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Questions tagged [quantum-statistics]

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Why is finite temperature many-body perturbation theory computed in the grand canonical ensemble?

Why does virtually every textbook and paper treat many-body perturbation theory at finite temperature in the grand canonical ensemble? Is it not possible to formulate a canonical theory where all ...
23
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2answers
3k views

Why do bosons tend to occupy the same state?

It is often said that, while many fermions cannot occupy the same state, bosons have the tendency to do that. Sometimes this is expressed figuratively by saying, for example, that "bosons are sociable"...
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0answers
23 views

Distribution of photons emitted by atoms

I am currently revising quantum gases, and a small but confusing thought experiment has been bugging me for a while. I understand the bookwork stuff on photons and how a photon gas in a blackbody ...
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1answer
35 views

Macroscopic population of excited states for Bose-Einstein condensation?

I am currently learning about Bose-Einstein condensation (BEC). I understand that the ground state is rapidly populated when the temperature goes below the critical temperature. This macroscopic ...
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0answers
18 views

Thermalization and structure formation, starting with a pure state coupled to heat bath

I am looking for tractable quantum mechanical systems which show thermalisation and/or structure formation. One idea is a small quantum system S, e.g. a quantum Heisenberg model in a pure state (the ...
3
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0answers
25 views

Spectral gap and local dynamics imply exponential decay, what about the reverse?

The paper "Spectral Gap and Exponential Decay of Correlations"(https://arxiv.org/abs/math-ph/0507008) proves that if a system has a spectral gap and is local, then ground state expectation values will ...
2
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0answers
34 views

Correlation length amplitudes in Ising 2D model

I am reading the article about Universal amplitude ratios in the 2D Ising model (https://arxiv.org/abs/hep-th/9710019) by G. Delfino. I have a question about page 3 of the paper. For a magnetic ...
0
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1answer
53 views

General formula for the variation of the chemical potential with temperature

For small temperatures $T$, such that $k_BT\ll \mu(T=0)\equiv \mu(0)$, the variation of chemical potential with temperature is given by $$\mu(T)=\mu(0)\Big[1-\frac{\pi^2}{12}\Big(\frac{k_BT}{\mu(0)}\...
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2answers
59 views

What happens when we cool down the gas of non-identical particles?

For gas of identical particles, when we cool it down to extremely low temperature we can see one of two types of behaviour depending on the symmetry of wavefunction with respect to argument ...
2
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1answer
84 views

distinguishable particles' Hamiltonian

Let us consider a classical Hamiltonian of a many body system \begin{equation*} H = \sum_{j=1}^N\frac{p_j^2}{2m}+V(\mathbf q) \end{equation*} and let us pass to quantum dynamics by promoting the ...
1
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1answer
62 views

Are the Fermi-Dirac, Bose-Einstein and Boltzmann distributions all probabilities, or are they ways to get to probabilities?

Hyper physics has a page for the energy distribution functions (here), they say that each of the distributions are the probabilites that a particle has a certain energy state E, but other websites ...
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3answers
62 views

In what conditions we can take ground state energy $E_0$ equal to zero?

Like in 3D harmonic oscillator , $E_m = {h} \omega {(m_x +m_y + m_z + 3/2)}$, At ground state energy $m=0$ and $ E_m = h \omega(3/2)$. While discussing Bose-Einstein condensation, for calculations we ...
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1answer
37 views

Local equilibrium of slow time varying thermal system

I'm trying to differentiate a thermal system in local equilibrium (and slow time varying) v/s a non-equilibrium system. For a thermal system which is slowly time varying, how does one define local ...
0
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1answer
75 views

Why do quantum effects of particles dominate when the thermal de Broglie wavelength becomes comparable to the inter-particle spacing? [duplicate]

Why do quantum effects of particles dominate when the thermal de Broglie wavelength becomes comparable to the inter-particle spacing?
0
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1answer
53 views

How to calculate the spin of an atom [duplicate]

If given an atom say ${^{108}_{47}Ag}$, what is the systematic way to determine its spin so that one knows whether it is a boson or a fermion?
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0answers
14 views

Atom in Radiation Reservoir (Cohen-Tannoudji formalism)

My task is to generalize the description in Cohen-Tannoudji's "Atom-Photon Interaction", p.284 from dipole only case onto forbidden transitions getting analogues of E.11 and E.13. Attempt: Took ...
3
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1answer
94 views

Finite temperature QFT: Can vertex correction function/ initial correlations induce short-time tunneling?

The collision term in the Kadanoff-Baym equation has the structure $I(\tau_1,\tau_2) = \int_C d \tau'\Sigma(\tau_1,\tau') G(\tau',\tau_2)$ where the contour $C$ is along the time-Forward $C_+$, time-...
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0answers
13 views

$- \mu / kT $ equal to $1/N$ in Bose Einstein Condesation

In what conditions $ -\mu/ kT = 1/N $ so that we can write $ e^{\mu n /kT} = e^{-n/N} $ in Bose Einstein condensation
2
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2answers
151 views

Partition function for an interacting bosonic mode

Let us assume a single bosonic mode, in equilibrium with a reservoir. For a non-interacting Bose gas, the partition function becomes $\mathcal{Z_\text{nonint}}=\sum_{N=0}^\infty e^{-\beta(\epsilon-\...
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1answer
205 views

How to find density of states in harmonic oscillator?

Density of state should be number of states per volume .Why weThe take derivative of "number of states " with respect to energy to get density of states ?
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0answers
18 views

Volume in first octant

Here we are finding number of state less than energy E for particle trapped in 3D harmonic potential .For large value of E as compare to hw , we assumed energy levels as continuous.And we introduces ...
0
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1answer
38 views

Is mean and average occupation number same?

In Bose distribution we have formula to find number of particles in quantum state .Is mean occupation number and average occupation number same ?
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0answers
103 views

Chemical potential in Bose-Einstein condensation

For a Bose gas, we know that when temperature goes to zero, chemical potential also reach to zero. At $T=0$ all bosons fall into ground state and thus chemical potential is also zero at $T=0$. Also ...
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1answer
91 views

Bose Einstein Condensation in Grand canonical ensemble

Why we develop formalism of Bose Einstein Condensation in framework of grand canonical ensemble ?
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3answers
1k views

Do distinguishable fermions obey the Pauli exclusion principle?

We know that fermions are identical particles and obey Pauli exclusion principle. But what is meant by distinguishable fermions? Does that mean, like proton and electron both are fermions but they are ...
3
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1answer
95 views

Derivation of relativistic pressure

As you can find in many cosmology textbooks, the relativistic pressure in quantum statistical mechanics can be witten as below: $$p=g \int \frac{d^3P}{(2 \pi \hbar)^3} \frac{c^2 |\mathbf{P}|^2}{3E(\...
1
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1answer
119 views

Thermal state of light with two modes

The thermal state corresponding to a photon mode $a$ can be written as $\rho_{a}(\beta) = \frac{e^{-\beta a^\dagger a}}{Tr(e^{-\beta a^\dagger a})}$. Since it is easy (I suppose) to talk in terms of ...
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0answers
39 views

Landau levels degeneration measurement units

I am still studying statistical mechanics. In the discussion on the Landau Diamagnetism, on various textbooks (I use the Pathria), notes taken during lessons and on-line resources, I found several ...
2
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1answer
120 views

Relationship between thermal (de Broglie) wavelength and Planck's constant

In statistical mechanics, the term $h^{3N}$ is introduced as a measurement unit for the phase space of a system of $N$ particles in $3$ dimensions, and is usually multiplied by the Boltzmann factor $\...
0
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1answer
35 views

Planck law near absolute zero

Is the Planck law of radiation valid even for $T$ near absolute zero? Why can we be sure that the mean photon number inside a black body is zero for $T\to 0$?
2
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1answer
326 views

What is the probability that a single-particle bosonic quantum state is occupied?

Unlike the Fermi-Dirac distribution function, the Bose-Einstein distribution function $$f(E)=\bar n_r=\frac{1}{e^{\beta(E-\mu)}-1}$$ can be greater than 1, and therefore, doesn't represent a ...
0
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1answer
89 views

Can we have Bose condensation for bosons satisfying a dispersion relation $E=Cp^s$ $\forall$ s?

Suppose a dispersion relation $E=Cp^s$ where $C$ is a constant is known for a collection of massive non-interacting bosons. What is the way to find out whether there will be Bose-Einstein condensation ...
8
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1answer
238 views

Non-Abelian anyons in the path integral formalism

Background Homotopy classes in the path integral Following the answer to this question about the role of homotopy classes in path integrals, it seems reasonable to me that, when calculating the ...
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0answers
94 views

Obtaining the number of quanta for a system of harmonic oscillators

So I need to find the entropy of a system made up of two harmonic oscillators having natural frequency $\omega_0$ and $2\omega_0$. The system is said to have a total energy of $E=(n+\frac12)\hbar\...
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2answers
165 views

Can the same density matrix represent two (or more) different ensembles?

Given an ensemble i.e, a collection of states and the respective probabilities $\{p_i,|i\rangle\}$, one can uniquely construct the density matrix using $\rho=\sum_ip_i|i\rangle\langle i|$. Is the ...
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4answers
430 views

Set of states $\{|\phi_n\rangle\}$ in the density operator $\rho=\sum\limits_n p_n|\phi_n\rangle\langle\phi_n|$

The set of quantum states $\{|\phi_n\rangle\}$ in the definition of the density operator $$\rho=\sum\limits_n p_n|\phi_n\rangle\langle\phi_n|$$ need not be orthonormal, and need not form a basis. But ...
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0answers
38 views

Matrix formulation of maximum entropy

In E.T Jaynes' book "Probability theory: the logic of science" the maximum entropy principle is discussed as the way to choose out of all possible hypotheses agreeing with constraints, the ones that ...
2
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1answer
30 views

What is the immiscibility condition in a Bose-Fermi mixture?

Also how can you approach that condition from a mean field approach? My intuition is you have to derive the free energy from the mean field energy and then do something with it, however, I'm not ...
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1answer
159 views

Apparent discrepancy between partition function from density matrix and partition function from counting microstates in finite level system

Consider a quantum two-level system indexed by states $|l\rangle = |0\rangle,|1\rangle$ and energies $\epsilon_l$, where $\epsilon_0 = 0$,$\epsilon_1 = \epsilon$. I throw in 2 bosons into the system ...
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0answers
52 views

spontaneous symmetry breaking within critical phases

There are many examples of the spontaneous symmetry breaking in discrete symmetries which result in the gapped phases, such as dimerization phase of the quantum antiferromagnetic spin-1/2 chain which ...
2
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1answer
128 views

Why is a collection of non-interacting bosons pathological?

In this lecture titled "Disorder and Interactions: From Spin Chains to Cold Atoms" the speaker Thierry Giamarchi claims that a collection of non-interacting bosons is totally pathological. His argues ...
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1answer
71 views

System of $N$ bosons

What precaution needs to be observed in writing down an expression for the total number of bosons $N$ valid at low temperatures?
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1answer
315 views

State equation in grand canonical ensemble

My teacher told us that $$ \ln Z = \frac{PV}{kT} $$ is the equation of state for an ideal gas, being $Z$ the grand canonical partition function and $k$ the Boltzmann constant. Where does this formula ...
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2answers
112 views

Difference between Fermions and electrons [closed]

I'm confused between fermions and electrons. For example say I have a system comprises of three electrons and there are three single particle energy level accessible to each of three electrons. What ...
4
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1answer
228 views

When is the Berry phase only dependent on path topology?

Background Suppose we have a Hamiltonian $H(\mathbf{R})$ which depends on some parameters $\mathbf{R}$. For each value of $\mathbf{R}$, the Hamiltonian will have some set of eigenvectors $\{ | \phi_{...
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1answer
83 views

Fluctuations of free energy in quantum statistical mechanics

I want to calculate the fluctuation of the mean value of the free energy, $\langle F \rangle$, which I denote as $(\Delta F)^2 = \langle F^2\rangle - \langle F\rangle^2$. Since I have calculated the ...
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3answers
704 views

Is the notion of Lebesgue Measure a necessary construct for statistical physics? [duplicate]

In chat last night a user and I were discussing the "physical" meaningfulness of the notion of lebesgue measure. In particular, we were curious as to whether physicists can "make do" without it. I ...
0
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1answer
194 views

Probability of microstate in einsteinsolid

This question from my course got me confused. "Consider a system of two Einstein solids, A and B, each containing 10 oscillators, sharing a total of 20 units of energy. Assume that the solids are ...
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2answers
219 views

Question about the existence of non-Abelian anyon

My question from reading this paper: Michael G. G. Laidlaw and Cécile Morette DeWitt, Feynman Functional Integrals for Systems of Indistinguishable Particles. Phys. Rev. D 3, 1375 (971). ...
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2answers
158 views

What is special about the indistinguishability of Boson and Fermions?

In the treatment of Bosonic or Fermionic systems that I'm familiar with, you start with a state containing at least two particles: $$ \left| a_{i}, a_{j} \right\rangle $$ And define a permutation ...