# Electron Energies in Atoms: Fixed Values or Expectation Values?

• RNA
In summary, the conversation discusses the concept of fixed energy levels in electronic systems, specifically in a hydrogen atom. The argument is made that electronic energy levels may have fixed expectation values rather than fixed values. However, it is pointed out that the ground state of a hydrogen atom is, in fact, an eigenstate of the Hamiltonian and thus has a fixed energy value. The conversation also touches on the variation of potential and kinetic energy in relation to radial distance and the collapse of the electron's wave function. Ultimately, it is concluded that the energy of the electron is not a well-defined quantity before it collapses into a fixed energy state.
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!

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RNA said:
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.

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?

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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.

## 1. What are electron energies in atoms?

Electron energies in atoms refer to the potential energy levels that electrons can occupy within an atom. These energy levels are quantized, meaning that they can only take on certain fixed values.

## 2. Are the electron energies in atoms fixed values or expectation values?

The electron energies in atoms can be described as both fixed values and expectation values. The fixed values refer to the specific energy levels that electrons can occupy, while the expectation values refer to the average energy of an electron in a given energy level.

## 3. How are the electron energies in atoms determined?

The electron energies in atoms are determined by the electron's quantum numbers, specifically the principal quantum number, which represents the energy level, and the orbital quantum number, which represents the sublevel within an energy level.

## 4. Can an electron have any energy level in an atom?

No, an electron can only occupy specific energy levels in an atom. These energy levels are determined by the energy of the electron's orbit around the nucleus, and are dependent on the atom's atomic number.

## 5. Why are electron energies in atoms important?

Electron energies in atoms are important because they determine the chemical and physical properties of an element. The arrangement and number of electrons in an atom's energy levels dictate how the atom will interact with other atoms, making electron energies crucial in understanding the behavior of matter.

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