(a) Show, for an appropriate value of a which you should find, that
??(x) = Ae??x²
is an eigenfunction of the time-independent Schrödinger equation with
potential
V(x) = \frac{1}{2}m?²x².
Hence find the associated energy eigenvalue.
Using the standard integral
$\int_{-?}^{?} e^{-Mx²}dx = \sqrt{\frac{?}{M}}$, M > 0,
find the value of A.
(b) The ladder operators associated with a particle under the potential V(x),
defined in part (a), are given by
$\hat{a} = \sqrt{\frac{1}{2L²}}(x + L²\frac{d}{dx})$, $\hat{a}^{†} = \sqrt{\frac{1}{2L²}}(x - L²\frac{d}{dx})$,
where L² = ?/m?.
i. Show that $\hat{a}??(x) = 0$ where ??(x) is the function in part (a). What
is the physical interpretation of this result?
ii. Derive the commutation relations for the ladder operators $\hat{a}$ and $\hat{a}^{†}$.
