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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Partition Function quantum identical particles
In Pathria and Beale Statistical Mechanics section 5.5, the book tries to compute the Partition function of a system of noninteracting, indistinguishable particles confined to a cubical box of volume $...
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Is there a relation between the density matrix and the density of the position probability? Are they the same concept?
We have a system of two bosons particles and we are interested in calculating the one-particle density and two-particle-density when both are in different states.
So, to do that, I consider the ...
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What if we charge an uncharged sphere by making contact with sphere bearing 3 electrons charge
Suppose say we have 2 identical conducting spheres. Then we have removed 3 electrons from one of the sphere. Now how will charge distribute on each sphere after charging the uncharged sphere by ...
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How does particle indistinguishability reconcile with the quantum Zeno effect?
In non-relativistic quantum mechanics, two particles are said to be indistinguishable if the wavefunction for the system consisting of just those two particles satisfies $|\psi(x_{1}, x_{2}, t)|^{2} =...
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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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Identical particles and the tensor product
To describe a quantum mechanical system, that consist out of a fixed number of particles we take advantage of the tensor product:
$\vert ab \rangle = \vert a \rangle \otimes \vert b \rangle \in H_1 \...
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Identical particles antysimmetric eigenfunctions with antisymmetric sum
In studying systems of 3 or more identical, non-interacting particles, fermions or bosons, I have read that to create eigenfunctions of the Hamiltonian that are antisymmetric or symmetric one creates ...
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Meaning of the exchange of identical particles
I don't quite understand how the exchange of two indistinguishable particles works. In the image I imagine I have two fermions of spin 1/2 in a kind of Hydrogen atom with two discrete energy levels, $...
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Single-particle wavefunction in Slater determinant
The ground state of $N$ non-interacting fermions can be written using a Slater determinant as:
$$ \Phi_{GS}(\textbf{r}_{1}, ..., \textbf{r}_{N}) = \frac{1}{\sqrt{N!}}
\begin{vmatrix}
\phi_{\mu_{1}}(\...
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Probability more than 1 when integrating the Electron density in Density functional theory
The electron density used in density functional theory for a system of $N$ electrons with wavefunction $\psi$ is defined as
$$\rho(r)=N\int \Psi^*(r,r_2,\dots r_N)\Psi(r,r_2,\dots r_N) d^3r_2\dots d^...
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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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Electron density question
On wikipedia it says that:
$$\rho(\mathbf{r})= \sum_{{s}_{1}} \cdots \sum_{{s}_{N}} \int \ \mathrm{d}\mathbf{r}_1 \ \cdots \int\ \mathrm{d}\mathbf{r}_N \ \left( \sum_{i=1}^N \delta(\mathbf{r} - \...
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Point out the wrong sentence about many identical particles systems
A friend asked me today about a question that appeared in the Spanish national examination for being a radiophysicist (in hospitals). He thinks this question is wrong.
It says the following:
In ...
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Are H$^1$ (1s) and H$^1$ (2s) identical particles?
As per Wikipedia,
Species of identical particles include, but are not limited to, elementary particles (such as electrons), composite subatomic particles (such as atomic nuclei), as well as atoms and ...
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Particle density vs. Probability Density in Quantum Mechanics
I am currently reading trough "Bose-Einstein Condensation and Superfluidity" by Pitaevksii and Stringari and noticed some inconsistencies in my reasoning.
In Chapter 5 (Non-uniform Bose ...
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Peskin and Schroeder, Feynman diagram for the Yukawa potential
On page 121 of Peskin and Schroeder, second paragraph, about fermion+fermion $\rightarrow$ fermion+fermion scattering, the book says:
"if the two interacting particles are distinguishable, only ...
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Second quantization in QFT
My first introduction to second quantization was in the context of condensed matter physics. The idea is if we have a system of $N$ indistinguishable particles then the $N$ fold tensor product of 1 - ...
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Why do we get extra symmetric terms/factors when applying annihilation operators to multiparticle states of definite momentum?
Sorry for the wordy title, but I wasn't sure how else to express what I want to ask succinctly.
My question is best illustrated with an example. Suppose we are calculating a scattering amplitude in ...
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Fermionic (or Bosonic) state vs Entangled state
One can see that the wavefunction for a system of two electrons (not very far apart) is one that cannot be written as a tensor product of individual states.
The same is true for a bosonic state.
For ...
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Entropy in a system of indistinguishable particles
(This question is very similar to this one which is left unanswered, and I didn't get much of the comments below that question.)
In a distinguishable system of $n$ particles in a grid of $C$ cells, ...
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Identical electrons after measurement
Let's take a system made of two electrons. Since they are identical, the spin state must be either symmetric or antisymmetric under exchange.
Let's suppose we previously have the following spin part :
...
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$\omega$ vs $\rho ^0$ meson decay into $e^+e^-$
The decay $\omega \rightarrow e^+e^-$ has a partial width that is around 10 times smaller than the partial width for the decay $\rho \rightarrow e^+e^-$. Why is this the case?
The only difference I ...
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Measure of Two Identical Particles
Suppose I have two bosons with symmetric wave function (I guess there should be tensor products?):
$$\psi(x_1,x_2)=\psi_a(x_1)\psi_b(x_2)+\psi_b(x_1)\psi_a(x_2).$$
Suppose now that I perform a ...
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Two-particle operators in QFT and the factor 1/2
I am learning about QFT through the book Quantum Field Theory for the Gifted Amateur and I am having trouble understanding the factor 1/2 in the definition of two particle field operators. In the book ...
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Confusion regarding Gibbs' paradox
I am trying to understand the following commentary I found in Wikipedia about this paradox:
Now a door in the container wall is opened to allow the gas particles to mix between the containers. No ...
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Can two particles have exactly the same statevectors?
I have a query:
Let us have 2 particles and 2 corresponding wavefunctions, under two incompatible Hamiltonians ( $H_1,H_2$).
$$\Psi_1(x_1,t_1)= e^{-x_1} e^{i\sin(\pi t_1/3)}+e^{-x_1^2}e^{i\cos(\pi ...
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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 ...
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Meaning of indistinguishability
As a mathematical tool, when dealing with something like a many-particle wave function, I understand well what indistinguishability means. It is encoded in the antisymmetry of the wavefunctions and is ...
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Linear combination of Slater Determinants
I've been reading about many particle states in quantum mechanics and came across the fact that combinations of fermions can be represented using a Slater determinant. This is quite easy to understand ...
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