QM: The time evolution of a Gaussian wave function

[DiracNotationKillsMe]

Heads up, I only recently got into quantum mechanics and don't feel like I got a solid grasp on the material yet.

1. Homework Statement
Given is the wave function of a free particle in one dimension:
\begin{equation}
\psi(x,0) = \left( \frac{2}{\pi a^2} \right)^{1/4} e^{i k_0 x} e^{-x^2/a^2}
\end{equation}
I already calculated the Fourier-transform (necessary in another part):
\begin{equation}
g(k) = ( 2 \pi )^{-1/4} \sqrt{a} e^{-(k_0 - k)^2 a^2/4}
\end{equation}
Now I am tasked with calculating the time evolution ψ(x,t).

Homework Equations

We were given hints by tutors for using the time-operator:
\begin{equation}
|\psi (x, t)> = e^{-i / \hbar \hat{H} t } | \psi(x,0) >
\end{equation}
Thus, we would only need to find the eigenvalue of the hamiltonian:
\begin{equation}
\hat{H} | \psi(x,0)> = \lambda | \psi(x,0) >
\end{equation}
However, against my intuition of just putting in my original wave function and calculating away ( which yields a λ depending on x ), they told us to use the Fourier-transform of g(k) instead:

\begin{equation}
\hat{H} | \psi(x,0)> = \hat{H} \frac{1}{\sqrt{2 \pi}} \int g(k) e^{ikx} dk
\end{equation}

The Attempt at a Solution

I tried calculating it the way they told us to, but the resulting integral seems endless and doesn't come to a head after hours of trying. Even if i were to finish it, it seems to me that the final term would still depend on x. I am also not that skilled in calculating eigenvalues, so allow me to ask: would it be fine if the eigenvalue for the Hamiltonian depends on x ( doesn't seem like it to me )? I'd also appreciate someone explaining to me why I'd have to use the double transformed function instead of the original one.

Thanks in advance.

 

[vela]

Thus, we would only need to find the eigenvalue of the hamiltonian:
\begin{equation}
\hat{H} | \psi(x,0)> = \lambda | \psi(x,0) >
\end{equation}

It doesn't make sense to write that since $|\psi(x,0)\rangle$ isn't an eigenstate of the Hamiltonian,.

What are the eigenstates of the Hamiltonian? Once you identify those, it might make more sense why it's been suggested you use the Fourier transform.

 

[blue_leaf77]

Thus, we would only need to find the eigenvalue of the hamiltonian:

No, the initial wavefunction is clearly not an eigenfunction of free space Hamiltonian.
Your starting point is the equation (3) but I would rewrite it in a more proper way.
$$
|\psi(t) \rangle = e^{iHt/\hbar} |\psi(t=0) \rangle
$$
where $H = \frac{\hat p ^2}{2m}$. Since the Hamiltonian is a function of only momentum operator, it will be helpful to use closure relation in momentum space $I = \int |p\rangle \langle p| dp$ where $I$ is the identitty operator. Now insert it to the right of the time propagation operator to get
$$
|\psi(t) \rangle = \int e^{ip^2t/(2m\hbar)} |p\rangle \langle p|\psi(t=0) \rangle dp
$$
Now $p$ in the exponential factor is just a number not an operator as it was before. Since you want the state in position space, you need to project it with $\langle x|$ to get
$$
\langle x|\psi(t) \rangle = \int e^{ip^2t/(2m\hbar)} \langle x|p\rangle \langle p|\psi(t=0) \rangle dp
$$
You have calculated $\langle p|\psi(t=0) \rangle$ as $g(k)$ above (note that $p=\hbar k$).

Note: didn't realize vela has posted his answer while I was writing mine.

 

[DiracNotationKillsMe]

Thanks for the replies, now it actually starts to make sense.

Regarding your reply, blue_leaf77, I generally understand your approach but am wondering what exactly I have to write for ⟨x|p⟩. Would it be fine to just write it as the inner product
\begin{equation}
\langle x | p \rangle = \hat{X} \hat{P}
\end{equation}
since the particle only exists in one dimension?

 

[blue_leaf77]

⟨x|p⟩ is the momentum eigenstate projected into position space, in other words it's the eigenfunction of the operator $-i\hbar \frac{d}{dx}$, shouldn't be too hard to calculate it or just make advantage of its popularity by searching in the internet.

 

[DiracNotationKillsMe]

Ah, obviously. Thanks for the help, you cleared a lot of things up for me.