#2 Proving the Second Law of Thermodynamics with QM | Statistical Mechanics series

Описание: #Thermodynamics #Entropy #Boltzmann

0:00 - Intro
3:52 - Intro to Quantum Mechanics
5:56 - Dirac’s Bra-ket Notation
9:19 - Quantum States as Complex Vectors
13:24 - Normalisation of a State Vector
14:29 - Different Observables are Different Vector Bases
16:31 - Observables as Hermitian Operators
21:44 - Eigenstates of an Observable
23:45 - Commutators between Observables
25:43 - The Identity Operator & Matrix Mechanics
28:51 - Dirac’s Delta Function
31:12 - Momentum wave function & de Broglie’s waves
34:54 - Normalisation of Momentum wave function
36:55 - Position-Momentum commutator
41:27 - Momentum wave function : Normalised wave packet
42:37 - Exercise : Normalising a wave packet
43:18 - Momentum wave function in a box
45:58 - From 1D to 3D
50:13 - Deriving the Ideal Gas Law
51:50 - Indistinguishability of Identical Particles
1:07:10 - Exercise : The Equipartition Theorem
1:08:20 - 1st Proof of 2nd Law : with Time Reversal Symmetry
1:13:16 - 2nd Proof of the 2nd Law : Time evolution in QM
1:24:19 - General Proof of the 2nd Law with QM
1:30:33 - Ending

I have decided to extend my first video on Statistical Mechanics into a new series. The goal is cover all the interesting topics on Statistical Mechanics, including the study of phase transitions and critical phenomena. We hope to introduce in details, topics like mean field theories, and the renormalisation group, which are indispensable tools if we wish to understand systems in thermal equilibrium, and their statistical fluctuations.

This path leads naturally to the topic of quantum field theory, which we hope to introduce in this way. Going beyond systems in thermal equilibrium, we enter into the domain of non-equilibrium physics. The most general approach to this problem is through the quantum theory of open systems. We shall cover this topic in some details, and make connections with the theory of decoherence, and the measurement problem in quantum mechanics; which is one of my pet topics…

This is the 2nd lecture of this series. In this video we are going to do 3 things : the first is to “fix” our proof of the 2nd Law of Thermodynamics from the last lecture. Recall that we have previously assumed that the microscopic processes involved, must possess time reversal symmetry. This time round we shall drop this assumption, and just use quantum mechanics to prove the same result…

Secondly, we shall apply the formalism derived in the last lecture, mainly the Boltzmann distribution and partition function, to derive the ideal gas law. This should be quite instructive on how to use the partition function to calculate the thermodynamic quantities of a system. We shall see that even here, we benefit from the use of quantum mechanics…

Thirdly, after mentioning quantum mechanics so profusely in the last two items, we must definitely provide an introduction to the theory. Although this must necessarily be a crash course, just enough for us to move along in Statistical Mechanics, but with sufficient details as not to cause confusion. More details are available in my series on quantum mechanics…

►Next(Lecture 3):    • #3 All About Free Energy / Lagrange Multip...  
►Solutions to exercises given in this video(Lecture 2) :    • Stat Mech Lecture 2 : Wave Packets, Quantu...  
►Previous(Lecture 1):    • Teach Yourself Statistical Mechanics In On...  

►Quantum Mechanics :
Lecture 1 :    • 1. Bras, Kets And Operators | Weinberg’s L...  
Lecture 2 :    • 2. Continuous States and Observables | Wei...  
Lecture 3 :    • 3. Symmetries : Unitary and Anti-unitary O...  
Lecture 4 :    • 4. Lie Algebra, Symmetries and Schrodinger...  

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