I'm slightly confused as to answer this question, someone please help:
Consider a free particle in one dimension, described by the initial wave function
$$\psi(x,0) = e^{ip_{0}x/\hbar}e^{-x^{2}/2\Delta^{2}}(\pi\Delta^2)^{-1/4}.$$
Find the time-evolved wavefunctions $\psi(x,t)$.
Now I know that since it is a free particle we have the hamiltonian operator as $$H = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2},$$ which yields the energy eigenfunctions to be of the form $$\psi_E(x,t) = C_1e^{ikx}+C_2e^{-ikx},$$ where $k=\frac{\sqrt{2mE}}{\hbar}$, and the time evolution of the Schrödinger equation gives $$\psi(x,t)=e^{-\frac{i}{\hbar}Ht}\psi(x,0)$$ but the issue I face is what is the correct method to find the solution so that I can then calculate things such as the probability density $P(x,t)$ and the mean and the uncertainty (all which is straight forward once I know $\psi(x,t)$.
In short – how do I find the initial state in terms of the energy eigenfunctions $\psi_E(x,t)$ so that I can find the time evolved state wavefunction.
Best Answer
For a free particle, the energy/momentum eigenstates are of the form $e^{i k x}$. Going over to that basis is essentially doing a Fourier transform. Once you do that, you'll have the wavefunction in the momentum basis. After that, time-evolving that should be simple.
Hint: The fourier transform of a Gaussian is another Gaussian, but the width inverts, in accordance with the Heisenberg uncertainty principle. The phase and the mean position will transform into each other -- that is a little more subtle and you need to work it out.
Also have a look at http://en.wikipedia.org/wiki/Wave_packet.