Time dependent dispersion (Quantum Mechanics)

[emol1414]

Homework Statement

The initial wave function $\Psi (x,0)$ of a free particle is a normalized gaussian with unitary probability. Let $\sigma = \Delta x$ be the initial variance (average of the square deviations) with respect to the position; determine the variance $\sigma (t)$ in a moment later.

Homework Equations

I'm not sure which equations should I start from... but, as for a gaussian distribution, I'm thinking of
$\Psi (x,t) = \int^{\infty}_{-\infty} a(k) e^{i(kx - \omega t)} dk$,
being

$a(k) = \frac{A \sigma}{\sqrt{2\pi}} \exp{\left[-\frac{(k - k_0)^2 {\sigma}^2}{2}\right]}$

The Attempt at a Solution

I don't know how to start it... once I describe the wave function, find A, I don't know how to construct this time dependent dispersion. I know the packet will 'spread', and its width will increase, but no ideas on how to describe this in terms of the dispersion... I know the result is $\sigma (t) = \sqrt{\sigma^2(0) + (\frac{\hbar t}{2m \sigma(0)})^2}$

 

[diazona]

Sorry you haven't gotten an answer on this yet, but if it's still relevant: I'd start by determining the dispersion relation $\omega(k)$ for the particle and plugging that in. The eventual goal is to take the explicitly time-dependent expression you have,
$$\Psi (x,t) = \int^{\infty}_{-\infty} a(k,\sigma) e^{i(kx - \omega t)} dk$$
and rewrite it as an implicitly time-dependent expression
$$\Psi (x,t) = \int^{\infty}_{-\infty} a(k,\sigma(t)) e^{ikx} dk$$
which should give you the expression you need.