# Questions tagged [identical-particles]

Questions related to the discernibility of many-body systems, its philosophical implications and its mathematical description.

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### Are photons with different frequency distinguishable?

When i learn statistical mechanic, the teacher told me that photons with different frequency are distinguishable, i confused. And the teacher say also photons with different polarization, direction ...
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### Identical particles in Bohmian quantum mechanics

Particles can be distinguished by their trajectories in Bohmian quantum mechanics and there is no natural reason for imposing symmetrization (or anti-symmetrization) of the wave function of the ...
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### Mixing identical gases in different states (Gibbs paradox)

Background It is well known that if we take a container separated into two volumes $V_1$ and $V_2$ by an insulating membrane and containing some moles $n_1$ and $n_2$ of the same gas at the same ...
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### Question regarding the degeneracy of energy states of Fermions

My professor during the lecture said exactly the following Let there be a system of non-interacting fermions. Since they are indistinguishable, they have the same Hamiltonian, and the single-particle ...
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### Partition function for the indistinguishable particles using symmetrization of states

In the derivation of the partition function for the the N particle ideal gas, the factor of $\frac{1}{N!}$ does not come naturally. We have to go for symmetrized and asymmetrized state. So, to derive ...
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### Conservation of symmetrization in quantum mechanics

I recently read about the symmetrization requirement, which my book states is axiomatic of quantum mechanics: $$\psi(\mathbf r_1, \mathbf r_2) = \pm \psi(\mathbf r_2, \mathbf r_1). \tag{*}$$ It ...
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### Derivation of partition function for $N$ identical quantum harmonic oscillators

What is the partition function $$\mathcal Z^{(N)}_\beta(H) : =\mathrm{Tr}\exp(-\beta H) \tag{Z}$$ $\left(\beta >0\right)$ for a system of $N$ indistinguishable and non-interacting bosons (e.g. ...
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### 2 fermions in a box (infinite potential well)

I have 2 fermions in a box. I know that they are in the state: $$|\psi\rangle = {1 \over \sqrt2}\, (|1\rangle |2\rangle -|2\rangle|1\rangle)\,|+,+\rangle$$ If I hadn't spin, I could find wave ...
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### Group theory proof of necessity of totally symmetric states

R. Shankar's Principles of Quantum Mechanics states in page 273 that, if we're working with three particles, antisymmetric states pick up a negative sign under all possible exchanges, and symmetric ...
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### Why should the wavefunction of two isolated bosons be symmetric?

The question is in the title, Let's say I have one boson (Mike) somewhere on earth and another one (Fatima) somewhere in proxima centauri. They are identical. My friend argues that whatever the ...
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### Group theory of tensor products of the harmonic oscillator

I learned that the symmetry group of the quantum isotropic harmonic oscillator in n-dimensions is $SU(n)$. Specifically, in two-dimensions, it is $SU(2)$ and hence the eigenstates are given by the ...
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### Kohn-Sham equations, Sakurai 3rd edition, possible typo?

In Sakurai's quantum mechanics book 3rd edition page 448, equation 7.88, the book writes "Kohn and Sham found a way to derive a self-consistent approximation scheme, based on single particle ...
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### Why can't you always write the anti-symmetric state as slater determinant?

For example, consider the following state $$|\Psi\rangle_a = [|r_1 r_2\rangle -|r_2 r_1\rangle ]\otimes \left[|\uparrow \downarrow\rangle +|\downarrow \uparrow \rangle \right]$$ You can't write this ...
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### Marginalisation of a joint probability distribution in bra-ket notation

Given a wave function $\Psi(\vec r_1, \vec r_2)$, where $\vec r_1$ and $\vec r_2$ are the positions of particle 1 and 2, respectively, the probability of finding particle 1 at position $\vec r$ (...
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### Exchange symmetry of spatial wavefunction of system of two particles with $\ell=1$

Consider a system of two identical particles. The combined system of these two particles has a total orbital angular momentum quantum number $\ell = 1$. I am aware that the spatial wavefunction of a ...
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### Homotopic Paths and Spin-Statistics

I am currently reading Schwartz' book on QFT, Section 12.2 on Spin and statistics. He shows, that in 3D there are only two inequivalent ways to exchange two indistinguishable particles. More formally, ...
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### Homework help: Energy levels of two distinguishable electrons

I suspect that the problem applies to spin 1-2 particles in general. But I am not sure how to find energy levels? How do you find the eigenvalues for Hamiltonians in matrix form? (I am a freshman ...
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### Why are photons being identical particles?

Recently, I study quantum optics and deal with quantization of EM field in a cavity. We know we can express/quantize vector potential in terms of $\hat{a},\hat{a}^{\dagger}$ to get a quantized EM ...
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### Why do the electron and position creation operators anti-commute?

I am learning QFT and is baffled by a minor problem. The electron and the positron should be distinguishable, as they have different charges. So why do their creation operators anti-commute? They ...
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### Argument for Fermions and Bosons as eigenfunctions of particle exchange operator $P$

Some time earlier, my prof took me through an argument leading to emergence of fermions and bosons by the application of particle exchange operator on the multiparticle wavefunction. I will try to ...
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### Exact eigenfunctions of two interacting identical particles [closed]

While I was reading about quantum states of $N$ interacting identical particles, I realized that I don't understand some fundamental things. So In order to clear my confusion, I decided to consider a ...
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### What is the difference between configuration and microstates in the distribution of fermions

Imagine a system of $3$ electrons distributed into $3$ energy levels ($E_1,E_2,E_3)$. I want to know the difference between the total number of configurations of the above system vs the total number ...
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### How to count the number of microstates in this example and in general

Suppose we have $3$ energy levels $E_1,E_2,E_3$, that we need to fill up with $3$ electrons. Each of these Energy levels can be filled with two electrons - spin up and spin down. How does one ...
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### Fermions in a micro-canonical ensemble

I've been reading statistical mechanics, and I read the following on Wikipedia, on the article on Fermi-Dirac Statistics derivation in the micro-canonical ensemble : Suppose we have a number of ...