Partition function Definition and 182 Threads

Why do the time-evolution operator in quantum mechanics ##\exp{iHt}## and the Gibbs-weight operator in statistical physics ##\exp{-H/T}## have the same functional form? – i.e. both exponentials of the Hamiltonian operator. The Matsubara trick/method just takes this as a fact in thermal QFT; but...

Why do they introduce the partition function. I have seen it in the derivation of the Boltzmann distribution. But I don't know the physical significance of it here? And how do they get to (L.11) after that? I get everything until L.7. Including L.7. The rest of the proof is here just in case...

Homework Statement Homework Equations I have question for (a) section. The Attempt at a Solution I have two answer for the question but I can't figure out which one is right. (1)Since the partition function is to sum up all the state in the system, I write down the answer (2)In other...

We have a partition function ## \displaystyle Z=\frac{1}{N! \: h^{f}} \int dq\: dp \:e^{-\beta H(q,p)} ## And we obtain, for example, the pressure by ##\displaystyle p = \frac{1}{\beta} \frac{\partial\: \ln Z}{\partial V}##. So if we do the transformation ##Z \rightarrow a Z## where ##a >0##...

Hi, maybe someone can help me with this problem? Homework Statement A system consist of N Atoms that have a magnetic moment m. The Hamiltonian in the presence of a magnetic field H is $$\mathcal{H}(p,q) - mH \sum_{i=1}^N cos(\alpha_{i})$$ where ##\alpha_i## is the angle between the magnetic...

Homework Statement With the Hamiltonian here: Compute the cananonical ensemble partition function given by ##\frac{1}{h} \int dq dp \exp^{-\beta(H(p,q)}## for 1-d , where ##h## is planks constant Homework EquationsThe Attempt at a Solution I am okay for the ##p^2/2m## term and the...

Looking for the heat capacity of ideal gas due to rotational degrees of freedom. If the temperature of the gas is much higher than the temperature corresponding to the energy differential between states,the partition function can be written as the integral over the density of states. If the...

i'm trying to understand the solution to this problem: http://physweb.bgu.ac.il/COURSES/StatMechCohen/ExercisesPool/EXERCISES/ex_2065_sol_Y13.pdf (link to the problem and the solution of it) All my questions come from the partition function: 1) From where the term (2*pi)^d comes from?, I...

Homework Statement In the real world, most oscillators are not perfectly harmonic. For a quantum oscillator, this means that the spacing between energy levels is not exactly uniform. The vibration levels of an ##H_2## molecule, for example, are more accurately described by the approximate...

Homework Statement Calculate the partition function, the entropy and the heat capacity of a system of N independent harmonic oscillators, with hamiltonian ##H = \sum_1^n(p_i^2+\omega^2q_i^2)## Homework Equations ##Z = \sum_E e^{-E/kT}## The Attempt at a Solution I am not really sure what to...

What is the difference between micro canonical Partition function and canonical Partition function? Is the mathematical expression of the above two Partition function are same? If it is then why?? [emoji29]

Homework Statement Consider the case when the three Ising spins are replaced by quantum spins 1/2's with a Hamiltonian H=-J(s1.s2+s2.s3+s3.s1) calcualte the quantum partition function Homework Equations Partition function is the sum of E^(-H*B) where B is 1/kt The Attempt at a Solution...

The canonical partition function in classical statistical mechanics is calculated by ## Q_N(V,T)=\frac 1 {N! h^{3N}}\int e^{-\beta H(\mathbf q,\mathbf p)}d^{3N}q \ d^{3N}p ##. The ## \frac 1 {N!} ## is there to prevent the Gibbs paradox. But now consider a system of N particles that have no...

There doesn't seem to be a forum that is specifically about statistical mechanics, so I'm posting this question here. I apologize for the long-winded introduction, but I think it's needed to provide context for my question: If you have a discrete collection of single-particle energy levels...

Hi! I need help with the following question: A system has two energy levels, ε and ε1 that can be occupied by fermions (spin=1/2) that are non-interacting from a reservoir at temperature T and chemical potential μ. Compute the avarage occupation number of the state with energy ε. I have...

Hi, I'm trying to calculate the partition function for a certain system and I arrived at an expression for the partition function $Z$, and have been stuck here for two weeks at the least. This is not a homework problem. If this is the wrong place to post a question like this, could you please...

Homework Statement For a diatomic gas near room temperature, the internal partion function is simply the rotational partition function multiplied by degeneracy ##Z_e## of the electronic ground state. Show that the entropy in this case is ## S = Nk\left[ \ln \left(...

Why is the partition function ##Z[J]=\int\ \mathcal{D}\phi\ e^{iS[\phi]+i\int\ d^{4}x\ \phi(x)J(x)}## also called the generating function? Is the partition function a q-number or a c-number? Does it make sense to talk of a partition function in classical field theory, or can we define...

Homework Statement Exercise 4 in the upload titled Dok1.pdf. Write down an expression for the canonical partition function for N ideal Na2 gas molecules, when the rotational contribution is treated classically, and all inner degrees of freedom are treated quantum mechanically. Use this and...

I was reading about partition function. I noticed that there are two approaches toward partition function. The first approach: Suppose we are dealing with a closed system where the system is composed of heat bath (R) and inside it there is a very small system (E), the two systems are in thermal...

I have a past paper question from statistical physics: By assuming that ##\hbar^2 k^2=p^2##, I arrived at the result: The interparticle spacing, ##a^3 =\frac{V}{N}## is $$ a^3 >> e^{-\frac{p^2}{2mk_B T}} \lambda_{deB}^3$$ Is my assumption correct? and does the result complete the purpose of...

Homework Statement Show that the partition function for the harmonic oscillator with an additional force H = \hbar \omega a^{\dagger} a - F x_0 (a + a^{\dagger}) is given by \frac{e^{\beta \frac{F^2 x_{0}^2}{\hbar \omega}}}{1-e^{\beta \hbar \omega}} and calculate \left<x\right> = x_0...

Hi, How did they break down the following summation? When finding the vibrational partition function ofa diatomic molecule it was approximated that the energy levels of the vibrational part of the diatomic molecule were harmonic and therefore the energy equation for a harmonic oscillator was...

Hi! The following image is taken from my note in Stat Mech. Please excuse my ugly handwriting... I copied this from my professor's note on a whiteboard, and I'm not so sure if it is correct. The equations for Z1 (partition function before mixing) and Z2 (partition function after mixing) seems...

Homework Statement The first excited state of the helium atom lies at an energy 19.82 eV above the ground state. If this excited state is three-fold degenerate while the ground state is non-degenerate, find the relative populations of the first excited and the ground states for helium gas in...

Homework Statement A system possesses three energy levels $$E_1=\varepsilon$$ $$E_2=2\varepsilon$$ $$E_3=3\varepsilon$$ with degeneracies $$g(E_1)=g(E_3)=1$$ $$g(E_2)=2$$. Find the heat capacity of the system. Homework Equations $$\beta=\frac{1}{kT}$$ $$Z=\sum_i g_ie^{-\beta \varepsilon_i} \...

I am reading an article by Tachikawa on the Nekrasov partition function ("A review on instanton counting and W-algebras"). The article is meant to be pedagogical but I have some trouble with what is supposed to be "baby" examples. The first one involves susy QM on \mathbb{C}^2 . He says the...

Homework Statement Ground state energy is set at 0. E_n=\left(1-\frac{1}{n+1}\right)\in with no degeneracy (\Omega(n)=1); (n=0,1,2...) Write down the partition function and look for its limit when kt \gg \in\\ kt \ll \in Homework EquationsThe Attempt at a Solution Partition function for this...

Hi all This is a fairly basic QFT question but it's bothering me. And Peskin/Schroeder fails at explaining basic stuff, so here I am. After calculating Z for a particular theory I know this can be used to calculate all kinds of correlation functions. Itself, however, is the probability...

Homework Statement A system in thermal equilibrium at temperature T consists of a large number of subsystems, each of which can exist only in two states of energy and , where . In the expressions that follow, k is the Boltzmann constant. For a system at temperature T, the average number of...

Homework Statement Why is it that the microcanonical partition function is ##W = Tr\{\delta(E - \hat{H})\}##? As in, for example, Mattis page 62? Moreover, what's the meaning of taking the Dirac delta of an operator like ##\hat{H}##? Homework Equations The density of states at fixed energy is...

Hi guys, New to the forum! Hope we have some fun with Physics in here! Cheers

Hi everibody, the other day in a stadistical physics lesson we were studyng Fermi Dirac and Bose Einstein stadistics and comparing it to the classical Maxwell Boltzmann's. We learned that in the quantum stadistics for indistinguishable particles the partition function of the whole system...

This question is in regards to the degeneracy of states for an Argon atom with just one missing electron. For hydrogen the problem of finding the partition function depends on finding the the ionized state of hydrogen divided by the non-ionized state... (please see Saha equation ->...

Hi all This is probably a naïve question to ask, but I am puzzled by it and need an answer. The first time I encountered the term 'partition function' that was in context of Boltzmann distribution. But the same formulas of manipulating a partition function ( to obtain free energy, temperature...

Homework Statement For a single molecule, derive the internal energy U = 3/2kBT In terms of the partition function Z, F = -kBTlnZ Where Z = V(aT)3/2 Homework Equations Thermodynamic identity: δF = -SδT - pδV p = kBT/V S = kB[ln(Z) + 3/2]The Attempt at a Solution U = F + TS δU = δF +...

Hi all, I am struggling to grasp the sense of the partition function. First of all, I had a look at a couple of derivations (which the relevant Wikipedia page follows) in which the concept of heat"energy of a thermal bath" is invoked. Well this is already confusing me: if the thermal bath has an...

Homework Statement An ideal monatomic gas at the temperature T is confined in a spherical container of radius R. There are N molecules of mass m in the gas. Molecules move in a spherically symmetric potential V(r) where r is the distance from the center of the container. The potential V(r) is...

Hi. 1) Does the Grand canonical partition function ##\Xi## behave in a similar fashion to the good old canonical partition function ##Z##? Do you calculate thermodynamical quantities (entropy, hemholtz free energy etc.) in similar fashions? 2) Is it possible for some kind soul here to...

First off, I'm glad I'm finally a member on this board. It has helped me TREMENDOUSLY over the past few years with various problems I've had. You guys/gals are awesome and hopefully I can make some contributions to this site. 1. Homework Statement A. Write down the partition function for...

My tutor asked us today to consider the partition function of the following model as an aside to our topic at the moment. I went to work out the maths of it today and I'm quite stuck for how the calculation can proceed. It's a 1d closed chain with some number, n, points. Each point has some...

So I'm trying to use the excitation law of Boltzmann and the ionisation law of Saha to calculate stuff about what percentage of a quantity of hydrogen in ionised and in what energy state. I have the temperature and energy levels values, so i still need the: -statistical weight (degeneracy)...

Hi guys, I'm studying a classical ideal gas trapped in a one-dimensional harmonic potential and I first want to write out the partition function for a single particle. This, I believe, requires two Gaussian integrations, like so: Z=\int_{-\infty}^{\infty} d\dot{x}...

In the semi-classical treatment of the ideal gas, we write the partition function for the system as $$Z = \frac{Z(1)^N}{N!}$$ where ##Z(1)## is the single particle partition function and ##N## is the number of particles. It is semi-classical in the sense that we consider the...

I have a partition function in euclidean quantum field theory. I have a parameter, let's say a charge, that I can change in the action that define the partition function. I found that for small charge the partition function is positive, but there is a critical charge, above the one the...

Greetings, I have been studying stat mech lately, and while I have gotten good at using partition functions to solve problems, I wanted to check my interpretation of what a partition function is, and especially to contrast it with the number of states. So, I'm just looking for a yes or no to...

Homework Statement A molecule has 4 states of energy -1, 0,0 and 1. Find its partition function and limit of energy as T → ∞. Homework Equations The Attempt at a Solution Z = \sum_r e^{-\beta E} = e^{-\beta} + 2 + e^{\beta} U = -\frac{\partial ln(Z)}{\partial \beta} = \frac{e^{-\beta} -...

Homework Statement If ##Z## is homogeneous function with property ##Z(\alpha T,\alpha^{-\frac{3}{\nu}}V,N)## and you calculate Z(T,V,N). Could you calculate directly ##Z(\alpha T,\alpha^{-\frac{3}{\nu}}V,N)##. Homework Equations ##Z(T,V,N)=\frac{1}{h^{3N}N!}(2\pi m k...

It seems like they have missed out a factor of ##(2S+1)## in the final expression for grand potential? I'm thinking it should be ##(2S+1)^2## instead.

Is there a way to derive entropy or free energy without using partition function?