Question

Solve Schrödinger equation for hydrogen atom?

Answer 1

The ingredients

To fill the Schrödinger equation,

ˆ

H

ψ=Eψ, with a bit of life, we need to add the specifics for the system of interest, here the hydrogen-like atom. A hydrogen-like atom is an atom consisting of a nucleus and just one electron; the nucleus can be bigger than just a single proton, though. H atoms, He+ ions, Li2+ ions etc. are hydrogen-like atoms in this context. We'll see later how we can use the exact solution for the hydrogen-like atom as an approximation for multi-electron atoms.

Fig.: Geometry of the hydrogen-like atom.

The potential, V between two charges is best described by a Coulomb term,

V(r)=−

Ze2

4πϵ0r

,

where Ze is the charge of the nucleus (Z=1 being the hydrogen case, Z=2 helium, etc.), the other e is the charge of the single electron, ϵ0 is the permittivity of vacuum (no relative permittivity is needed as the space inside the atom is "empty").

With the system consisting of two masses, we can define the reduced mass, i.e. the equivalent mass a point located at the centre of gravity of the system would have: μ=

mM

m+M

, where M is the mass of the nucleus and m the mass of the electron.

Thus, the hydrogen atom's Hamiltonian is

ˆ

H

=−

ℏ2

2μ

∇2−

Ze2

4πϵ0r