Ensemble Definition and 125 Threads

So from what I understood from some coure notes I've been reading, a microcanonical ensemble is a situation where we have an isolated system in thermal equilibrium with a constant given N,V,E - particles, volume,total energy. I'm a bit confused. How I understood 'ensemble' is as a set of all...

Hello everybody :D My question is: given the distribution of the canonical ensemble, how do we get the helmoltz free energy? I think we can't use A = U-TS because we don't know how to write S. So what's the solution? Thanks

Hi, In QED it is stated that an EM field can be written as a sum of quantized oscillators (the photons). In "classical" Electrodynamics, it can also be shown that the EM field decomposes into normal modes. But both the quantized oscillators (in the Heisenberg picture) and the classical normal...

Suppose that we have a system of particles (I am assuming a general system), and I want to find the ground state energy ##E_0##. We know that we can consider our system by canonical ensemble formalism OR by grand canonical ensemble so that ##H_G=H-\mu N## (in which ##H## is the Hamiltonian in...

Ref: R.K Pathria Statistical mechanics (third edition sec 5.2A) First it is argued that the density matrix for microcanonical will be diagonal with all diagonal elements equal in the energy representation. Then it is said that this general form should remain the same in all representations. i.e...

Homework Statement We have a quantum rotor in two dimensions with a Hamiltonian given by \hat{H}=-\dfrac{\hbar^2}{2I}\dfrac{d^2}{d\theta^2} . Write an expression for the density matrix \rho_ {\theta' \theta}=\langle \theta' | \hat{\rho} | \theta \rangle Homework Equations...

An ensemble is a collection of systems, all prepared in the same way. Does this mean that all the systems are in the same state? I have seen some authors create ensembles where 30% of the systems are in a state, s, and 70% of the systems are in a state, t . As far as measurements go, this...

I have a problem in understanding the quantum operators in grand canonical ensemble. The grand partition function is the trace of the operator: e^{\beta(\mu N-H)} (N is the operator Number of particle) and the trace is taken on the extended phase space: \Gamma_{es}= \Gamma_1 \times \Gamma_2...

Homework Statement Statistical Mechanics by Pathria. Problem 3.1 Homework Equations (1) <(△nr)2>=<nr2>-<nr>2=(wrd/dwr)(wrd/dwr)lnΓ, for all wr=1 How to derive above equation from these equations? <nr>=wrd/dwr(lnΓ), for all wr=1 <nr2>=(1/Γ)(wrd/dwr)(wrd/dwr)Γ, for all wr=1 (2) Also, if you...

I have a short question which I have been discussing with a fellow student and a professor. The question (which is not a homework question!), is as follows: If you shift all the energies E_i \to E_i + E_0 (thus also shifting the mean energy U \to U + E_0), does the entropy of the system remain...

I have a question that has puzzled me during the last couple of days. Suppose that there is a system (e.g. a small box filled with gas) that is connected to a heat bath (much larger than the system) at a constant temperature T. The studied system and the heat bath are thought to be isolated from...

I think I understand it but I prefer a response from a sci advisor rather than whatever someone put up on wikipedia. I also searched for it here but I didnt find a proper explanation. Any help would be greatly appreciated.

Homework Statement Part (a): Using grand canonical distribution, show ideal gas law ##P = nkT## holds, where ##n = \frac{\overline{N}}{V}##. Part (b): Find chemical potential of diatomic classical ideal gas in terms of ##P## and ##T##. The rotational levels are excited, but not the...

What is ensemble? I've read about this in blundell's book, and It is said that it's used to control microscopic properties. I don't understand this statement. Somebody please help me ..

http://arxiv.org/pdf/1104.2822.pdf "Abstract: A new ensemble interpretation of quantum mechanics is proposed according to which the ensemble associated to a quantum state really exists: it is the ensemble of all the systems in the same quantum state in the universe. Individual systems within...

Homework Statement I have a system of N non-interacting anharmonic oscillators whose potential energy is given by, V(q) = cq^2 -gq^3 -fq^4 where c,f,g > 0 and f,g are small. Homework Equations The Hamiltonian is given by, H = \sum_{i=1}^N \big ( \frac{p^2_i}{2m} + V(q_i) \big )...

Hi, can I know why the number of partitions separating different states have to be taken into account for the derivation of number of states in an ideal Bose Gas but not in the Fermi Gas? What is the physical significance of this "partition"? In what ways can they vary?

Recall the famous "Shut up and calculate interpretation". Should it be called an interpretation? The content of the interpretation is "don't interpret quantum mechanics". I argue that for certain purposes it would make sense to distinguish between interpretations (like Many Worlds and...

Hi guys :cry:, I regard an ideal diatomic gas which is in a Volume x,z and got a angular \phi: 0 \le x \le L, ~~~~~~~~~~~~ 0 \le z \le \infty, ~~~~~~~~~~~~0 \le \phi \le 2 \pi The hamiltonian for the single particle is: ~~~~~~H= \frac{p^{2}_{z} + p^{2}_{x} }{2M} +...

Hello guys, I would really need some help on the following problem. Consider a non-interacting & non-relativistic bosonic field at finite temperature. We are all aware of the fact that such a statistical system is well described by the grand-canonical ensemble in the limit N→∞. However...

Homework Statement I have the equation Z=1/N!h3N∫∫d3qid3pie-βH(q,p) How can I get the entropy from this equation assuming a classical gas of N identical, noninteracting atoms inside a volume V in equilibrium at T where it has an internal degree of freedom with energies 0 and ε What...

Optical density of an atomic ensemble and its linewidth Hi I've been thinking about something for a while now. If I take a two-level atom, completely at rest, then I can probe it with a quasi-resonant laser. The absorption of the atom is a nice Lorentzian, nothing fancy here. However, now...

In Statistical Mechanics, the key step in the derivation of the Canonical Ensemble is that the probability of S being in the m-th state, P_m , is proportional to the corresponding number of microstates available to the reservoir when S is in the m-th state. That is P_m=c\Omega(E_0-E_m), where...

I have some question about Ensemble Average. The macroscopic variable of fluid is average of microscopic variable of gas molecule. And if time average is same with spatial average I know as it called edgodic. My question is that if two different microscopic variables which have same...

Hello, The entropy of the Grand Canonical Ensemble (GCE) is: S = KB ln ZG + (\bar{E}/T) - μo\bar{N}/T Helmholtz function is: F = \bar{E} - TS = \bar{E} - TKB ln ZG - \bar{E} + μo\bar{N} = -TKB ln ZG + μo\bar{N} But \partialF/\partialT = -S (From thermodynamics). Then...

Hey, Here is the problem: The method by which we solve is by Langrange Multipliers, and so I believe I found the derivative of f with respects to P(i) but I have two quantities I'm sure what they equal: Summations over i=1 to N : Ʃln(P(i)) and ƩE(i) Thanks for any help, S

Hello, I was investigating a system with N indistinguishable particles, each of which can have an energy \pm \epsilon, and using the grand canonical ensemble, i.e. \Xi = \sum_{N=0}^{\infty} e^{\beta \mu N} Z_N. But my entropy formula is S = \left( \textrm{a couple of $\sim N $ positive...

Hello, I was wondering if it is a well known fact that the microcanonical ensemble (i.e. W = \int \delta(\mathcal H(\vec x, \vec p) - E) \mathrm d \vec x \mathrm d \vec p) does not weight every point equally, in the sense that in the integral (which was just quoted) some points on the energy...

Hi Guys I have recently been reacquainting myself with Ballantines - Quantum Mechanics - A Modern Development. He is pretty big on the Ensemble interpretation, and I must admit I am rather attracted to it as well - none of this collapse of a wave function stuff, many worlds etc. He also makes...

I have trouble distinguishing the two, what's the physical difference between a fock state|p1;p2> and a mixed ensemble described by density matrix 0.5|p1><p1|+0.5|p2><p2|?

Can somebody explain to me the differences between the ensembles, and how does this differences refer to experiment? I know that: Microcanonical ensemble is a concept used to describe the thermodynamic properties of an isolated system. Possible states of the system have the same energy and...

Homework Statement I'm looking for a closed form expression for the partition function Z using the Canonical Ensemble Homework Equations epsilon_j - epsilon_j-1 = delta e Z = Sum notation(j=0...N) e^(-beta*j*delta e) beta = 1/(k_B*T) t = (k_B*T)/delta e N is the number of excited...

Can we take the grand canonical ensemble and then switch the roles of the thermodynamic conjugate variable pair (P, V) making P (pressure) the parameter and V (volume) the variable and allowing it to fluctuate in the system. The macrostate would then be defined by the pressure temperature and...

Dear friends, I have a question regarding thermodynamic integration in the NPT ensemble. Suppose I vary the parameter a from 0 to 1, where 0 is the reference state whose free energy I know and 1 corresponds to the state whose free energy I want to find out. then is the...

Hi everyone, I've recently entered a slightly different subfield of research in which ensemble averages are used profusely. I haven't yet taken stat mech and we didn't discuss this in my quantum courses, so I am wondering - can anyone please give me a clear definition of how one computes...

Hi folks, since the volume V is fixed in a canonical ensemble I'm a bit confused about the fact, that the pressure is calculated as the derivation of the internal energy U with respect to the volume V. Sure, P = dU/dV comes from dU = dQ + dW = tdS - pdV + ... But what does it mean to derivate...

hi, usually the density operator for the microcanonical ensemble is given by \rho = \sum_n p_n|n><n| where |n> are energy eigenstates and p_n is the probability that our system is in this state. p_n = const. if the energy corresponding to |n> is in the energy inteval (E,E+∆E)...

I was trying to get answer to question if it's possible to construct non-contextual ensemble from contextual particles. As a result I constructed this simple toy model in excel. It is using kind of "quantum memory" as polarizer. "Quantum memory" of polarizer evolves coherently with horizontal...

b]1. Homework Statement [/b] I have the soloution to this question, but am confused as to what has been done between each step between lines 2,3,4. Can anyone explain how they have been simplified (espicially what happened to the operator) and what the value of the intergral is? I think I am...

Homework Statement A system consists of 3N (N >> 1) independent, identical, but distinguishable one-dimensional oscillators. This is relevant in that the atoms in a solid are sitting around their equilibrium positions. Assume that every atom constitutes an independent oscillator and all...

Homework Statement In deriving <E2>-<E>2 starting from <E>=U=sum(Eiexp(-beta Ei))/sum(exp(-beta Ei). the taking derivative of U with respect to beta, the book always notes E (thus Volume) is held constant. what i am trying to do is taking the derivative of U with respect to beta or T...

In derivation of probability of system at energy E with canonical ensemble, one assumes that the probability of system in a microstate Ei is proportional to the multiplicity of reservoir. Is this probability the conditional probability by knowing that system is at energy Ei with knowing it is at...

Homework Statement Some systems are adequately described by a one-dimensional potential in the form of an asymmetric double well. To good accuracy each can assumed to be harmonic with potential energies: V_L(x)=\frac{1}{2}k_Lx^2, V_R(x)=\epsilon+\frac{1}{2}k_R(x-a)^2 Here, \epsilon=V_R(a)>0. N...

Homework Statement To calculate internal energy for a system of non interacting S.H.O's in 1 dimension at constant T & u(chem potential) using grand partition function The Attempt at a Solution L(Grnd prtition fn)= Summation(z^N)*[Z(TVN)] where z = fugacity , Z = partition fn of...

To get a distribution of some dynamic variable of a wavefunction, we actually need to prepare an ensemble of particles, in which all the particles have the same wavefunction, right? And no-cloning theorem states that it's impossible to copy an unknown quantum state. So is this a contradiction？...

I posted this once already without seeing the rule that HW questions must go here--sorry :redface: So, the problem: I'm really lost on where to get started here. It's a two state system, one with energy 0 and the other with energy ε. I already have ensemble average, <E>, found to be: ε / (e^βε...

I'm really lost on where to get started here. It's a two state system, one with energy 0 and the other with energy ε. I already have ensemble average, <E>, found to be: ε / (e^βε + 1) , where β is thermodynamic beta, 1/KbT. How do I convert this to an expression for the variance of the...

Does anyone actually use the term "ensemble interpretation"? People have been asking questions about the "ensemble interpretation" here lately. It's mentioned on the Wikipedia page "interpretations of quantum mechanics" and has its own page. But is that term actually used in books and articles...

Hi all, My question is about obtaining the force between two particular atoms in an ensemble. Please excuse my ignorance on this topic. I can calculate all the forces in the system by the Hellmann–Feynman theorem using any quantum mechanical code. This is the print out from such a code...

My understanding of this interpretation is that (in the context of the electron double slit experiment) the experiment is one particular outcome of an ensemble of equally prepared experiments. This explains the statistical nature of QM. But how does this explain the interference pattern...