Lecture 20 Identical Particles

1
Lecture 20 Identical Particles Chapter 6,
Monday February 25th

• Symmetry and antisymmetry
• Bosons
• Fermions
• Implications for statistics
• Calculating partition function for identical
  particles

Reading All of chapter 6 (pages 128 -
142) Assigned problems, Ch. 6 2, 4, 6, 8,
(1) Homework 6 due on Friday 29th 1 more
homework before spring break Exam 2 on Wed.
after spring break
2
Quantum statistics and identical particles
Indistinguishable events
1.
1.
Heisenberg uncertainty principle
h
2.
2.
The indistinguishability of identical particles
has a profound effect on statistics. Furthermore,
there are two fundamentally different types of
particle in nature bosons and fermions. The
statistical rules for each type of particle
differ!
3
Bosons
This wave function is symmetric with respect to
exchange.
4
Bosons

• Easier way to describe N particle system

• The set of numbers, ni, represent the occupation
  numbers associated with each single-particle
  state with wave function fi.
• For bosons, these occupation numbers can be zero
  or ANY positive integer.

5
Fermions
This wave function is antisymmetric with respect
to exchange.
6
Fermions
This wave function is antisymmetric with respect
to exchange.

• It turns out that there is an alternative way to
  write down this wave function which is far more
  intuitive

• The determinant of such a matrix has certain
  crucial properties
• It changes sign if you switch any two labels,
  i.e. any two rows.
• It is ZERO if any two columns are the same.
• Thus, you cannot put two Fermions in the same
  single-particle state!

7
Fermions
This wave function is antisymmetric with respect
to exchange.

• As with bosons, there is an easier way to
  describe N particle system

• The set of numbers, ni, represent the occupation
  numbers associated with each single-particle
  state with wave function fi.
• For Fermions, these occupation numbers can be
  ONLY zero or one.
• This is the basis of the Pauli exclusion
  principle.

8
Fermions

• As with bosons, there is an easier way to
  describe N particle system

• The set of numbers, ni, represent the occupation
  numbers associated with each single-particle
  state with wave function fi.
• For Fermions, these occupation numbers can be
  ONLY zero or one.

2e
e
0
9
Fermions

• For Fermions, these occupation numbers can be
  ONLY zero or one.

2e
e
0
10
Bosons

• For bosons, these occupation numbers can be zero
  or ANY positive integer.