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 atoms unique?

Do atoms have any uniquely identifying characteristic besides their history? For example, if we had detailed information about a specific carbon atom from one of Planck's fingerprints, and could time-...
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Are black holes indistinguishable?

In the standard model of particles it is understood that besides characteristics like momentum, spin, etc., two electrons are indistinguishable. Are two black holes, in the same sense, ...
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Theoretically, could there be different types of protons and electrons?

Me and my friend were arguing. I think there could theoretically be different types of protons, but he says not. He says that if you have a different type of proton, it isn't a proton, it's something ...
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Are all electrons identical?

Why should two sub-atomic (or elementary particle) - say electrons need to have identical static properties - identical mass, identical charge? Why can't they differ between each other by a very ...
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Are protons and neutrons affected by the Pauli Exclusion Principle?

I'm very confused about the Pauli exclusion principle. Wikipedia states it as "two identical fermions cannot occupy the same quantum state in a quantum system". I understand this for electrons that ...
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Why are particles in Quantum Mechanics indistinguishable?

I'm currently reading about tensor products in Quantum Mechanics and composite systems and I've read that in Quantum Mechanics particles are indistinguishable while in Classical Mechanics that's not ...
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Why doesn't the entropy increase when two similar gases mix with each other?

Entropy increases when two substances mix with each other. For example, the entropy of mixing of two different gases are given by $$\Delta S= 2Nk\ln\frac{V_f}{V_i}\;.$$ But, the entropy doesn't ...
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Same photon or different photon?

Consider a typical optical focusing system: A small light source, then a collimating lens, then a focussing lens, and then a detector (e.g. CCD). Assume that source intensity is so low that only one ...
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Geometric quantization of identical particles

Background: It is well known that the quantum mechanics of $n$ identical particles living on $\mathbb{R}^3$ can be obtained from the geometric quantization of the cotangent bundle of the manifold $M^...
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Why is the partition function divided by $(h^{3N} N!)$?

When computing partition functions for classical systems with $N$ particles with a given Hamiltonian $H$ I've seen some places writing it as $$Z = \dfrac{1}{h^{3N} N!}\int e^{-\beta H(p,q)}dpdq$$ ...
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How are anyons possible? (another version)

I know that this question has been submitted several times (especially see How are anyons possible?), even as a byproduct of other questions, since I did not find any completely satisfactory answers, ...
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Is the $N$ factorial in the Partition function for $N$ indistinguishable particle an approximation?

I suspect that the $N$ factorial in the partition function for N indistinguishable particles $$ Z = \frac{ Z_0^N } {N!} $$ is an approximation. Please someone correct me if I am wrong and why or why ...
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For ideal classical gasses, in terms of the energy levels why do we ignore whether the particles are fermions or bosons?

I am confused as to why when dealing with ideal classical gasses, the dependency of the particles being either fermions or bosons is ignored. How does this relate to the energy levels within the ...
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Why is the phase picked up during identical particle exchange a topological invariant?

I've been wondering about the standard argument that the only possible identical particles in three dimensions are bosons or fermions. The argument goes like this: Consider exchanging the positions ...
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What are distinguishable and indistinguishable particles in statistical mechanics?

What are distinguishable and indistinguishable particles in statistical mechanics? While learning different distributions in statistical mechanics I came across this doubt; Maxwell-Boltzmann ...
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1answer
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Why must fermions be antisymmetric? [closed]

I have read that fermions cannot exist in the same state simultaneously. I understand why indistinguishable particles with an antisymmetric superposition of states can't exist in the same state ...
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How can we “exchange” particles, since they are “identical”?

Doubt 1: If two particles are identical, you can not distinguish between them. Then, I think, permutation operation is meaningless. Because you can not distinguish them, how can you tell if they are ...
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Quantum entanglement for indistinguishable particles

When I encounter the definition of the mathematical definition of quantum entanglement. System composed by many parts $A$, $B$,.., $N$ can be described by a density matrix operator $\hat{\rho}$ acting ...
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When are two fermions considered identical?

I am a bit confused about the terminology concerning identical fermions. In quantum mechanics, identical fermions need to obey certain anticommutation relations i.e. have an antisymmetric total ...
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Identical particles seem to reduce probability

$\newcommand{\ket}[1]{| #1\rangle}$This question basically has two very related parts. This came up in the context of trying to verify something my professor said a while ago: that if the wave ...
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How axiomatic is the symmetrization requirement (i.e. the Pauli principle)? (in QM)

I've so far always been told, that the symmetrization requirement is an axiom on the level of the Schrödinger equation and the statistical interpretation of the wave function (or it's absolute value). ...
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2answers
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What is the resolution to Gibb's paradox?

This question is essentially a duplicate of Gibbs Paradox - why should the change in entropy be zero?. The question concerns the following situation: I have some gas of identical particles and they ...
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598 views

Making indistinguishable particles distinguishable?

I am wondering if we fix the locations of indistinguishable particles, will the identical particles become distinguishable? Say, put each one of the indistinguishable particles into a small box. ...
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Do bosons and fermions produce the same interference pattern in a double slit experiment?

I have read that when bosons interfere they do so by adding the probability amplitudes, then I read that when fermions interfere they do so by subtracting the probability amplitudes. The usual double ...
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Grand canonical partition functions for Bose-Einstein statistics vs. Maxwell-Boltzmann statistics

In Bose-Einstein statistics, the grand canonical partition function is $$\mathcal{Z}=1+e^{-\beta(\epsilon-\mu)}+e^{-2\beta(\epsilon-\mu)}+e^{-3\beta(\epsilon-\mu)}+\cdots$$ In Maxwell-Boltzmann ...
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Symmetric Under Particle Exchange?

Usually, in undergrad QM, we see states such as: $$|\psi\rangle_{\pm}=\frac{1}{\sqrt 2}(|01\rangle\pm|10\rangle)$$ Where trivially the + state is symmetric, and the $-$ state is antisymmetric. However,...
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Bose Enhancement Factor

How may one explain the fact that the probability of a boson transferring to a state with an occupation number n is 'enhanced' by a factor of (1+n), compared to the classical case? (In the classical ...
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Gibbs Paradox - why should the change in entropy be zero?

The Gibbs paradox deals with the fact that for an ideal gas with $N$ molecules in a volume $V$ seperated by a diaphragm into two subvolumes $V_1,V_2$ with $N_1,N_2$ particles in each subvolume, ...
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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 ...
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4answers
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Distinguishable, indistinguishable paramagnetic ideal gas

In the canonical ensemble, the partition function for an ideal gas is given by: $$\frac{Z}{N!}$$ The factor $N!$ accounts for the indistinguishability of the particles of the ideal gas. What ...
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3answers
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Is there is a reason for Pauli's Exclusion Principle?

As a starting quantum physicist I am very interested in reasons why does Pauli's Exclusion Principle works. I mean standard explanations are not quite satisfying. Of course we can say that is because ...
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What are the differences between indistinguishable and identical?

What is the difference between indistinguishable particles and identical particles?
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How to determine whether an eigenstate of total spin is symmetric or antisymmetric?

Here we have two identical paticles with spin $I$, integer or half-integer, and there are $(2I+1)^2$ states. Each one of them can be uniquely determined by total spin and its orientation, we can use $...
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2answers
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Why aren't Maxwell-Boltzmann statistics used in general cases?

From Probability Theory Vol. 1 Feller Section 2 Chapter 5: Maxwell-Boltzaman distribution: consider $r$ indistinguishable balls and $n$ cells. Assuming that all $n^r$ possible placements are ...
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Why does exchanging coordinates produce a phase of $\pm 1$ in an identical particle wavefunction?

Consider a system of two identical particles described by a wavefunction $\psi(x_1, x_2)$. There are two kinds of exchange operators one can define: Physical exchange $P$, i.e. swap the positions of ...
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Representation of indistinguishability in quantum mechanics

I was wondering that if particles are indistinguishable in quantum mechanics, then why do we still express their states $\left| \uparrow \downarrow \right\rangle$, as meaning particle 1 (in the first ...
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2answers
870 views

Normalising multi-particle wavefunctions

For a quantum mechanical system of $n$ particles the state of the system is given by a wave function $\Psi (q_1, \dots , q_n)$. If the particles are indistinguishable, we demand that the swapping of ...
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Quantum Field Theory is equivalent to QM of identical particles for free fields?

This question focuses on just free fields. The point is, studying Merzbacher's Quantum Mechanics book, in the chapter about identical particles, the author shows how Quantum Fields appear naturally ...
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Is there any “singlet state” for 3 or more spin 1/2 particles?

Every system with $N$ or more electrons lies in a Hilbert space $H=H_{\text{space}} \otimes H_{\text{spin}}$, with $H_{\text{space}}=H_{\text{space}}^{1}\otimes\cdots\otimes H_{\text{space}}^{N}$ and $...
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1answer
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Interchange symmetry for states with identical particles

I was reading this web page about interchange symmetry for states with identical particles here: http://quantummechanics.ucsd.edu/ph130a/130_notes/node317.html The article states that the highest ...
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Classical and Semi-classical treatments of the ideal gas

In the semi-classical treatment of the ideal gas, we write the partition function for the system as $$Z = \frac{Z(1)^N}{N!}$$ where $Z(1)$ is the single particle partition function and $N$ is the ...
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Indistinguishable particles when wavefunctions don't overlap

My question is the following : Imagine that we study two electrons, one has spin up and the other down. If the two wavefunctions overlap, then I have the symetrisation postulate that occurs, the ...
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1answer
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When is separating the total wavefunction into a space part and a spin part possible?

The total wavefunction of an electron $\psi(\vec{r},s)$ can always be written as $$\psi(\vec{r},s)=\phi(\vec{r})\zeta_{s,m_s}$$ where $\phi(\vec{r})$ is the space part and $\zeta_{s,m_s}$ is the spin ...
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Does quantum gases obey ideal gas equation $ PV= nRT$?

At extremely low temperature, does an ideal gas of bosons or fermions obey the ideal gas equation, $PV= nRT$?
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Proving that the free energy is extensive

If I have two system of an Ideal gas $A$ and $B$ each of these system has a partition function: $Z_{A,B} = \left ( \frac{V_{A,B}}{\lambda_T} \right )^{N_{A,B}}$ Where: $\lambda_T = \left ( \frac{m}...
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2answers
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“distinguishability” of 1D identical particles

Recently when I deal with 1D electron system, it occurred to my mind that since these electrons are not able to bypass each other during the scattering processes, we can actually label them as the 1st,...
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1answer
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Why does a quantum gas lose its “quantum nature” in the limit $(\epsilon-\mu)/k_BT\gg1$?

Mathematically, the Fermi-Dirac (FD) distribution and Bose-Einstein (BE) distribution coincides with the Maxwell-Boltzmann (MB) distribution in the limit $(\epsilon-\mu)/k_BT\gg 1$. Therefore, in this ...
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About the factorial N! in the partition function

After reading these posts: Why is the partition function divided by $(h^{3N} N!)$?, What is the resolution to Gibb's paradox?, and these articles: The Gibbs Paradox and the Distinguishability of ...
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Spin of two identical particles

I read that when I have two identical particles with spin 1/2 there are 4 possibilities: |↓↓⟩,|↑↑⟩,|↑↓⟩,|↓↑⟩. Then since there is the symmetrization requirement I can take as eigenvalues the ...
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Electrons, spins, and degeneracy

In an atom, two electrons can have the same set of $n,\ell,m$ quantum numbers as long as they have opposite spins. My introductory physics and chemistry courses have all introduced this as two ...