Questions tagged [identical-particles]
Questions related to the discernibility of many-body systems, its philosophical implications and its mathematical description.
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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 $$|s_1p_1n_1,...,s_3p_3n_3\rangle \tag{1}$...
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The Indistinguishability Postulate and the Copenhagen Interpretation
The indistinguishability postulate states that any (normalized) state vector $|\psi\rangle$ for a system of $N$ identical particles should satisfy $\langle \psi | \hat{O} | \psi \rangle = \langle \hat{...
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Why are all quarks and leptons of this universe the same?
We know the composition of stars by spectroscopic analysis. The EM waves generated by them are blue- or redshifted. We could have said, "Look, the wavelength is slightly different so it may be ...
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Is there a unique way to construct the overall spatial wavefunction for identical particles?
While studying the quantum mechanics of $N$ identical particles, I stumbled upon formulas for generalizing the spatial wavefunction for bosons:
$$\psi(x_1,...,x_N)=\frac{1}{\sqrt{N!\prod_\alpha N_\...
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Bosonic Fock space
Given the single particle Hilbert space $H$ with basis $(|e_\alpha\rangle)_\alpha$, consider the free Fock space $$\mathcal{F}(H)=\bigoplus_{n\ge 0} H^{\otimes n}.$$ According to experimental facts, ...
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'Physical realizability' in the Symmetrization Postulate
I have two questions about the $Symmetrization \ Postulate$:
In a system with $N$ identical particles, physical states aren't arbitrary states in $V^{\otimes n}$. Rather, they're totally symmetric (...
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Do we have to treat ideal gas as identical particle?
In Quantum Mechanics 2nd edition by Griffiths, he says we can get Maxwell-Boltzmann distribution as we deal with distinguishable particles. And I think ideal gas follows MB distribution; therefore, I ...
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Extrapolating partition functions to $N$ atoms
Consider the partition function of an ideal monoatomic gas with electronic and nuclear spin degrees of freedom,
$$z = \sum_{n_x,n_y,n_z} e^{-\beta (n_x^2 + n_y ^2 + n_z ^2)\hbar ^2 /2mL^2} (2S+1)(2I+1)...
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Wave function of a proton and a neutron with and without isospin
Suppose I have a proton and a neutron in an external harmonic oscillator potential well. Let us first neglect all interactions between the two: since they are distinguishable particles I conclude ...
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What is the difference between indistinguishability that leads to Gibbs correction and indistinguishability that leads to quantum Statistics?
In Gibbs paradox, Ideal gas is considered indistinguishable, and the partition function is divided by N factorial, how does the indistinguisabilty differ from quantum statistics?
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The Eigenstates of a Symmetric Operator
Good Afternoon,
By definition, an observable $O$ for a system of N identical particles is symmetric just in case $\langle\psi|O|\psi\rangle = \langle\psi|P^{\dagger}OP|\psi\rangle$ for any permutation ...
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The Indistinguishability Postulate
What is often called the Indistinguishability Postulate is expressed in (at least) two different ways depending on the textbook.
For any normalized composite states of N identical particles $|\psi \...
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Why must the eigenstate of an operator correspond to a particle?
I was studying identical particles in Quantum Mechanics, when I came across the notion of the 'exchange operator' acting on a two-particle wavefunction, $\psi_(x_1, x_2)$, in one dimension: $$ P_{12}\,...
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Entanglement Entropy for indistinguishable Particles in a Fock Basis
Given some total system $ H = H_A \otimes H_B$, we can find the entanglement entropy for a subspace by taking $S = -tr(\rho_A ln \rho_A)$ with $\rho_A = tr_B \rho$, where $\rho$ is the density matrix ...
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Do electrons have no hair, like black holes?
Does John Wheeler's conjecture that black holes have no hair apply to electrons?
Can the electrons have some hair that I can't see?
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Why do we care about indistinguishability?
I am reading material related to statistical physics, and I am having a problem understanding why we care about the notions of distinguishably and indistinguishability. I have found What are ...
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1answer
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Invariant subspaces of multi-particle Hamiltonians
I haven't found anywhere that discusses multi-particle quantum mechanics exactly like this, and I'm not sure the correct thing to Google. If anyone has any pointers to a reference that might help, I'd ...
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What is the relationship between distinguishability and the wave function of particles?
I've started to learn Quantum mechanics and statistical mechanics on my own. There is a line in my book, which says, that
In an ideal gas, to remove the Gibbs paradox, the gas molecules are treated ...
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Anti-symmetric property of Fermionic wave function
I am reading Quantum Statistics from 'Fundamentals of Statistical and Thermal Physics' by Frederick Reif. I have questions in two places.
I understand the following paragraph:
Particles with half-...
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If electrons are identical and indistinguishable, how can we say current is the movement of electrons?
When we talk about current, we say electrons are "flowing" through a conductor. But if electrons are identical particles, how does it make sense to talk about them flowing?
To expand on that:...
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Coupling of two spin half particles
If there are two electrons coupled by interaction having hamiltonian H=A*S1*S2 where S1 and S2 are spin angular momentum operators of two electrons, we know we have four possible eigenstates for the ...
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2answers
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Identical particle state with spin
We construct identical particle state by symmetrizing or antisymmetrizing the tensor product of single partice states. When considering spin, a two fermions state should be $$|\psi\rangle=\frac{1}{\...
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Distinguishable particles/atoms in classical and quantum mechanics
Classical Mechanics It is often said that the particles/atoms in classical mechanics are distinguishable because we can keep track of their trace. First how is this possible? If we have 3 atoms how ...
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Normalizing symmetric wavefunctions
Wikipedia and other sources say that the normalized symmetric ket for $N$ particles with quantum numbers $n_1, n_2, ...,n_N$ is
$$|n_1n_2...n_N;S\rangle=\sqrt{\frac{\Pi_km_k!}{N!}}\sum_P|n_{P(1)}\...
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LSZ Reduction formula and interacting identical particles
I am studying the derivation of LSZ reduction formula on the book "A modern introduction to quantum field theory" by Maggiore. Here it is derived considering a single species of neutral scalar ...
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Why two spin-$1$ bosons could not be in a spin $\frac{1}{\sqrt{2}}(|1,0\rangle-|0,1\rangle)$ state?
Consider two boson of spin $1$ without angular momentum.
I'm seeing an argument that
"because those two particles were bosons, they must be symmetric under the exchange $m_1,m_2$. Thus they could ...
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What is the range of Pauli's exclusion principleļ¼
In many introductions to the pauli's exclusion principle, it only said that two identical fermions cannot be in the same quantum state, but it seems that there is no explanation of the range of those ...
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1answer
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Probability that two identical particles are somewhere
Let's say we have two identical particles, $r_1$ is the position of the first particle and $r_2$ is the position of the second particle. The wave function is $\psi(r_1,r_2)$. Since these particles are ...
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1answer
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Two-body problem with identical particles
In the two body problem we have two particles interacting via a potential $V(| r_1 - r_2 |)$:
$$ H = \frac{p_1^2}{2m_1} + \frac{p_2^2}{2m_2} + V(|r_1 - r_2|) \, .$$
It is well known that, with a ...
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3answers
259 views
Measuring the position of identical particles and wavefunction collapse
I'm working through Shankar's Principles of Quantum Mechanics, and I think I have hit a confusion over identical particles.
The book refers to 'measuring the position' of two bosons to be $x_1$ and $...
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3answers
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Are subatomic particles, such as electrons, truly indistinguishable?
In thermodynamics and statistical mechanics, we learn that many subatomic particles, such as electrons, are indistinguishable. But are they really? I can understand from one perspective that if we ...
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Number Operator on the Product State of Identical Bosons
Suppose that we have a single photon (or any elementary boson) with the state $$\Phi_{1} = |n\rangle.$$
Suppose also that there is a two-particle system whose state is given by $$\Phi_{2} = |n\rangle_{...
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Indistinguishability in the Ising Model
I am recently studying Statistical mechanics (In equilibrium). And get confused about that $\frac{1}{N!}$ factor when calculating the phase-space-volume.
When calculating the entropy for an ideal gas ...
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How do we know that the two indistinguishable particles in the same infinite well have different energies?
I'm reading an example in which we have two identical particles in the same infinite well. They have different quantum numbers "n", which means that they have different energies. This example is used ...
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1answer
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Tell one atom from another
It is said you can't tell one atom from another like you can't tell one electron from another, like Are atoms unique?, but:
(1) Most physics about atom system is based on BornāOppenheimer ...
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2answers
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Why can two particles in a box have different quantum numbers and still be indistinguishable? [duplicate]
I'm reading from the "Modern Physics" textbook by Randy Harris.
I'm Chapter 8 on Spin and Atomic Physics, and the book has just introduced how to solve for the wave function of 2 particles in a box. ...
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Classical, identical particles which are distinguishable
Aren't classical, identical particles, always indistinguishable?
Consider monitoring the trajectory of one particle. After it collides with an identical particle how would one continue to keep track ...
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Should entropy be always extensive?
My question came up from the discussion in class which is about whether we should treat a specific system distinguishable or indistinguishable. To figure out what will happen in these two ...
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When two identical fermions exchange, the wavefunction changes sign. Then why the statement is no new state is created?
When two identical fermions exchange, the wavefunction changes sign. Then why the statement is no new state is created now that the wavefunction is changed?
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When are particles distinguishable?
I'm just revisting some basics from statisitical mechanics for an exam. One of the exercises asks the reader to calculate the canoncial partition functions of $N$ harmonic oscillators. How should I ...
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1answer
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Are electrons and holes distinguishable particles?
In condensed matter physics: If we describe e.g. an exciton as a combination of an electron and a hole, do we need to combine them in a Slater determinant or into a simple product state? What happens ...
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Partition function of two spin 1/2 particles - Distinguishable or indistinguishable?
Suppose I have some fermions with spin 1/2 on a harmonic potential. Then the energy of each particle is given by:
$$
E_i=\hbar\omega(n_{x_i}+n_{y_i}+n_{z_i}+3/2)
$$
By definition the partition ...
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On the indistinguishable photons
Reading articles related to quantum optics, I have encountered the term "indistinguishable two photons" so many times.
I could roughly understand what that means: two photons are very similar w.r.t. ...
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Can two fermions occupy the same energy level on a harmonic potential? [closed]
Suppose that we have a harmonic potential $\hat{V}(\hat{X})=\frac{1}{2}k\hat{X}^2$ which we will, for simplicity, consider to be one dimensional. Now let's place two fermions within this potential, ...
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Does permutation symmetry follow the Wigner's theorem?
The permutation symmetry for boson/ferminon is
$$
P_{12} | 12 \rangle = \pm |21\rangle
$$
I may think it as a linear unitary transformation, since
$$
P_{12}^{\dagger} P_{12} =P_{12} P_{12} =1
$$
...
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1answer
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What does Baym mean here in his Lecture on Identical Particles?
I'm reading Lectures on Quantum Mechanics by Gordon Baym (1969). In his discussion of 3-identical fermions Baym writes: "One way to make $\Psi(1,2,3)$ [the total wave-function] antisymmetric is to ...
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If the wave function of two identical fermions is antisymmetric, how can they be identical? [duplicate]
If the wave function of a system of two identical fermions is antisymmetric, how can they be identical?
I replace two 'identical' particles and get a different system. This must mean they are not ...
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2answers
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How does the repulsion due to equal spin fermions show up mathematically?
I expect that in many-body problems of electrons, spin should cause same-spin-electrons to repel more strongly than opposite spin electrons because the Pauli exclusion principle is the observation ...
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The antisymmetrisation of two identical single particle wave functions is identically zero, why is this important?
Let $f_1,f_2$ be two $\mathbb{R}^3 \to \mathbb{C}$-functions and
$$\mathrm{asym}(f_1,f_2)(x_1,x_2) = f_1(x_1)f_2(x_2) - f_1(x_2)f_2(x_1).$$
If $f_1=f_2$ then $\mathrm{asym}(f_1,f_2)$ is identically ...
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Identical particles in infinite potential well [closed]
I have a question about identical particles in an infinite potential well.
In Zettili's quantum mechanics textbook, Section 8.5, problem 8.1(c), the Pauli exclusion principle is used to find the ...
