Is it possible to prepare experimentally a quantum state which is a superposition of two states with the different number of particles? For example $|\psi\rangle=|N=1\rangle+|N=2\rangle$
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1$\begingroup$ I guess you can form $|N=1 \rangle \otimes | \mathrm{vacuum} \rangle_2 $ for $|N=1 \rangle$ for consistency $\endgroup$– user26143Commented Aug 24, 2013 at 11:31
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3$\begingroup$ Because QM is linear, nothing bad happens if you form a superposition of states chosen completely freely. But such a superposition may not be very interesting if it's not possible for its different parts to interfere. $\endgroup$– user4552Commented Aug 24, 2013 at 14:20
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5$\begingroup$ @richard In absence of any mechanism for production or annihilation of particles, time evolution will not be able to mix up the states with different particles numbers and hence you can study them separately too. $\endgroup$– user10001Commented Aug 24, 2013 at 16:14
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5$\begingroup$ @jjcale: I would stress that the ground state of a superconductor doesn't actually consist of a superposition of different particle numbers. This may be clearer in the case of nuclear physics, where the particle number fluctuations in the BCS ground state are usually quite large, and obviously can't represent anything real. This is an example where the states with different particle numbers can't physically interact in reality, but we introduce such an interaction purely as an approximation scheme in order to make the many-body problem more tractable. $\endgroup$– user4552Commented Aug 24, 2013 at 20:06
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5$\begingroup$ @jjcale: The BCS state is not just an approximation but the true ground state in the thermodynamic limit. Yes, but the thermodynamic limit is obtained with $N\rightarrow\infty$. In that same limit the fluctuations in the particle number become negligible in the sense that $\sigma_N/<N>\rightarrow0$. That's why I used the example of nuclear physics, where we only get $N\sim100$, and it is very clear and uncontroversial that the BCS ground state is an approximation. Condensed matter physicists don't usually have to worry about this because $N$ is on the order of Avogadro's number. $\endgroup$– user4552Commented Aug 25, 2013 at 21:26
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2 Answers
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The key to answering this question are the words "superselection observable." This concept, and that of "superselection rules", was introduced by three of the most notorious W's of quantum physics, Wick, Whitman, and Wigner. In a nutshell, superselection rules are strict limits to the possibility of building quantum superpositions that are physically meaningful.
Examples of superselection observables you can find in electric charge (or any other gauge charge) and mass. While charge is very fundamental, mass is apparently some derived concept that we probably don't fully understand as yet.
Mass:
Curiously enough, while there is no problem in having quantum superpositions of states with different energy or momentum: $$ c_{1}\left|E_{1}\right\rangle +c_{2}\left|E_{2}\right\rangle $$ $$ c_{1}\left|\boldsymbol{p}_{1}\right\rangle +c_{2}\left|\boldsymbol{p}_{2}\right\rangle $$ In QFT, we build mass as a so-called Casimir, which is a polynomial combination of the generators of the group that commutes with all the generators, and is an invariant under the group. In the case of the Poincaré group, this is mass: $$ M^{2}c^{4}=H^{2}-c^{2}\boldsymbol{P}^{2} $$ For reasons (perhaps) not fully understood, we can never have superpositions of states with different mass: $$ c_{1}\left|M_{1}\right\rangle +c_{2}\left|M_{2}\right\rangle $$
Charge:
Electric charge is much more fundamental in the theory: Not only no superpositions of states with different charge are allowed, but not even transitions between states of different charge are allowed. So we don't allow, $$ c_{1}\left|Q_{1}\right\rangle +c_{2}\left|Q_{2}\right\rangle $$
Observation: Because both mass and charge are diagonal in the number operator, you cannot have superpositions of different eigenstates of such operator without having a superselection rule violated. Not so with photons:
Photons:
As pointed out by @tparker, there is no problem for photons to be in coherent superpositions of states with different values of (expected numbers) of them. A well-known case is coherent states, which are eigenstates of the creation/annihilation operators.
I hope that helped. This question kept me awake at night for years when I was a student. I cannot claim by any means that this is a clinch-case answer. The "why" still stands.
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2$\begingroup$ Doesn't the impossibility to have superpositions of different mass already arise from the Galilei invariance of the non-relativistic Schroedinger equation? $\endgroup$ Commented Mar 24, 2021 at 16:03
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2$\begingroup$ You're absolutely right. Thanks a lot for the observation. That's Bargmann's superselection rule. I hope I can edit my answer tomorrow and improve the answer. In that case, the superselection rule is indeed a consequence of a symmetry principle, isn't it? $\endgroup$– joigusCommented Mar 24, 2021 at 22:07
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2$\begingroup$ Interesting, I did not know it was linked to a name! (Learned it from a colleague who in turn learned from it in Galindo & Pascual.) $\endgroup$ Commented Mar 24, 2021 at 22:29
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2$\begingroup$ Galindo & Pascual indeed! Volume I, p. 292 $\endgroup$– joigusCommented Mar 24, 2021 at 22:37
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The comments already give the hint answer, each number state would evolve independent if there are no mechanism of creation or annihilation. One situation is that the number operator commutes with the Hamiltonian $\mathcal{\hat{H}}$, so the number state in the superposition would preserve over time with extra phase on each of them:
$|\psi(t)\rangle = e^{-i\mathcal{\hat{H}}t/\hbar}|\psi(0)\rangle = \sum_N e^{-i\mathcal{\hat{H}}t/\hbar}c_N|N\rangle = \sum_N e^{-iE_Nt/\hbar}c_N|N\rangle \tag{1}$
The state with different number will not mix in this case.
For the preparation, suppose the system can absorb photons and create excitation. If the photon state is something like $|\psi\rangle = a|1\rangle+b|2\rangle$, then the electron excitation in the system could get the same state and evolve like equation (1) when there is no creation and annihilation. (I am not sure what systems have such properties.)
