Ensemble Definition and 105 Threads

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...

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...

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...

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...

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...

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 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...

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...

In the last step to get result from a quantum computer, measurement is required to collapse the quantum state. This can be done by the measuring a large ensemble average of same computed result. One realization I know is to use NMR to measure billions of spin. So, what is the minimum number of...

Given a macro-state M of a system, let S denote the potion of the phase space that has the macro-state M. A micro-canonical ensemble is one in which the probability of finding the micro-state in any part of S is equally likely (the density function is constant over S). In Pathria's...

Is it true that partition function for canonical ensemble is Z=\sum_{\{states j\}}e^{-\beta E_j}=e^{-\beta F} where F is Helmholtz free energy?

The ensemble interpretation of QM essentially ignores the wave function collapse, and apparently decoherence as well, by stating that only the statistical distributions within ensembles of systems matter (each system being in just one state). My questions are: Does the preparation of...

Homework Statement 1.a. analyze an ideal gas in a two dimensional world using micro-canonical ensemble. Specifically, find the equation of state (surface tension and area will replace pressure and volume) and also the energy as a function of temperature. b. modify the equation to create a “van...

Hi all, I was brushing up on statistical ensembles, and I found something apparently weird in microcanonical treatment of the ideal "classical" gas. I'm mainly following K. Huang's Statistical Mechanics. So there's a first approach to the problem in which the MC entropy is evaluated via an...

Homework Statement In the canonical ensemble, the probability that a system is in state r is given by P_i = \frac{g_i \exp (-\beta E_i)}{\sum_i g_i \exp( -\beta E_i)} where g_i is the multiplicity of state i. This is confusing me because I thought P_i = \frac{g_i}{\sum_i g_i} = states...

Homework Statement Grand Canonical ensemble problem- A surface of N sites that can have 0,1,2 atoms. It costs no energy to adsorb an atom. Grand canonical problem therefore in contact with particle reservoir. Assume \mu chem potentiol and temp T. What is probability for site to be empty,1,or 2...

Hello! I have a question arising from my course in statistical physics: Describing microcanonical, canonical and grandcanonical ensemble you take three variables (like E,V,N for microncanonical) and get the "rest" from derivatives of the free energies. Is there a deeper meaning why there are...

Four spin1 particles are rigidly fixed at four corners of a square. 1. What will be the entropy. 2. A field is now applied that produces an energy difference between m=0 and m=1,-1 states of each of each particle. Take m=0 as the ground state and find the free energy at temperature T. 3. Find...

I've been studying and thinking about statistical physics for a couple days now... and what bothers me is the grand canonical partition function. Namely that for a system with fixed chemical potential and energy \epsilon_i the probability of having N_i particle in that state is proportional to...

Homework Statement In a microcanonical ensemble is entropy constant? Since there is only one macrostate of energy. The Attempt at a Solution I think so.

im a bit confused about which interpretation is experimentally favored,wave function collapse or the 'minimalist ensemble interpretation (ala L.E Ballentine' (as Doc Al wrote somewhere)?All the older literature talks about 'reduction of wave packet' but reading Ballentine has left me confused.

Hey folks, I really need help setting up a canonical ensemble. I am using the einstein model and have an energy: E_\nu=A + \sum_{i=1}^{2N} \hbar\omega n_i+\sum_{j=1}^{N} \hbarh\omega n_j where A is some constant. Now, I need to build the partition function which I 'think' looks like...

Hi there, I have a system with the following energy using the einstein model: E_\nu=\sum_{i=1}^{2N} h\omega n_i+\sum_{j=1}^{N} h\omega n_j I need to set up a canonical ensemble for this. How would I write the partition function please?

Hi, I'm taking a course in Stat Mach using Kerson and Huang's Statistical Mechanics book. I am quite confused with their treatment of a Gibbsian Ensemble. They say imagine an infinite copies of the same system whose state can be represented by a point in phase space. Then \rho (p,q,t) =...