Coulomb potential

Consider a particle of charge $- 𝑞$ and mass $𝑚$ bound to a stationary charge $𝑄$ at the origin. The potential experienced by the particle, given by Coulomb's law, is spherically symmetric with

𝑉 (𝑟) = - 𝑄 𝑞 4 𝜋 𝜖 0 1 𝑟

giving the radial equation

- ℏ 2 2 𝑚 𝑑 2 𝑢 𝑑 𝑟 + (- 𝑄 𝑞 4 𝜋 𝜖 0 1 𝑟 + ℏ 2 2 𝑚 ℓ (ℓ + 1) 𝑟 2) 𝑢 = 𝐸 𝑢

which has bound states

𝜓 𝑛 ℓ 𝑚 (𝑟, 𝜃, 𝜙) = \sqrt (2 𝑛 𝑎 𝑄 𝑞) 3 (𝑛 - ℓ - 1)! 2 𝑛 (𝑛 + ℓ)! 𝑒 - 𝑟 / 𝑛 𝑎 𝑄 𝑞 (2 𝑟 𝑛 𝑎 𝑄 𝑞) ℓ 𝐿 2 ℓ + 1 𝑛 - ℓ - 1 (2 𝑟 𝑛 𝑎 𝑄 𝑞) 𝑌 𝑚 ℓ (𝜃, 𝜙)

where $𝑌 𝑚 ℓ$ is a spherical harmonic, $𝐿 2 ℓ + 1 𝑛 - ℓ - 1$ is an Associated Laguerre polynomial, and

𝑎 𝑄 𝑞 = 4 𝜋 𝜖 0 ℏ 2 𝑚 𝑄 𝑞

which in the case of hydrogen is the Bohr radius. The allowable energies are

𝐸 𝑛 = - [𝑚 𝑒 2 ℏ 2 (𝑄 𝑞 4 𝜋 𝜖 0) 2] 1 𝑛 2

which are each $𝑛 2$ -degenerate.

Quantum numbers

$𝑛 = 1, 2, \dots$
$ℓ = 0, 1, \dots, 𝑛 - 1$
$𝑚 = - ℓ, - ℓ + 1, \dots, ℓ - 1, ℓ$

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