# 398: Particle in a One-dimensional Egg Carton

Numerical Solutions for Schrödinger's Equation

Integration limit: xmax = 10 Effective mass: $$\mu$$ = 1

Potential energy: Vo = 2 atoms = 2 $$V(x) = V_{o} ( \cos (atoms 2 \pi \frac{x}{x_{max}}) +1)$$

Numerical integration of Schrödinger's equation:

Given $$\frac{-1}{2 \mu} \psi (x) + V(x) \psi (x) = E \psi (x)$$ $$\psi (0) = 0$$ $$\psi '=0.1$$

$$\psi = Odesolve (x, x_{max})$$ Normalize wave function: $$\psi (x) = \frac{ \psi (x)}{ \sqrt{ \int_{0}^{x_{max}} \psi (x)^{2}dx}}$$

Enter energy guess: E = 0.83583

Fourier transform coordinate wave function into momentum space.

p = -10, -9.9 .. 10 $$\Phi (p) = \frac{1}{ \sqrt{2 \pi}} \int_{0}^{x_{max}} exp(-i p x) \psi (x)~dx$$