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If there were a particle in a box, how could one measure its energy?

I understand the theory behind quantum measurements: the Hamiltonian operator represents the energy observable, so you perform an "energy measurement" that returns an eigenvalue of the operator, and that eigenvalue is the energy. But how would you actually physically do this?

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3 Answers 3

If the particle is an ion in an ion trap (which is essentially a kind of box with electromagnetic fields forming its "walls"), there are ways to measure which electronic state it's in. You can shine a laser on it and see whether it fluoresces. There are similar ways to measure its kinetic energy. See measurement in the wikipedia article on ion trap quantum computers.

If the atom is one hydrogen atom in a very large metal box, I think you're out of luck.

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To your first question, it depends on how you pose your Hamiltonian. In its usual form H=K+V, where V is the bonding among atoms and/or intermolecular forces, indeed the energies are inferred from spectroscopic studies, as anna pointed out. Identifying the peaks to the associated energy levels (and subsequently inferring other physical parameters of the molecule such as bond length, etc) was a major sub-discipline of Chemical Physics in the earlier years.

So relating to your second question, we first postulate that the world is expressed by (separable) vector spaces and interactions by (Hermitian) operators. Once we do this, we find ourselves only able to measure in the subspace of some eigenvector. The "eigenvalue is energy" part is more of a generalization of the experiments we have done and we now do. In the spectroscopy case, we find the energies first and then infer what the Hamiltonian must look like. So naturally, the Hamiltonians are found so that their eigenvalues match the energies we measure. (Btw, we rarely infer the actual form of the Hamiltonian from spectroscopic studies. We usually tie the data to some simple models and do Taylor expansions of terms.)

@Jan: It does not matter whether we are doing spectroscopy on a population of molecules/atoms or a single one (the latter is NOT a non-existing concept). That is because the energy levels of identical species are the same. As long as the energy levels (ie. the eigenvalues of the Hamiltonian) are the same, there will be dominant transitions which result in the peaks in our spectra.

Further, I would like to point out that it is possible to prepare a very localized wavepacket and do experiments on it, if that alleviate some of your concern about the reality of particle in a box.
Hydrogen atom, on the other hand, does not have much to do with particle in a box physically. It terms out that we like to assume separability in solving differential equations. And in solving the Hamiltonian between a single proton and a single electron, we did exactly that and one component of the solution has similarities with the solution of particle in a box. That's about it.
A better example is quantum dots, where you can sometimes change their colors by manipulating their sizes, because size affects spacings between energy levels! Look it up! It was of much interest when it was first proposed and made. Not sure how it goes now.

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Thanks, Argyll! When we do spectroscopy, don't we see photons that come from transitions? If so, wouldn't we be seeing the gaps between energy levels, rather than the energy levels themselves? –  daunpunk Jan 11 at 6:03
    
Indeed, duanpunk, energy gap is what we see! It is good to have the concept super clear as you do! Once you are familiar with the concept though, it is customary to equate both "identify the energy levels" and "identify the transition" to finding the energy level a transition occurs from and the level to. Further, the energy levels are in a sequence of increasing energies. So we often just say "assign peak numbers" too. –  Argyll Jan 11 at 18:03
    
So if there were a system with two different pairs of energy levels that are the same distance apart, this type of measurement wouldn't tell you what the energy of the system is? –  daunpunk Jan 14 at 6:39
    
That is correct! This is the reason we cannot distinguish species with identical interactions along their electrons and nuclei. Try search if chiral compounds can be discerned by spectroscopic measurements? –  Argyll Jan 29 at 22:51

The hydrogen atom is modeled very well quantum mechanically as a "particle in a box", a particle in a potential well.

The electron around the proton is in a specific energy level, usually the lowest one. How do we measure the energy levels? By observing the spectrum emitted by an electron falling in them ( emission spectrum) or by the energy of the photon absorbed to kick an electron to a higher energy state.

Edit after comments: The spectra in the links are an accumulation of photons from the same boundary conditions. Single photon emission can be measured. For example it is used in information technology and for cameras used in astrophysics. One can know the energy level an excited electron falls in for a single atom, the accumulated width is due to the uncertainty principle.

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But how would you measure $energy$ of one atom? We cannot find this purely from the spectrum, as this is due to $many$ atoms. Also, we cannot contrive single absorption by the given atom. –  Ján Lalinský Jan 9 at 21:54
    
Thanks, Anna! I have basically the same objection as Jan. I'd like to perform one measurement of the energy of a single particle in a box. –  daunpunk Jan 10 at 2:31
    
@daunpunk please see edit –  anna v Jan 10 at 4:48

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