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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!

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!

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