#### A. Neumaier

Commonly there is a lot of imprecision in talking about ”indistinguishable” (or ”identical”) particles, even in serious work. This Insight article clarifies the issues involved in a conceptually precise way.
Classical mechanics. Historically, indistinguishable particles were introduced in order to explain the failure of the thermodynamics of a Newtonian $N$-particle system to describe the absence of an entropy increase when mixing two volumes of the same substance. This assumption, which has no logical basis in classical mechanics but appears as an ad hoc device to save the theory, drastically changes the state space of the multiparticle system, the phase space of functions of positions and momenta, to a much smaller space of symmetric functions of positions and momenta.
Nonrelativistic quantum mechanics. The first quantum manifestation of indistinguishable particles was the Pauli exclusion principle, which states that wave functions of a nonrelativistic...

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#### PeterDonis

Mentor
Excellent article!

Regarding the discussion of entanglement and when it makes sense to use that term: consider an atom with electrons in multiple energy levels in its ground state (for example, beryllium, with two 1s electrons and two 2s electrons). Would it make sense to say that the electrons in a particular energy level (e.g., 1s) are indistinguishable and therefore it's not useful to talk about them being entangled, whereas the electrons in different energy levels (e.g., 2s vs. 1s) are distinguishable and therefore it is useful to talk about them being entangled (so the overall ground state of the Be atom could be described as an entangled state of two 1s electrons and two 2s electrons)?

#### A. Neumaier

Would it make sense to say that the electrons in a particular energy level (e.g., 1s) are indistinguishable and therefore it's not useful to talk about them being entangled, whereas the electrons in different energy levels (e.g., 2s vs. 1s) are distinguishable and therefore it is useful to talk about them being entangled (so the overall ground state of the Be atom could be described as an entangled state of two 1s electrons and two 2s electrons)?
The notion of an elecron being in a particular energy level is already a notion of a particular approximation scheme that describes the multielectronic wave function in the full Hilbert space in terms of a reduced, simplified description based on a tensor product of Hilbert spaces for electrons with fixed energy levels - a degenerate extension of the Hartree-Fock approach.

In this reduced description, the two electrons in a particular degenerate energy level are indistinguishable, and the electron pairs in different energy level are distinguishable and entangled. However, it is not clear to me how useful the terminology is in this case.

#### HomogenousCow

For example, multiplication by xkxk, the standard position observable for particle kk in the distinguishable case, does not have this property. And if one correctly (anti)symmetrizes the result, the resulting operator only describes the center of mass of the system, not a property of individual particles.
By this, do you mean that if we took an operator acting on two antisymmetric particles like

$$X_2 = X\otimes I + I\otimes X,$$ and calculated its exepctation value on an arbitrary antisymmetric state $|A\rangle = |\psi\rangle|\phi\rangle - |\phi\rangle|\psi\rangle,$ we would get $$\langle X_2\rangle = 2 \langle\psi|X|\psi\rangle + 2 \langle\phi|X|\phi\rangle$$

And that it's impossible to conjure up any legal operator which could be used to probe just one of the factor states $|\psi,\phi \rangle$?

#### A. Neumaier

By this, do you mean
I mean that the physical Hilbert space $\cal H$ is not a dense subspace of the tensor product, and the position operator of the first particle, as an operator on the tensor product, does not map a dense subspace of $\cal H$ into $\cal H$. But the latter is required for operators with a physical meaning.

"Clarifying Common Issues with Indistinguishable Particles - Comments"

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