Hydrogen Atom in 1D: Schrödinger Equation Solution

Step 1: Time-Independent Schrödinger Equation (TISE)

The 1D Schrödinger equation for a hydrogen-like atom is:

$\frac{d^2\psi}{dx^2} + \frac{2m}{\hbar^2} \left( E + \frac{e^2}{4\pi\epsilon_0 |x|} \right) \psi = 0$

Using the dimensionless variable $\xi = x/a_0$ , where $a_0$ is the Bohr radius:

$a_0 = \frac{4\pi\epsilon_0 \hbar^2}{m e^2}$

The equation becomes:

$\frac{d^2\psi}{d\xi^2} + \left(\lambda + \frac{1}{|\xi|}\right) \psi = 0$

For large $\xi$ , the equation simplifies to:

$\frac{d^2\psi}{d\xi^2} - \kappa^2 \psi = 0$

which has the solution:

$\psi(\xi) \sim e^{-\kappa \xi}$

Expanding $\psi(\xi)$ as a power series:

$\psi(\xi) = e^{-\kappa \xi} \sum_{n=0}^{\infty} a_n \xi^n$

The recurrence relation for coefficients is:

$a_{n+1} = \frac{(2\kappa n - 1)}{(n+1)^2} a_n$