Lecture 22 Lecture 22 ——The Maxwell Distribution 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 gasReading: Reading: All of chapter 7 (pages 144 All of chapter 7 (pages 144 --157)157)Assigned problems, Assigned problems, Ch. 6Ch. 6: 2, 4*, 6, 8, (+1): 2, 4*, 6, 8, (+1)Homework 6 due today by 5pmHomework 6 due today by 5pmAssigned problems, Assigned problems, Ch. 7Ch. 7: 6, 8, 12, 14, 16: 6, 8, 12, 14, 16Homework 7 due next FridayHomework 7 due next FridayExam 2 on Wed. after spring breakExam 2 on Wed. after spring breakSpinSpin351222: , , ,....: 0, , 2 , 3 ,....FermionsBosons======12 space spinψχΨ=()()1234χχχχ=↑↑=↑↓+↓↑=↓↓=↑↓−↓↑SymmetricAntisymmetric()() ()(){}()() () ()(){}()12 1 2 1 212 1 2 1 2ij jiij jixx xxxx xxφφ φφφφ φφΨ= − ↑↓+↓↑Ψ= + ↑↓−↓↑Diatomic molecules: ortho and Diatomic molecules: ortho and parapara11HH22LxLyLz12 3sin sin sinixyznx nynzALLLπππφ⎛⎞⎛⎞⎛⎞=⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠22 2 2 21232222ixyznnnmL L Lπε⎛⎞=++⎜⎟⎜⎟⎝⎠=Particle (standing wave) in a boxParticle (standing wave) in a box3/222Bxyzmk TZLLLπ⎛⎞=⎜⎟⎝⎠=/iBkTiienNpNZε−==Boltzmann probability:kkyykkxxkkzzDensity of states in Density of states in kk--spacespace123xxyyzznkLnkLnkLπππ===dVkkkkkkkyykkxxkkzzConvert to spherical coordinatesConvert to spherical coordinatesdk222xyzk kkk=++2sinkdV k dk d dθθφ=×n(u)uMaxwell speed distributionMaxwell speed
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