Scattering of a gaussian wave packet at a potential

[KaiserBrandon]

Homework Statement

start with the wave function

$\Psi(x,0) = Ae^{-cx^{2}}e^{ikx}$

where A,c, and k are real constants (and c is positive)

i) Normalize $\Psi(x,0)$
ii) Determine $\Psi(x,t)$ and $|\Psi(x,t)|^{2}$

Homework Equations

The Attempt at a Solution

I normalized it to get $A = (\frac{2c}{\pi})^{1/4}$

And now to determine $\Psi(x,t)$, I'm fairly sure that I have to make the wave function as a superposition of the energy eigenvectors of the wave-function. However, I am unsure of how to go about doing this.

 

[collinsmark]

Homework Statement

start with the wave function

$\Psi(x,0) = Ae^{-cx^{2}}e^{ikx}$

where A,c, and k are real constants (and c is positive)

i) Normalize $\Psi(x,0)$
ii) Determine $\Psi(x,t)$ and $|\Psi(x,t)|^{2}$

Homework Equations

The Attempt at a Solution

I normalized it to get $A = (\frac{2c}{\pi})^{1/4}$

That's what I got

And now to determine $\Psi(x,t)$, I'm fairly sure that I have to make the wave function as a superposition of the energy eigenvectors of the wave-function. However, I am unsure of how to go about doing this.

I don't think you can go any further unless the potential is specified (actually, what you really need are the energy eigenvectors, which you can solve for if you know the potential). The thread title is "scattering of a gaussian wave packet at a potential," however the potential isn't specified in the problem statement.

Assuming the potential is given somewhere else in the book/assignment, you can use it to solve Schrödinger's equation.

You'll end up with something of the form, (In my notation, $\psi = \Psi (x, 0)$.)
$$\left| \psi \right> = \sum_i c_i \left| \psi_i \right>$$
where $\left| \psi_i \right>$ are the energy eigenvectors.
You can find the $c_i$s via
$$c_i = \left< \psi_i | \psi \right>.$$
Where did I get that last equation you may ask? Note that from a previous equation,
$$\left| \psi \right> = \sum_i c_i \left| \psi_i \right>$$
and bringing to the bra of each energy eigenvector (denoted this time with a k subscript) to each side of the equation,
$$\left< \psi_k | \psi \right> = \left< \psi_k \right| \left( \sum_i c_i \left| \psi_i \right> \right)$$
$$= c_0 \left< \psi_k | \psi_0 \right> + c_1 \left< \psi_k | \psi_1 \right> + c_2 \left< \psi_k | \psi_2 \right> . \ . \ .$$
But remember we're working with Hilbert space. All the energy eignvectors are orthogonal to one another. That means all the above terms on the right side of the equation are zero except where i = k. And in that case, $\left< \psi_k | \psi_i \right> = 1$. Thus,
$$\left< \psi_i | \psi \right> = c_i.$$
But of course, you can't really solve for the $c_i$s without knowing the general solution to the particular potential at hand, since you'll need to know that to find the $\psi_i$s. Good luck!

(By the way, this problem probably qualifies to go in the Advanced Physics homework forum.)