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

A particel is moving along the x-axis in a potential,V(x). The ground state of the wavefunction is:

[tex]\[\psi \left( x \right)={{e}^{-\frac{1}{2}{{\left( \frac{x}{a} \right)}^{4}}}},\][/tex]

whereais a constant of dimension length. We also know thatV(a) = 0.

DetermineV(x).

2. Relevant equations

The stationary Schrödinger equation:

[tex]\[-\frac{{{\hbar }^{2}}}{2m}\frac{{{d}^{2}}\psi }{d{{x}^{2}}}+V\psi =E\psi \][/tex]

3. The attempt at a solution

Well, first I derive the wavefunction two times and get:

[tex]\[\begin{align}

& \frac{d}{dx}\psi \left( x \right)=-\frac{2}{{{a}^{4}}}{{x}^{3}}{{e}^{-\frac{1}{2}{{\left( \frac{x}{a} \right)}^{4}}}} \\

& \frac{d}{dx}\psi '\left( x \right)=\left( \frac{4{{x}^{6}}}{{{a}^{8}}}-\frac{6{{x}^{2}}}{{{a}^{4}}} \right){{e}^{-\frac{1}{2}{{\left( \frac{x}{a} \right)}^{4}}}} \\

\end{align}\]

[/tex]

Then I insert it in the Schrödinger equation.

And here is where I get confused. It's been a while since I've had QM, so I can't really remember what's next.

I figure that the potential is NOT just what I get when I isolate V in the Schrödinger equation. So what's is my next step ?

Regards

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# Homework Help: Determine the potential V(x)

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