# Homework Help: Infinite system of wells

1. Jan 26, 2012

### v_pino

1. The problem statement, all variables and given/known data
For the (unnormalized) wave function ψ(x) = sech(ax), ﬁnd the potential energy V (x), and show that the ground-state energy E1 is V(0)/2. The energies are in units of (hbar)^2a^2/2m.

2. Relevant equations

$$- \frac{\hbar}{2m}\frac{d^2 \psi}{dx^2}+V(x) \psi)=E \psi$$

3. The attempt at a solution

I differentiated the SE twice and sub it back in to SE. Does that seem right to you?

$$\frac{d^2 \psi}{dx^2} = 0.5a^2 (cosh(2ax)-3)sech^3(ax)$$

$$\frac{-\hbar^2}{4m} a^2 (cosh(2ax)-3)sech^2(ax)+V(x)=E=\frac{\hbar^2 a^2}{2m}$$
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Jan 26, 2012

### v_pino

Also, I forgot to say that I got this :

$$V(0)/2=\frac{3 \hbar^2 a^2}{8m}$$