Electron Energies in Atoms: Fixed Values or Expectation Values?

[RNA]

Hi. First post. I'm trying to understand if electronic energy levels have fixed values, or merely fixed expectation values (in the latter case, orbital electrons could have any energy and it's only the average that would be fixed). Here's my argument for the latter. If it's incorrect, could you please tell me the flaw in my physical picture?:

Let's consider the ground state (gs) of an isolated hydrogen atom.

1. The gs of a hydrogen atom is not an eigenfunction of the position operator. Thus the radial distance (of the electron from the nucleus) does not have a fixed value -- any position is allowed (according to a probability distribution). The only fixed value corresponding to the radial distance is its average (the expectation value).

2. The gs is an eigenfunction of the momentum operator, and thus does have a fixed value for kinetic energy.

3. The energy of the electron in each state is determined by the sum of its potential (V) and kinetic (T) energy. Closer to the nucleus, V decreases but T (because of confinement) increases. The gs energy represents the minimum of this sum.

1-3 should (I hope!) be fine. Now here's my argument:

4. V is determined by the radial distance. Thus we don't have a fixed value for V, only an expectation value. Thus we don't have a fixed value for E (E=V+T); again, only an expectation value. Thus the electrons in a hydrogen atom (and in any other atom) are not confined to fixed energy levels, it's only the average that is fixed. E.g., for hydrogen, there is a distribution of gs energies; it's only the average that is -13.6 eV.

Of course, one obvious problem with this argument is that if the radial position can take on any value, then that should allow variation not only in V but also in T; but T is supposed to be fixed.

Thanks!

 

[SpectraCat]

Hi. First post. I'm trying to understand if electronic energy levels have fixed values, or merely fixed expectation values (in the latter case, orbital electrons could have any energy and it's only the average that would be fixed).

Well, the H-atom energy levels are eigenstates of the H-atom Hamiltonian, so their are fixed values by definition. So, it's definitely the first one ...

Here's my argument for the latter. If it's incorrect, could you please tell me the flaw in my physical picture?:

Let's consider the ground state (gs) of an isolated hydrogen atom.

1. The gs of a hydrogen atom is not an eigenfunction of the position operator. Thus the radial distance (of the electron from the nucleus) does not have a fixed value -- any position is allowed (according to a probability distribution). The only fixed value corresponding to the radial distance is its average (the expectation value).

That's fine.

2. The gs is an eigenfunction of the momentum operator, and thus does have a fixed value for kinetic energy.

That's wrong .. what makes you think the H-atom functions are eigenfunctions of the momentum operator? Have you tried to prove that mathematically?

Perhaps you are confused because they actually *are* eigenfunctions of the *angular* momentum operator? That is a very different thing than what you are claiming here ...

3. The energy of the electron in each state is determined by the sum of its potential (V) and kinetic (T) energy. Closer to the nucleus, V decreases but T (because of confinement) increases. The gs energy represents the minimum of this sum.

That seems ok, but should tell you that why your 2 above is incorrect.

1-3 should (I hope!) be fine. Now here's my argument:

4. V is determined by the radial distance. Thus we don't have a fixed value for V, only an expectation value. Thus we don't have a fixed value for E (E=V+T); again, only an expectation value. Thus the electrons in a hydrogen atom (and in any other atom) are not confined to fixed energy levels, it's only the average that is fixed. E.g., for hydrogen, there is a distribution of gs energies; it's only the average that is -13.6 eV.

Of course, one obvious problem with this argument is that if the radial position can take on any value, then that should allow variation not only in V but also in T; but T is supposed to be fixed.

Thanks!

Well, hopefully you will understand why the above analysis is wrong, now that you know that your condition 2 is false.

 

[RNA]

SpectraCat: Ah, yes, of course (this indicates the dangers of trying to think about QM long after you last studied it). Thanks for your detailed reply (I also appreciated the pedagogy of guiding me to figure it out for myself).

So the resolution is that, as in a quantum harmonic oscillator, both T and V can vary, but their observed sum is fixed.

Likewise, when we observe the hydrogen electron (for instance, through a spectral transition), we find that it always collapses to a state with a fixed energy.

But what about the wave function before it's collapsed? Is the electron allowed to have any (total) energy? For instance, suppose you looked only at the part of the gs orbital in which the electron is very close to the nucleus. Here, can one calculate an integrated average T for the electron when it is within this region? If so, would it be enormously high, because of its proximity to the nucleus? But then would the integrated average V be sufficiently negative to just compensate for this, still giving us a fixed value for T+V?

Or what if you instead looked at the instantaneous energy? Is its value unrestricted?

I'm guessing the resolution here is that the energy of the electron is simply not a well-defined quantity prior to observation. Thus rather than saying QM tells us that orbital electrons are confined to fixed energy levels, it would be more precise to say QM tells us that these electrons will always be observed as having certain energies. Of course, an empiricist could then retort: "What's the difference?" Thus is my inquiry about the energy of the electron prior to observation getting me into the realm of the different philosophical interpretations of quantum mechanics (Copenhagen, etc.) --- which, as we know, are empirically indistinguishable?

 

[Parlyne]

If the electron is in an energy eigenstate, there is no ambiguity of any sort. By whatever measure you choose, the value of its energy will be the one corresponding to that state.

However, the electron is also allowed to be in any linear superposition of energy eigenstates. In that situation, there is no well-defined instantaneous energy. The expectation value of the energy will vary over time (never falling either below the lowest energy eigenvalue of any state included in the superposition or above the highest); but, any measurement of energy will simply find one of the system's energy eigenvalues.