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Quantum Stat­ist­ics is a fun­da­ment­al tool for the study of thermal phys­ics, sol­id state phys­ics, opto­elec­tron­ics, lasers, optics, quantum mech­an­ics and pretty much any piece of mod­ern phys­ics you can think of. Although I found the top­ic to be par­tic­u­larly dif­fi­cult when it was first intro­duced to me, now I feel that

 it’s actu­ally a sur­pris­ingly intel­li­gible sub­ject. Per­haps the main obstacle to over­come is the lengthy math­em­at­ic­al deriv­a­tions and manipulations.

I have endeavored to present the top­ic in a struc­tured fash­ion, cov­er­ing each item com­pre­hens­ively. There are nat­ur­ally some typo­graph­ic­al errors and I would appre­ci­ate them noted in the video comments!

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Return to Video Tutori­als G-Z

The Quantum Stat­ist­ics video tutori­als may be viewed as part of the fol­low­ing playl­ists, or as lis­ted later on.

Max­well Boltzmann Stat­ist­ics Playl­ist Fermi Dir­ac Stat­ist­ics Play­ist Bose Ein­stein Stat­ist­ics Dens­ity of States 

Mul­ti­pli­city Rules Applic­a­tions of Quantum Stat­ist­ics Som­mer­feld Expan­sion Black­body Radiation

Video # Video Tutori­al Title Remarks
1 Macro and Micro States Unless you under­stand these you’re going to struggle at a later stage
2 The Mul­ti­pli­city Function The num­ber of ways of arran­ging the particles
3 Dis­tin­guish­able VS Indis­tin­guish­able Particles Clas­sic­al particles, Bosons and Fermions!
4 Clas­sic­al Particles, Bosons & Fermions Defin­ing each type of particle
5 Mul­ti­pli­city Rule 1 All particles are dis­tin­guish­able and inter­act­ing (4 species)
6 Mul­ti­pli­city Rule 2 Mix­ture of dis­tin­guish­able and indis­tin­guish­able particles — all inter­act­ing (3 species)
7 Mul­ti­pli­city Rule 3 Mix­ture of dis­tin­guish­able and indis­tin­guish­able particles — all inter­act­ing (3 species)
8 Mul­ti­pli­city Rule 4 Mix­ture of dis­tin­guish­able and indis­tin­guish­able particles — all inter­act­ing (2 species)
9 Mul­ti­pli­city Rule 5 Non inter­act­ing particles (could be either Bosons or Clas­sic­al Particles)
10 Mul­ti­pli­city Rule Summary A sum­mary of the first five mul­ti­pli­city rules
11 Mul­ti­pli­city Rule 6 All particles are indis­tin­guish­able and non-interacting
12 Max­well Boltzmann Mul­ti­pli­city Function A deriv­a­tion which is used later to derive the occu­pancy function
12 a Dens­ity of States, Num­ber Dens­ity and Occupancy Intro­du­cing fun­da­ment­al quant­it­ies of quantum statistics
13 Bose Ein­stein Mul­ti­pli­city Function A deriv­a­tion which is used later to derive the occu­pancy function
14 Fermi Dir­ac Mul­ti­pli­city Function A deriv­a­tion which is used later to derive the occu­pancy function
15 The Boltzmann Factor A deriv­a­tion of a most import­ant quantity
16  The Par­ti­tion Function A deriv­a­tion of yet anoth­er import­ant and use­ful quant­ity. (The video light­ing here isn’t great 😐 )
17 Aver­age Energy Via Boltzmann Factor An extremely use­ful trick IOT greatly sim­pli­fy stat­ist­ic­al calculations
18 Deriv­a­tion of Equipar­ti­tion Theorem One of the most import­ant the­or­ems from clas­sic­al mechanics!
19 Dens­ity of States 1/7 Sch­rodinger Equa­tion for particle in infin­ite poten­tial well
20 Dens­ity of States 2/7 Vec­tor ‘k’ space and Vec­tor ‘n’ space
21 Dens­ity of States 3/7  Scal­ar ‘k’ and scal­ar ‘n’ space
22 Dens­ity of States 4/7  Dens­ity of states in momentum space
23 Dens­ity of States 5/7  Dens­ity of space in velo­city space
24 Dens­ity of States 6/7  Dens­ity of states in energy space
25 Why the # ‘Dots’ in n Space = The Volume in n Space Why you can cal­cu­late the num­ber of n space states by cal­cu­lat­ing the volume
26 Max­im­ising the Occu­pancy Function How to cal­cu­late the most prob­able distribution
27 Meth­od of Lag­range Multipliers Learn why the Meth­od of Lag­range Mul­ti­pli­ers Works
28 Max­well Boltzmann Dis­tri­bu­tion / Occu­pancy Function Deriv­a­tion of the occu­pancy func­tion for clas­sic­al particles
29 Eval­u­ation of α and the Ther­mo­dy­nam­ic β Max­im­ising the occu­pancy func­tion res­ults in α and β: I eval­u­ate them here
30 Bose Ein­stein Dis­tri­bu­tion Func­tion 1/2 Deriv­a­tion of the occu­pancy func­tion for Bosons
31 Fermi Dir­ac Dis­tri­bu­tion Func­tion 1/3 Deriv­a­tion of the occu­pancy func­tion for Fermions
32 a Max­well Boltzmann Scal­ar Speed Distribution Deriv­a­tion of scal­ar speed dis­tri­bu­tion for clas­sic­al particles such as air molecules
32 b Max­well Boltzmann Aver­age Speed Cal­cu­late the aver­age speed of particles in a gas (say)
32 c Max­well Boltzmann Vrms Cal­cu­late the root mean squared velo­city of a gas (say)
33 a Deriv­a­tion of the Fermi Level 1/2 Free elec­tron the­ory IOT derive a func­tion for the Fermi Level
33 b Deriv­a­tion of the Fermi Energy 2/2 A sim­il­ar video to # 33 a but much quick­er and with less back­ground theory
33 c Dens­ity of States Writ­ten as a Func­tion of E-Fermi Using the res­ult from video # 33 b
33 d Total Elec­tron Con­tri­bu­tion to a Solid’s Energy  
34 Aver­age Energy Via Par­ti­tion Function A net trick! Use it to derive all the occu­pancy func­tions in a flash!
35 Power Series Res­ult­ing from 1/(1-x)  Anoth­er neat trick for using the Par­ti­tion Func­tion to derive the B.E. Occupancy
36 a Aver­age Energy of a Quantum Har­mon­ic Oscillator Ein­stein mod­elled solids as LHOs to dis­cov­er quant­ised phonons
36 b Heat Capa­city of Har­mon­ic Oscil­lat­or System Einstein’s LHO approach to solids dis­covered that they obeyed B.E. statistics
36 c Ein­stein For­mula for the Spe­cif­ic Heat Capacity A sim­il­ar video to # 36 b : Ein­stein, his phon­ons and Bose Ein­stein statistics
37 Par­ti­tion Func­tion and Free Energy Why is the Par­ti­tion Func­tion to useful?
38 The Gibbs Factor An improve­ment upon the Boltzmann Factor to account con­ser­va­tion of particles
39 Fermi Dir­ac Dis­tri­bu­tion Func­tion 2/3 Deriv­a­tion using the Gibbs Factor rather than multiplicities
40 Bose Ein­stein Dis­tri­bu­tion Func­tion 2/2 Deriv­a­tion using the Gibbs Factor rather than multiplicities
41 Fermi Dir­ac Dis­tri­bu­tion Func­tion 3/3 Deriv­a­tion using the Grand Par­ti­tion Function
42 Dens­ity of States 7/7 Where no peri­od­ic bound­ary con­di­tions exist (pre­vi­ous 6 videos used peri­od­ic BCs)
43 Spher­ic­al Polar Co-ordinates Trans­form from Cartesian to Spher­ic­al Polar co-ordinates
44 The Planck Dis­tri­bu­tion 1/2 Black­body Radi­ation and the Ultra­vi­olet Catastrophe
45 Photon Elec­tro­mag­net­ic Energy Density Energy dens­ity in the Planck Dis­tri­bu­tion (light as photon quanta)
46 Black­body Radi­ation : Con­tinu­ous Energy Levels The Ultra­vi­olet Cata­strophe occurred when con­tinu­ous energy levels were assumed
47 The Planck Dis­tri­bu­tion 2/2 Planck’s for­mula for Black­body Radi­ation via quant­isa­tion of energy
48 Wien’s Dis­place­ment Law Deriv­a­tion
49 Stefan Boltzmann Law  I=σT4
50 The Sol­id Angle or Steradian Vital for flux calculations
51 Photon Flux Cal­cu­la­tion of irradiance
52 Som­mer­feld Expan­sion 1/3 The neces­sary Taylor Series expansion
53 Som­mer­feld Expan­sion 2/3 A full deriv­a­tion of Sommerfeld’s expansion
54 Som­mer­feld Expan­sion 3/3 The con­tri­bu­tion of elec­trons to a solid’s energy
55 Laser Co-effi­cients Ein­stein LASER absorp­tion and emis­sion co-efficients
56 Laser Gain Co-efficient Deriv­a­tion
57 Elec­tron Orbitals Intro­duc­tion to the Peri­od­ic Table
58 Helm­holtz The­or­em 1/2 Deriv­a­tion of scal­ar potential
59 Helm­holtz The­or­em 2/2 Deriv­a­tion of vec­tor potential



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