 ( r 11 , r 2 2 )   ( r 2 2 , r 11 )
i  r1 , 1 i  r 2 , 2   i  r2 , 2 i  r 1 ,1 
2
1
2
1
2
2
2
Simple product ansatz introduces unphysical labels to indistinguishable particles
What we need is a property like this
2
 ( r 11, r 2 2 )  e  ( r 2 2 , r 11 ) to fulfill  ( r 11 , r 2 2 )   ( r 2 2 , r 11 )
i
Nature picks to simple realizations for e
  ( r 2 2 , r 1 1 )


 ( r 1 1 , r 2 2 )  

  ( r 2 2 , r 1 1 )

ei  e0  1, ei  1
2
i
bosons
fermions
These symmetry requirements regarding particle exchange are fulfilled by
bosons


1 
 ( r 11 , r 2 2 ) 
 i1  r1 , 1 i2  r 2 , 2  i1  r2 , 2 i2  r 1 ,1  

2

fermions
Let’s summarize properties of antisymmetry product ansatz for fermions
 ( r 1 1 , r 2 2 ) 

1
i1  r1 ,  1  i2  r 2 ,  2   i1  r2 ,  2  i2  r 1 ,  1 
2

1 Solves Schroedinger equations for non-interacting particles
2 Eigenenergies E are given by E   i   i
1
2
3 Antisymmetry of wave function  ( r 11, r 2 2 )   ( r 2 2 , r 11 )
4 Pauli principle fulfilled: for identical single particle quantum numbers i1  i2
 ( r 11 , r 2 2 )  0 
2 identical fermions cannot occupy the same single particle state
Antisymmetric wave function for N identical fermions
Slater determinant
i  r 1 ,  1 
1
i  r 2 ,  2  ... i  r N ,  N 
1
1
1 i2  r 1 ,  1  i2  r 2 ,  2  ... i2  r N ,  N 
 i1 ,i2 ,...,iN ( r 1 1 , r 2 2 ,..., r N N ) 
N!
i
N
 r , 
1
1
i
N
r
2
,  2  ... iN  r N ,  N 
Check N=2
1 i1  r 1 ,  1  i1  r 2 ,  2 
 i1 ,i2 ( r 1 1 , r 2 2 ) 
2! i2  r 1 ,  1  i2  r 2 ,  2 


1
i1  r 1 ,  1  i2  r 2 ,  2   i1  r 2 ,  2  i2  r 1 ,  1 
2

Using occupation numbers to characterize N-particle states
Let ni be the # indicating how often the single particle state i is occupied
within the N-particle state described by 
fermions
ni  0,1 only possibility in accordance with Pauli principle
bosons
ni  0,1, 2, 3, ...
A few examples:
N=2 particles
bosons


1
2  r 1 ,  1  3  r 2 ,  2   2  r 2 ,  2  3  r 1 ,  1 
2
fermions

n1  0, n2  1, n3  1, n4  0, ...

1
3  r 1 ,  1  3  r 2 ,  2   3  r 2 ,  2  3  r 1 ,  1 
2
n1  0, n2  0, n3  2, n4  0, ...
E  2 3

1
2  r 1 ,  1  3  r 2 ,  2   2  r 2 ,  2  3  r 1 ,  1 
2
n1  0, n2  1, n3  1, n4  0, ...
E   2  3


E   2  3


Summary occupation number representation:
1
N   ni
i
2
E    i ni  E (n1 , n2 ,..., ni ,...)
i
3 N-particle state characterized by set of
occupation numbers of single particle states
4
fermions
ni  0,1
bosons
ni  0,1, 2, 3, ...
(n1 , n2 ,..., ni ,...)
i labels set of single particle quantum numbers
Partition functions with occupation numbers
Partition function of the canonical ensemble
Z   e  E 


( n1 ,n2 ,...,ni ,...)
e  E ( n1 ,n2 ,...,ni ,...)
Partition function of the grandcanonical ensemble

ZG   e  N
N 0






  
    ni  i    ni  
 


   E ( n1 , n2 ,..., ni ,...)   N 
i
 i




Z (N )  
e

e


 N 0  ( n1 ,n2 ,...,ni ,...)

N  0  ( n1 , n2 ,..., ni ,...)
 N  ni

 N  ni

i
i




We use the grandcanonical ensemble to derive <ni>
the average occupation of the single particle state i
Let’s consider how the summation works for an example of N=0,1,2,3 fermions
N=0
(0,0,0) meaning all single particle states are unoccupied
N=1
(1,0,0) (0,1,0) (0,0,1)
N=2
(1,1,0) (0,1,1) (1,0,1)
N=3
(1,1,1)




    ni  i    ni  
3 


i
 i


ZG  
e


N  0  ( n1 , n2 , n3 )
 N  ni

i


 1 e
   1   
e
e
   1  2  2  
e
   1  2  3 3  
   2   
e
e
   3   
  1  3  2  
e
    2  3  2  






    ni  i    ni  
    ni  i    ni 
3 




i
i
 i

 i


ZG  
e

e




N  0 ( n1 , n2 , n3 )
n1 , n2 , n3
 N  ni

i

 independent summation
Next we show
Let’s first look at
e

 


 ni i   
i
i

ni 


over the ni
      n     n      n
 e  1 1 2 2 3 3
and do a summation over n1
e

 



i
ni  i  

i

ni 


       0    2    n2   3    n3 
        2    n2   3    n3 
e  1
e  1
n1
      n    n
         2    n2    3    n3 
 e  2 2 3 3  e  1
Now summation over n1 and n2
e

 



 nii    ni 
i
i


      n     n
        2    n2   3    n3 
  e  2 2 3 3  e  1
n1 , n2
n2

      0    3    n3 
        2    0  3    n3 
       3    n3 
       2     3    n3 
e  2
e  1
e  2
e  1
      n
         3    n3 
        3    n3 
        2     3    n3 
 e  3 3  e  1
e  2
e  1
And finally summation over n1 , n2 and n3

e

 



 nii    ni 
i
i
n1 , n2 , n3


      n
        3    n3 
       3    n3 
       2     3    n3 
  e  3 3  e  1
e  2
e  1
n3
     
     
        2    
 1 e  1   e  2   e  1
Compare with
     
        3    
        3    
        2      3    
e  3  e  1
e  2
e  1




    ni i    ni  
3 


i

ZG     e  i

N 0  ( n1 ,n2 ,n3 )
 N  ni

i


 1  e    1     e     2     e     3     e     1   2  2    e     1   3  2    e     2   3  2    e     1   2   3  3  





    ni  i    ni  
 


i
 i


ZG  
e


N  0  ( n1 , n2 ,..., ni ,...)
 N  ni

i




e
  1    n1    2    n2
e
... e


e

 



ni i  
i

i

ni 


n1 , n2 ,..., ni ,...
   i    ni
...
n1 , n2 ,..., ni ,...

  1    n1 
   2    n2  
   i    ni 
  e
 ...
  e
 ...   e
 n1
 n2
  ni

ZG    e
i
    i    ni
ni
Holds for fermions and bosons with the only obvious difference
fermions
ni  0,1
bosons
ni  0,1, 2, 3, ...