Lecture 21 — Identical Particles Lecture 21 — Identical Particles Chapter 6, Chapter 6, Wednesday February 27Wednesday February 27thth

•Review of Lecture 19•Calculating partition function for identical particles•Dilute and dense gases•Identical particles on a lattice•Spin and rotation in diatomic molecules (if time)

Reading: Reading: All of chapter 6 (pages 128 - 142)All of chapter 6 (pages 128 - 142)Assigned problems, Assigned problems, Ch. 6Ch. 6: 2, 4*, 6, 8, : 2, 4*, 6, 8,

(+1)(+1)Homework 6 due on Friday 29thHomework 6 due on Friday 29th1 more homework before spring break 1 more homework before spring break

(Fri.)(Fri.)Exam 2 on Wed. after spring breakExam 2 on Wed. after spring break

BosonsBosons 2,Bose 1 2 1 2 2 1 2,Bose 2 1

1, ,2 i j i jx x x x x x x x

3,Bose 1 2 2 1 2 3 2 1 3

2 3 1 3 2 1

1 1 2 1 3 2

, , i j k i j k

i j k i j k

i j k i j k

x x x x x x x x x

x x x x x x

x x x x x x

• Wavefunction symmetric with respect to exchange. There are N! terms.• Another way to describe an N particle system:

1 2 3

1 1 2 2 3 3

, , ,i

i

n n nE n n n

• The set of numbers, ni, represent the occupation numbers associated with each single-particle state with wavefunction i.

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

FermionsFermions 2,Fermi 1 2 1 2 2 1 2,Fermi 2 1

1, ,2 i j i jx x x x x x x x

• Alternatively the N particle wavefunction can be written as the determinant of a matrix, e.g.:

1 1 1

3,Fermi 1 2 3 2 2 2

3 3 3

( ) ( ) ( ), , ( ) ( ) ( )

( ) ( ) ( )

i j k

i j k

i j k

x x xx x x x x x

x x x

• The determinant of such a matrix has certain crucial properties:1. It changes sign if you switch any two labels, i.e. any two rows.

It is antisymmetric with respect to exchange2. It is ZERO if any two columns are the same.

• Thus, you cannot put two Fermions in the same single-particle state!

FermionsFermions• As with bosons, there is another way to describe N particle system:

1 2 3

1 1 2 2 3 3

, , ,i

i

n n n

E n n n

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

0

/ 2 / 3 /Fermi

B B Bk T k T k TZ e e e

BosonsBosons

1 1 2 2 3 3iE n n n • For bosons, these occupation numbers can be zero or ANY positive

integer.

/ 2 / 3 / 4 /Bose 1 2B B B Bk T k T k T k TZ e e e e

A more general expression for A more general expression for ZZ• First consider just two particles, and make a guess:

31 2

1 31 2 1 4

2 3 2 52 4

//2

1 1

2 /2 / 2 /

// /

/ //

2 2 2

2 2 2

j Bi B

BB B

BB B

B BB

M Mk Tk T

i j

k Tk T k T

k Tk T k T

k T k Tk T

Z e e

e e e

e e e

e e e

• Terms due to double occupancy – correctly counted.• Terms due to single occupancy – double counted.

VERY IMPORTANT: if the two particles are distinguishable, the counting is fine, i.e. (x1,x2) and (x1,x2) represent distinct quantum states. The states are indistinguishable if the particles are identical.

A more general expression for A more general expression for ZZ• What if we divide by 2 (actually, 2!):

31 2

1 31 2 1 4

2 3 2 52 4

//2

1 1

2 /2 / 2 /1 1 12 2 2

// /

/ //

12!

j Bi B

BB B

BB B

B BB

M Mk Tk T

i j

k Tk T k T

k Tk T k T

k T k Tk T

Z e e

e e e

e e e

e e e

• Terms due to double occupancy – under counted.• Terms due to single occupancy – correctly counted.

SO: we fixed one problem, but created another. Which is worse?•Consider the relative importance of these terms....

A more general expression for A more general expression for ZZ• What if we divide by 2 (actually, 2!):

31 2

1 31 2 1 4

2 3 2 52 4

//2

1 1

2 /2 / 2 /1 1 12 2 2

// /

/ //

12!

j Bi B

BB B

BB B

B BB

M Mk Tk T

i j

k Tk T k T

k Tk T k T

k T k Tk T

Z e e

e e e

e e e

e e e

Dilute gases (what does dilute mean in this context?):• Particle spacing large compared to average de-Broglie wavelength.• Energy levels are sparsely occupied.

Dense versus dilute gasesDense versus dilute gases

•Either low-density, high temperature or high mass

•de Broglie wave-length

•Low probability of multiple occupancy

•Either high-density, low temperature or low mass

•de Broglie wave-length

•High probability of multiple occupancy

Dilute: classical, particle-like Dense: quantum, wave-like

D

D (mT )1/2 D (mT )1/2

A more general expression for A more general expression for ZZ• What if we divide by 2 (actually, 2!):

31 2

1 31 2 1 4

2 3 2 52 4

//2

1 1

2 /2 / 2 /1 1 12 2 2

// /

/ //

12!

j Bi B

BB B

BB B

B BB

M Mk Tk T

i j

k Tk T k T

k Tk T k T

k T k Tk T

Z e e

e e e

e e e

e e e

Dilute gases (what does dilute mean in this context?):• Particle spacing large compared to average de-Broglie wavelength.• Energy levels are sparsely occupied.• In the dilute limit, the error associated with doubly occupied states

turns out to be inconsequential.

A more general expression for A more general expression for ZZ• Therefore, for N particles in a dilute gas:

1

!

N

N

ZZ

N

1ln ln 1BF Nk T Z N

and

VERY IMPORTANT:VERY IMPORTANT: this is completely incorrect if the gas is this is completely incorrect if the gas is densedense..• If the gas is dense, then it matters whether the particles are bosonic If the gas is dense, then it matters whether the particles are bosonic

or fermionic, and we must fix the error associated with the doubly or fermionic, and we must fix the error associated with the doubly occupied terms in the expression for the partition function.occupied terms in the expression for the partition function.

• Problem 8 and Chapter 10.Problem 8 and Chapter 10.

Identical particles on a latticeIdentical particles on a latticeLocalized Localized → Distinguishable→ Distinguishable

1 1and lnNN BZ Z F Nk T Z

DeDelocalized localized → → InIndistinguishabledistinguishable

11and ln ln 1

!

N

N B

ZZ F Nk T Z N

N

SpinSpin3 51

2 2 2: , , ,....: 0, , 2 , 3 ,....

FermionsBosons

12 space spin

1

2

3

4

Symmetric

Antisymmetric

}

12 1 2 1 2

12 1 2 1 2

i j j i

i j j i

x x x x

x x x x

Diatomic molecules: ortho and para Diatomic molecules: ortho and para 11HH22