(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

This was a test question I just had, and I'm fairly certain I got it wrong. I'm confused as to what I did wrong, though. We were told that our potential was infinite when x<0, and Cx where x>0. We were asked to approximate the ground state potential using the variational method with a test function [tex]xe^{- \alpha x}[/tex].

2. Relevant equations

The variational method states that

[tex]\frac{\langle \psi \mid \hat{H} \mid \Psi \rangle}{\langle \psi \mid \Psi \rangle} \ge E_0.[/tex]

3. The attempt at a solution

I calculate

[tex] \langle \psi \mid \hat{H} \mid \Psi \rangle = \frac{3\hbar^2}{8m} + \frac{c}{4 \alpha^3}.[/tex]

and

[tex]\langle \psi \mid \Psi \rangle = \frac{1}{4 \alpha^3}.[/tex]

In total this gives me

[tex]\frac{\langle \psi \mid \hat{H} \mid \Psi \rangle}{\langle \psi \mid \Psi \rangle} = \frac{3 \hbar^2 \alpha^3}{2m} + c.[/tex]

Now assuming I had gotten this far correctly, I was a little confused where to go from here. Presumable I need to minimize this with respect to alpha. However, this would mean that alpha will be negative and unbounded, correct? Or do I need to assume that alpha is greater than 0. In that case, wouldn't the minimum energy just be c? This seems wrong to me. Any advice would be appreciated.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Variational method approximation for half-space linear potential

**Physics Forums | Science Articles, Homework Help, Discussion**