Finding Potential Energy for Sech Wave Function

[v_pino]

Homework Statement

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.

Homework Equations

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

The Attempt at a Solution

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

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

$$\frac{-\hbar^2}{4m} a^2 (cosh(2ax)-3)sech^2(ax)+V(x)=E=\frac{\hbar^2 a^2}{2m}$$

 

[v_pino]

Also, I forgot to say that I got this :

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