Answer to Question #282726 in Quantum Mechanics for BIGIRMANA Elie

Question

6. Show that xe−x

2

is an eigenfunction of the linear operator d

2

dx2 − 4x

2

.

What is the eigenvalue?

Answer 1

We have the linear operator $\hat A=\frac{d^2}{dx^2}-4x^2$ .

$\psi (x)=xe^{-x^2}$ is an eigenfunction of $\hat A$ , if $\hat A\psi =\lambda \psi$ .

$\hat A\psi=\frac{d^2}{dx^2}\big(xe^{-x^2}\big)-4x^2\cdot xe^{-x^2}=\frac{d}{dx}\big( e^{-x^2}-2x^2e^{-x^2}\big)-4x^3e^{-x^2}= -2xe^{-x^2}-4xe^{-x^2}+4x^3e^{-x^2}-4x^3e^{-x^2}=-6xe^{-x^2}=-6\psi$

So, $\psi(x)$ is an eigenfunction of $\hat A$ and the eigenvalue is $\lambda=-6$ .