Schrödinger's Equation — Question 5

Question 5

A specified Gaussian stationary wave determines the potential that supports it

A nodeless stationary state is $\psi(x)=A\exp[-x^2/(4a^2)]$ with $a>0$. Choose the potential-energy zero so that $V(0)=0$. Use $-(\hbar^2/2m)\psi\prime\prime+V\psi=E\psi$.

1. Infer $V(x)$ and E by differentiating the prescribed state.

2. Express the resulting potential as $m\Omega^2x^2/2$ and relate E to $\Omega$.

Question 5 - Solution

A specified Gaussian stationary wave determines the potential that supports it

(a) Method and calculation

$$\frac{\psi\prime\prime}{\psi}=\frac{x^2}{4a^4}-\frac1{2a^2},\quad V=E+\frac{\hbar^2}{2m}\frac{\psi\prime\prime}{\psi}.$$$$\boxed{E=\frac{\hbar^2}{4ma^2}},\qquad\boxed{V(x)=\frac{\hbar^2x^2}{8ma^4}}.$$

(b) Interpretation and further calculation

$$\frac12m\Omega^2=\frac{\hbar^2}{8ma^4}\quad\Rightarrow\quad\boxed{\Omega=\frac{\hbar}{2ma^2}}.$$$$\boxed{E=\tfrac12\hbar\Omega}.$$

Physical check

This is an inverse check of one supplied state. It does not assume or derive the full trapped-particle spectrum, which belongs to the following chapter.