Questions tagged [identical-particles]

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

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Creating multiparticle states from vacuum in QFT

In creating multiparticle states from the vacuum we apply the creation operators, \begin{align} &|p_1,s_1\rangle=a^{\dagger}_{p_1,s_1}|0\rangle,\\ &|p_1,s_1;p_2,s_2\rangle=a^{\dagger}_{p_1,s_1}...
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Factor of 1/2 in the cross section of Møller scattering

I have seen everywhere (like Wikipedia) that in Møller scattering the cross section is calculated with the formula: $$\frac{\mathrm d \sigma}{\mathrm d \Omega} = \frac{1}{64 \pi^2 E_{CM}^2 } \frac{|\...
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Momentum and photon indistinguishability

Suppose two photons have the same frequency and polarization, and their wavepackets also have identical temporal and spatial (along their propagation directions, respectively) profiles. The two ...
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Degeneracy in $N$-particle Quantum System [closed]

I was recently introduced to the concept of $N$ particle systems in Quantum Mechanics, and the concept of indistinguishable and distinguishable particles. While reading the following material online, ...
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Normalization of Slater Determinant and Antisymmetry

Consider a system consisting of two electrons with Slater-Determinant $\vert\chi_1\chi_2\rangle$, where $\chi_1$ and $\chi_2$ are one-electron orbitals (spin-orbitals). The Slater Determinant is ...
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When is it necessary to use more than one Slater determinant to write the wave function of a quantum system?

Consider a system of $N$ non-interacting identical fermions of spin $s$, spin quantum number $m_i=m_{s_i}$and position coordinates $\mathbf{r}_i$. Let $\alpha$ denote a state specified by the numbers $...
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Quantum Mechanics - Prove that the permutation operators do not commute in general

Previously, in my Quantum Mechanics exam, We have asked to "Prove that the permutation operators do not commute in general." And I have answered it as follows But my teacher marked it as ...
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Identical particles placed on the energy levels of a 1D harmonic oscillator

If the particles are 6 spinless bosons, would they tend to occupy the ground state together and make the lowest total energy of the system $E=6E_0=3 \hbar \omega$? And If the particles are 6 spin 1/2 ...
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Adding quantum numbers breaks exclusion principle?

Consider 2 non-interacting Fermions with spin $s\ne 0$ trapped in a 1D harmonic potential with ground state $\left|\phi_0\right\rangle$. Due the Pauli exclusion principle I would expect that only 1 ...
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Indistinguishability in statistical mechanics

I have two questions about using the concept of indistinguishability to determine the partition function in statistical mechanics, like for instance when determining the partition function of an ideal ...
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Identical particles

I am reading Quantum Mechanics from Griffiths (for the first time); in chapter 5 (2nd ed.) page-179 it is written: Now I really is not able to understand what Griffiths exactly wants to say by the ...
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Finding the number density of a Fermi gas

When calculating the number density of a Fermi-Dirac or Bose-Einstein distributed particle: $$n=\frac{g}{2\pi^3} \int f(\vec{p})d^3p$$ Apparently one can change variables and get $$n=\frac{g}{2\pi^3} \...
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What exactly does an exchange of particle labels for identical particle wave functions mean *physically*?

We know that the wavefunction of identical particles behaves as follows: $$\Psi(1,2)=\begin{cases}-\Psi(2,1) & \text{for fermions} \\ +\Psi(2,1) & \text{for bosons} \end{cases}$$ Now, what ...
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$N$ bosons in two energy levels

What is the partition function for $N$ bosons in a two state system with $E_1=0$ and $E_2= E$? I know that bosons don't obey the Pauli exclusion principle and they are indistinguishable but I am ...
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In thermodynamic limit, how is $\frac{V}{(2 \pi)^{3}} \int \mathrm{d}^{3} k= \int \frac{\mathrm{d}^{3} r \mathrm{~d}^{3} \boldsymbol{P}}{h^{3}}$?

In the book Intro. Statistical Physics by K.Huang, on page 106, it is given that Because of indistinguishability; the $N$ -body wave function is labelled by the set $\left\{\alpha_{1}, \cdots, \...
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Ground state energy for fermions with different spin orientation [closed]

Electrons are subject to a harmonic potential in one dimension, described by one-particle Hamiltonian $$H = \frac{P^2}{2m}+\frac{1}{2}m\omega X^2,$$ where $P$ are the momentum operator, $X$, is the ...
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Partition function

I am trying to calculate the partition function of a system that has 2 bosons (spin 0) in thermal equilibrium with a reservoir. The particles are independent and each one admits as possible orbitals ...
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Question about the Partition Function For 2 Fermions In a 2-Level System

So say we have $2$ indistinguishable electrons that can occupy $2$ energy states $E=0$ and $E=\Delta$. If we only had one fermion then are both states doubly degenerate since an electron can be spin ...
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Fermions and Pauli Exclusion Principle

I started to study quantum ideal gas and I am reading Salinas "Introduction to Statistical Physics". In chapter 8 , he states that when speaking of fermions, only one particle can occupy an ...
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Why is a 2-electron wavefunction antisymmetric? [closed]

Why does a 2-electron system have an antisymmetric wavefunction when the combination should be bosonic? I.e. If it's an overall bosonic combination, shouldn't the overall wavefunction be symmetric?
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Don't the four quantum numbers make two electrons distinguishable?

From the Pauli's Exclusion Principle no two electrons in a bound system have all same quantum numbers. This means that an electron can be uniquely specified by the four quantum numbers and hence can ...
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What happens when two 0-spin identical bosons interact?

Suppose we have a system of two identical 0-spin bosons, with a certain symmetric, time independent hamiltonian, which involves some general angular momentum for each particle (I am not currently ...
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Identical Particles in Quantum Mechanics

I'm reading Chapter 5 of Griffiths' Quantum Mechanics book about "Identical Particles". He says that: The state of a two-particle system is a function of the coordinates of particle one ($...
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General counting of microstates for a specific energy. Indistinguishable vs distinguishable

I have the following problem: I have two systems $A$ and $B$, each one with 5 distinguishable particles, $N_A=N_B=5$. The systems have infinite energy levels with energy $E_m = m \cdot \epsilon$ with ...
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Exchange symmetry between mixed species

I am reading Weinberg Vol.1 chapter 4, in section 4.1 page 171 last paragraph he says that for interchanges of particles to different species can be taken to be symmetric for any two bosons or one ...
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Why must systems of identical particles be either totally antisymmetric or totally symmetric? Why can there not exist a mixture?

I am reading chapter 6 of Sakurai's Modern Quantum Mechanics and have come across the 'symmetrization postulate', which tells me that for any given system of identical particles, all states must ...
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Identicalness and Indistinguishability in quantum mechanics

I've been reading chapter 10.3 'Identical Particles' in Shankar's book on quantum mechanics and also looked through some of other books on this subject and one rather subtle objection started ...
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Problem regarding Eigenfunctions of Identical Particles in Quantum Physics

Suppose I have 2 electrons under an harmonic potential: $$H=\frac{1}{2m}(p_1^2+p_2^2)+\frac{1}{2}m\omega ^2 (x_1^2+x_2^2)$$ Now let's think about the eigenvalue and the eigenfunctions of energy of the ...
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Slater determinant as (zero order) approximation to Schrodinger equation

I understand how the Slater determinant for N particle works and its application in writing an antisymmetric wavefunction. However the question asks me to show that the Slater determinant is a zero ...
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Motivation behind the principle that all electrons are not distinguishable

EDIT: It is usually claimed without providing much motivation that elementary particles of the same kind, e.g. electrons, are not distinguishable in principle: there is no way to distinguish between ...
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Properly accounting for indistinguishability of (homonuclear) diatomic molecules in “internal” partition function

I have always liked Schroeder's take on the partition function being a product of translational and internal degrees of freedom: $$ Z_1 = Z_{\text {trans}} Z_{\text{int}} $$ where $Z_{\text{int}}$ can ...
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Does this wave function violate any rules of Quantum mechanics?

I am studying bosons and fermions in quantum mechanics. We learned that for a two-particle system with identical bosons, the wave function can have the form (assume symmetric spin) $$ \frac{1}{\sqrt2} ...
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Contradictory statements on product states for distinguishable particles in Quantum Mechanics

Page no. $5$ in Many-Body Theory Exposed! by Willem H Dickhoff & Dimitri Van Neck states the following: The complex vector space, relevant for N particles, can be constructed as the direct ...
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Distinguishability and tagging of particles

I was reading Sakurai's book and here is an extract: In classical physics it is possible to keep track of individual particles even though they may look alike. When we have particle 1 and particle 2 ...
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Indistinguishability in Spin-1/2-system

In terms of statistical physics I thought the microcanonical partition function can be interpreted as summing over all possible quantum numbers. Neglecting indistinguishability in the case of two ...
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Different types of magnetic exchange

I am a bit confused about the mechanisms of different types of exchange. Mainly, I am trying to understand mechanisms that lead to magnetic ordering and I came across several different types of ...
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Does it make sense to ask how many of the molecules you are inhaling Caesar exhaled in his last breath?

There is a nice example of a Fermi estimation question where you ask: How many of the molecules which Julius Caesar (or Jesus, Muhammad, etc. - anyone who died a long time ago) exhaled in their dying ...
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Interfering alternatives and identical particles in Feynman and Hibbs

I am currently self-studying Feynman and Hibbs, and in his first chapter, Feynman talked about 'alternatives' like the various possibilities or paths an experiment can take. He defined two different ...
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Can identical particles be “distinguished” by their states?

Let's say we have an atom with three electrons in the lowest energy states. So two electrons are at energy level 1 and one at energy level 2. Now we measure the positions of those electrons, and found ...
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Valid state of electron systems

I would like to show that the state $|\psi \rangle = |\varphi_1,\varphi_2,\varphi_3 \rangle$ is not a valid state for 3 electrons (without spin) for $|\varphi_1\rangle,|\varphi_2\rangle,|\varphi_3 \...
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Scattering of identical particles

I am studying scattering of identical particles using the book of Joachain and I am having some difficulties. Consider the scattering of an electron by an atom with $N$ bound electrons. The asymptotic ...
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Measuring a superposition state of identical particles

Suppose that the state of a system of two identcial particles (say, two photons) is given by the following: $$\frac{1}{\sqrt{}2}(|\psi_{1}\rangle|\psi_{2}\rangle + |\psi_{2}\rangle|\psi_{1}\rangle) \, ...
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Confused about partition function of identical particles

The total partition function of N identical, independent particles is $Z^N/N!$ where Z is the partition function of a single particle. To find out the correct magnetization due to N identical atoms of ...
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Making indistinguishable particles distinguishable 2.0

An extended discussion of the selected best answer to the following SE question. In part 2 of the answer concerning 'far-apart' particles, I don't see how the equality can follow in the first integral....
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Permutation of Identical Fermions - Spatial and Spin Decomposition

As far as I know, fermions are the particles which exhibit antisymmetric states: $\hat{P}\left|n_1\right>\otimes\left|n_2\right> = -\left|n_2\right>\otimes\left|n_1\right>$. Often times, ...
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Spin wave function of two identical spin-1/2 particles in a 1D potential box

I came across this problem in N. Zettili's Quantum Mechanics book (Chapter 9, Problem 16): Two identical particles of spin 1/2 are enclosed in a one-dimensional box potential of length L with walls ...
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Why is every electron in the universe not entangled with every other electron?

According to the principles of identical particles, the wavefunction of a collection of fermions must be antisymmetric and such a state is entangled. Doesn't this mean that any given electron in the ...
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Identical Particles in Quantum Field Theory

In Quantum Field theory by M. Schwarz, the author in the introduction of chapter 12 on Spin Statistics theorem says, while describing identical Particles: Let $$|s_1p_1n_1,...,s_3p_3n_3\rangle \tag{1}$...
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The Indistinguishability Postulate and the Copenhagen Interpretation

The indistinguishability postulate states that any (normalized) state vector $|\psi\rangle$ for a system of $N$ identical particles should satisfy $\langle \psi | \hat{O} | \psi \rangle = \langle \hat{...
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Why are all quarks and leptons of this universe the same?

We know the composition of stars by spectroscopic analysis. The EM waves generated by them are blue- or redshifted. We could have said, "Look, the wavelength is slightly different so it may be ...

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