Property and time evaluation of Gaussian wavefunction

Question

I faced some problems while learning the time evolution of Gaussian wavepacket . I started the part with a wavefunction in momentum space which was like $ \ \phi(p) = c.e^\frac{-(p-p_0)^2}{2.\sigma_p^2} \ $ . Then found the position space wavefunction by Inverse Fourier transform . And found a $\psi(x) \ $. I was asked to find out the $\psi(x,t)$. I thought that like the bound state problems I will multiply the $ \ \psi(x) \ $by $ \ e^\frac{-iEt}{\hslash} \ $and find the time evolution of the wavepacket. But in the solution they invoked a solution with integrals like : $ \ \psi(x,t) = (2\pi \hslash)^\frac{1}{2}.\int_{-\infty}^{+\infty} e^{(kx-\omega t)}.\phi(p) \ dp \ $ and solved this to find the time dependent wavefunction . Can anyone tell me that why they followed this approach ? I am suspecting that the Gaussian case is not always bounded so they didn't followed the $ \ e^\frac{-iEt}{\hslash} \ $ approach . Can the Gaussian wavepacket represent a free particle ?

Answer 1

The time evolution of a system is actually given by $e^{-iHt/\hbar}$ operating on the initial condition, where $H$ is the Hamiltonian describing the system. You have to know how the Hamiltonian operates on the state your system is in. IF your system is in an energy eigenstate, then the $H \phi_E = E \phi_E$ and you get the term you were expecting. However if its not, then you need expand the state in the basis of energy eigenstates. This can involve an integral over all the energy eigenstates. IF the momentum eigenstates are also energy eigenstates, and you know the relationship between the momentum in a given eigenstate and the energy in that eigenstate, then that integral can become an integration over all possible momenta.

Try to figure out what the momentum eigenstates are for your system. Can you write your initial condition in terms of an integral over all the momentum eigenstates? Do you know what the energy is when you operate your Hamiltonian on a single momentum eigenstate?