- #1

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Now my question is this:

In deriving the energy levels of the hydrogenic atom, we assume that the nucleus is infinitely heavy, so that the reduced mass becomes the mass of the electron. Why do we do so and what follows next?

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- Thread starter spaghetti3451
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- #1

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Now my question is this:

In deriving the energy levels of the hydrogenic atom, we assume that the nucleus is infinitely heavy, so that the reduced mass becomes the mass of the electron. Why do we do so and what follows next?

- #2

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I hope one can make sense of what I wrote :uhh:

- #3

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In deriving the energy levels of the hydrogenic atom, we assume that the nucleus is infinitely heavy, so that the reduced mass becomes the mass of the electron. Why do we do so and what follows next?

You don't even need to assume that it is infinite, just know that the mass of the proton is a lot larger than the mass of the electron. This basically means that the reduce mass becomes the mass of the electron and that you can treat the system as having a center of mass on the proton. I think this simplifies the problem because your electrostatic potential will begin at the center of mass which makes the hamiltonian a bit easier to write (or is it a lot easier?).

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alxm

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If you start with the Schrödinger equation for the an atom or molecule, then you have 3-4 sets of terms: Nuclear kinetic energy, electronic kinetic energy, the nuclear-electronic potential, and if you have more than one electron, the electronic-electronic potential. You have the nuclear coordinates and the electronic coordinates as variables. For other systems than the hydrogenic atom, you can't solve this equation analytically, even if they only have one electron, since the electronic coordinates depend on the nuclear coordinates and vice-versa. Through the potential, the nuclear kinetic energy is coupled to the electronic kinetic energy, so it's a many-body problem.

The first approximation/simplification you can do here is the Born-Oppenheimer approximation. The nuclei are much heavier than the electrons and correspondingly move much slower. That means the potential 'felt' by the electrons varies very slowly. If a potential varies sufficiently slowly, there's no change of state for the electrons - That's the Adiabatic Theorem (adiabatic meaning 'no transfer of energy' in this case). The B-O approximation is an application of that. Another way of looking at this, is that the nuclei are so slow that they can be viewed as stationary

So if you get rid of that coupling, you can separate the Schrödinger Equation into two PDEs, for the nuclear and electronic motion, respectively. In the electronic equation, the nuclear coordinates are no longer variable, but constants set to their average locations. So you can then solve the two equations separately, and the total energy is the sum of the electronic and nuclear energies.

Now, there's

Okay, so that's how it works in general. But you have additional simplifications that apply in the case of the hydrogenic atom. First there's the situation for single atoms in-general: In the case where you have a single atom, it's a free particle. The nucleus only has translational degrees of freedom. The electronic energy is the same no matter how fast the nucleus is moving, and the translational kinetic energy of the nucleus is arbitrary and set to zero. So the clamped nucleus approximation isn't an approximation for a single atom, where the nucleus doesn't have any kinetic energy. It comes into play for molecules, where the nuclei are bound and have vibrational degrees of freedom. Due to zero-point vibrations, their nuclear kinetic energy is always greater than zero (but small).

The other thing that applies for the hydrogenic atom (but not in general), is that you only have

So you have to be careful about assuming the nuclei are stationary/infinitely massive, because it can mean two different approximations depending on the context. And you have to be a bit careful with the hydrogenic atom, since it happens to be separable without the B-O approximation, which isn't the case in general.

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