Lecture 22 — The Maxwell Distribution Lecture 22 — The Maxwell Distribution Chapter 7, Friday February 29Chapter 7, Friday February 29thth

•Final points from Ch. 6 – molecular hydrogen

•Maxwell-Boltzmann probability

•Density of states in k-space

•Single-particle density of states in energy

•Maxwell distribution of speeds in a classical gas

Reading: Reading: All of chapter 7 (pages 144 - 157)All of chapter 7 (pages 144 - 157)Assigned problems, Assigned problems, Ch. 6Ch. 6: 2, 4*, 6, 8, : 2, 4*, 6, 8,

(+1)(+1)Homework 6 due today by 5pmHomework 6 due today by 5pmAssigned problems, Assigned problems, Ch. 7Ch. 7: 6, 8, 12, 14, : 6, 8, 12, 14,

1616Homework 7 due next FridayHomework 7 due next FridayExam 2 on Wed. after spring breakExam 2 on Wed. after spring break

SpinSpin3 51

2 2 2: , , ,....

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

Fermions

Bosons

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

LxLy

Lz

1 2 3sin sin sinix y z

n x n y n zA

L L L

2 2 2 2 21 2 32 2 22ix y z

n n n

m L L L

Particle (standing wave) in a boxParticle (standing wave) in a box

3/ 2

22B

x y z

mk TZ L L L

/i Bk T

i i

en Np N

Z

Boltzmann probability:

kkyy

kkxx

kkzz

Density of states in Density of states in kk-space-space

1

2

3

xx

yy

zz

nk

L

nk

L

nk

L

dVk

k

k

k

k kkyy

kkxx

kkzz

Convert to spherical coordinatesConvert to spherical coordinates

dk

2 2 2x y zk k k k

2 sinkdV k dk d d

n(u)

u

Maxwell speed distributionMaxwell speed distribution