Hydrogen atom in the momentum representation

The wave functions for the hydrogen atom are determined in the momentum representation, using momentum variables

p_{r}

p_{θ}

p_{φ}

which are associated with the spherical polar variables

r

t (= e^{i π cos θ})

and

v (e^{i φ})

. The functions may be written as

{φ_{n, l, m} (p_{r}, p_{θ}, p_{φ}) = α_{n, l} (p_{r}) β_{l}^{m} (p_{θ}) δ (p_{φ} + m),} {α_{n, l} (p_{r}) = {(\frac{n}{i p_{0}})}^{3 l + z} Σ \overset{n - l - 1}{k = 0} ({n - l - 1} {k}) \frac{2^{k - l + 1}}{(2 l + k + 1) (2 l + k) \dots (l + k + 2)} {(\frac{\frac{i p_{0}}{n}}{p_{r} - \frac{i p_{0}}{n}})}^{l + k + 2},} {β_{l}^{m} (p_{θ}) = Σ \overset{E (\frac{l}{2})}{j = E (\frac{m}{2})} {a_{j} p_{θ}}^{- \frac{(m + 1)}{2}} J_{\frac{j - (m - 1)}{2}} (p_{θ}),}

where

δ

is the Dirac delta function and

p_{0} = Z a_{0}^{- 1}

(

a_{0}

is the Bohr radius).

J_{ν} (p_{0})

are Bessel functions of integer or half-integer order. The radial functions...

This publication has 19 references indexed in Scilit: