Why is the probability for say the Ising model to be found in state of energy E proportional to $e^{-E/kT}$ ?  Is this some postulate or can it be derived from simpler principles?
76868
I'm looking for an intuitive understanding of the factor $$e^{-E/kT}$$ so often discussed. If we interpret this as a kind of probability distribution of phase space, so that $$\rho(E) = \frac{e^{-E/kT}}{\int_{0}^{\infty}e^{-E'/kT}dE'}$$ then what, precisely, does this probability correspond to? Is its physical significance obvious? Specifically, why is it largest for energies close to zero?  Edit: I'd like to know how it is possible that for increasing $T$ the lower energy states grow more populated. The naive thing to think is "If I stick my hand in an oven and turn up the temperature, I'm assuredly more likely to be burned." This raises the question, exactly what does it mean for the lower energy states to grow more populated relative to the higher energy ones?
50221
Suppose we have a particle of mass $m$ that is in an eigenstate $|\psi\rangle$ of the Hamiltonian $\hat{H}=\hat{T}+\hat{V}$, where $\hat{T}$ is the kinetic energy operator and $\hat{V}=V(\hat{r})$ is the potential energy operator. If the potential has lower bound $V_0$, then is it necessary for the energy eigenvalue $E$ of $|\psi\rangle$ to be greater than $V_0$? Classically this is true, since we regard negative kinetic energy as physically unrealizable/meaningless. However, I don't know if I can necessarily say the same in the quantum case. For example, what I'm tempted to do is write  $$\langle\psi | \hat{T}|\psi \rangle = \langle\psi | E-\hat{V}|\psi\rangle$$  and say "If $E&lt;V_0$, then the RHS is necessarily negative, implying the LHS is as well, which we will regard as physically meaningless. If $E=V_0$, then $|\psi\rangle$ is trivial.", but I'm not sure if that's right.   Confused by that, I then wanted to show that, if $E&lt; V_0$, then a non-trivial $|\psi\rangle$ is non-normalizable. However, I'm not entirely sure how to do this.
118719
I know that according to law of equipartion of energy ,energy per molecule per degree of freedom is 1/2 Kb T I wanted to know if it is true for a real gas ?
10264
I am starting with Quantum Mechanics, learning online. I can't seem to find the reason for $|\Psi|^2$ being the probability density of finding an electron. They've just taken it for granted everywhere. I am learning all this math but I am not able to fully grasp the intuitive idea behind all of it. If anyone could explain this with proper reasoning, I would be grateful.
86622
What would be the proof for $\langle q| p \rangle = e^{ipq}$?  Is it derived from canonical commutation relation?  ($|q \rangle $ represents the position eigenstate, while $|p \rangle$ represents the momentum eigenstate.)
6561
After deriving the entropy of an ideal gas we get to : $$S = Nk \left[\ln(V) + \frac{3}{2}\ln(T) + \frac{3}{2}\ln\left(\frac{2\pi mk}{h^2}\right) - \ln(N) + \frac{5}{2} \right]$$  In the zero temperature limit, we expect to have $S=0$, however, we get infinity. How can we overcome this mathematical inconsistency?
112675
I am wondering about the partition function of the classical microcanonical ensemble. It contains Planck's constant and also an indistinguishability argument about the particles I am looking at and I find this confusing for the following reasons:  If I describe for example an ideal gas using classical mechanics, there is no reason why I should assume that there is a problem to distinguish the gas molecules or why $\hbar$ should occur there.   I mean, how could people like Gibbs etc. derive this equation if they did not know about QM?   In some sense, there has to be a reason why we have these two QM properties in there and maybe it is just to get the limit to QM right, but is this really the only reason for this?  I would love to receive a few ideas about this strange definition by this community, in order to get impressions, why we model a (NVE) ensemble this way?
49357
I'd like some suggestions for good reading materials on the Keldysh formalism in the condensed matter physics community. I'm familiar with the imaginary time, coherent state, and path integral formalisms, but lately, I've been seeing Keldysh more and more in papers. My understanding is that it is superior to the imaginary time formalism at least in that one can evaluate non-equilibrium expectations.
41735
Consider the Ising model with nearest neighbours interactions on a rectangular lattice $L\times M$.  If $L=M$ (2-dimensional square lattice), it is known (e.g. by Peierls argument or Onsager explicit solution) that the model exhibits a phase transition when $L=M\to\infty$.  If instead we fix $L=1$ (1-dimensional line) and let $M\to\infty$, the model does not exhibit a phase transition.  My question is: which relation among the side lenghts $L,M$ guarantes the presence/absence of a phase transition? For example what about the case $L=\log M$ ?
119493
Can someone summarize as to what are the problems and/or the open questions with the ? (in the context of understanding black-holes or as a computer science question)It gives a framework to see how the black-hole can thermalize in a $log (entropy)$ time scale. What next?   To start off,   The sort of problem with the Hayden-Preskill circuit is that it works via random unitary transformations on disjoint pairs of qubits and hence doesn't have an Hamiltonian interpretation. I believe one has always believed that matrix models (of matrix size $\sqrt{entropy}$) can saturate this logarithmic thermalization bound via Hamiltonian evolution.    Has this above belief about the matrix models been proven yet?  Is there an intuitive explanation as to why a theory with finite dimensional matrices should behave like an infinite dimensional system?
45834
I'm working through a problem regarding expected values in Markov chains, and at some point it says:     Recall from probability that if $X$ is a positive integer valued random variable, then $\mathbb{E}[X] = 1 + \sum^\infty_{k=1}\mathbb{P}(X &gt; k)$.   I know that by definition $\mathbb{E}[X] = \sum_a a\mathbb{P}(X = a)$, but I can't see how the above equality follows from this, nor am I sure if this is how to approach the problem.
86806
I have the following problem that I was unable to solve for class, but I had a couple first steps that I started with that I am unable to finish.  I know I can't get this since it's already been turned in, but I would like to see if this was even a viable option for doing this proof in the first place.  ``Given an orthogonal set of states, $\{\phi_{n}\}$, and a Hamiltonian, $\hat{H}$, show that the partition function, $Q_{\beta}$, satisfies the following $$ Q_\beta \geq \sum_{n}\exp\{-\beta \langle \phi_n|\hat{H}|\phi_n\rangle \}$$ with equality holding when the $\phi_n$ states are eigenstates of the Hamiltonian.''  I started by dropping in the identity in the exponential (as eigenstates of the Hamiltonian) $$ \sum_n \exp\{ -\beta\sum_k \langle \phi_n|\psi_k\rangle\langle \psi_k |\hat{H}|\phi_n\rangle\}=\sum_n \exp\{ -\beta \sum_k E_k |c_{nk}|^2\}$$  Then I am left with showing that $$\sum_n \exp\{ -\beta \sum_k E_k |c_{nk}|^2\} \leq \sum_k \exp\{ -\beta E_k\}$$ with equality showing up again the same way, with a Kronecker delta $\delta_{nk}$ collapsing the sum in the exponential.  I realize I didn't make it very far, so this might not be the best way to show this, but it seems manifestly true just by looking at it, but I can't actually show it.  Does anyone have any hints about how to continue with this?
15504
As history of thermodynamics say, it was a mystery that what is the required condition for a given energy conversion to take place? Like there are two possible events each conserving energy but only one is chosen. So, in order to resolve this Clausius introduced a quantity called  which was given by $\int dq/T$. But can I know the reason for which Clausius chose this integral or quantity, why not any other quantity (changes in whom, positive or negative, would decide the occurrence of a given event)? I hope there lies an explanation to this which does not use statistical mechanics.
42958
So, as your usual physics undergrad, I read Griffiths's derivation of the general   uncertainty principle. I understood it but there was no physical intuition given behind it in the book. It was basically a bunch of manipulations and inequalities used to get just the right result.  Other than "It's because of the statistical interpretation." is there a more intuitive reasoning behind uncertainty. Like commutators, behavior of particles, etc.  Thank you for the replies and as always, much obliged.
118049
Would like to ask a question, but first i would like to say Hello Everybody in a way that plays the system, since some geniouses decided that one should not be able to say hello in a question.  The  in quantum mechanics is well known and considered one of most basic properties of natural reality. The  is also well known and also considered one of the most basic processes of natural reality.  The uncertainty principle uses and is related to . Planck's constant has the dimensions of action and in a statistical mechanics approach, also relates nicely with the partitioning of the phase-space providing the basic measure for the entropy functional ( provides a nice outline of this).  Apart from that, there are  which relate the Heisenberg Uncertainty Principle in quantum mechanics directly and intuitively to the 2nd Law of Thermodynamics.  Is this relation correct? And if so can we derive one from the other?  Thank you  PS. One can also check , which although not the same, is related in an interesting way.  UPDATE:  anna's answer is accepted since by mentioning the derivation of (part of) the 2nd law from unitary dynamics, answers the question at least in one way. Please consider this as still open so you can add another answer. There are more alternatives (and one of which is my stance, ie thermodynamics -> uncertainty)
71005
As we know, if an ideal gas expands in vacuum, as its energy is unchanged, the temperature remains the same. An ideal gas's energy does not depend on volume. In general, the energy is $kT$ times the total degrees of freedom, like in an ideal gas, the total degrees of freedom is $N$ particles plus three dimensions, $3N$.    Then if the total energy of the universe is $kT$ times the total degrees of freedoms of the universe, as the universe expands, its energy and entropy should not change, but if the temperature falls, the number of degrees of freedom should grow. It is quite puzzling to me that the universe is having more and more new degrees of freedom. It seems to be contradictory to the entropy argument.
67065
Suppose that we observe the discrete random variable $ X = (X_1, \dotsc , X_n)$ with state space S, whose distribution we do not know but we are assuming that its joint p.m.f. belongs to a known family $\{f_θ : θ ∈ Θ\}$. We derive a sufficient statistic $T(X)$.  Is it correct to write $\Bbb E[X]$? If not, why? Would it be more correct to write $\Bbb E_{\theta}[X]$?