Scattering of a gaussian wave packet at a potential

In summary, the problem involves a wave function with real constants A, c, and k, and the task is to normalize the function and then determine its value at a given time. The solution to the normalization yields A = (\frac{2c}{\pi})^{1/4}. However, further progress cannot be made without knowing the potential, which is necessary to solve Schrödinger's equation and find the energy eigenvectors. The final equation for determining the c_i coefficients is c_i = \left< \psi_i | \psi \right>, where \psi is the wave function and \psi_i is the corresponding energy eigenvector. This problem may be better suited for the Advanced Physics homework forum.
  • #1
KaiserBrandon
54
0

Homework Statement


start with the wave function

[itex]\Psi(x,0) = Ae^{-cx^{2}}e^{ikx}[/itex]

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

i) Normalize [itex]\Psi(x,0)[/itex]
ii) Determine [itex]\Psi(x,t)[/itex] and [itex]|\Psi(x,t)|^{2}[/itex]

Homework Equations





The Attempt at a Solution



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

And now to determine [itex]\Psi(x,t)[/itex], 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.
 
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  • #2
KaiserBrandon said:

Homework Statement


start with the wave function

[itex]\Psi(x,0) = Ae^{-cx^{2}}e^{ikx}[/itex]

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

i) Normalize [itex]\Psi(x,0)[/itex]
ii) Determine [itex]\Psi(x,t)[/itex] and [itex]|\Psi(x,t)|^{2}[/itex]

Homework Equations


The Attempt at a Solution



I normalized it to get [itex]A = (\frac{2c}{\pi})^{1/4}[/itex]
That's what I got :approve:
And now to determine [itex]\Psi(x,t)[/itex], 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. :confused:

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, [itex] \psi = \Psi (x, 0) [/itex].)
[tex] \left| \psi \right> = \sum_i c_i \left| \psi_i \right> [/tex]
where [itex] \left| \psi_i \right> [/itex] are the energy eigenvectors.
You can find the [itex] c_i [/itex]s via
[tex] c_i = \left< \psi_i | \psi \right>. [/tex]
Where did I get that last equation you may ask? Note that from a previous equation,
[tex] \left| \psi \right> = \sum_i c_i \left| \psi_i \right> [/tex]
and bringing to the bra of each energy eigenvector (denoted this time with a k subscript) to each side of the equation,
[tex] \left< \psi_k | \psi \right> = \left< \psi_k \right| \left( \sum_i c_i \left| \psi_i \right> \right) [/tex]
[tex] = c_0 \left< \psi_k | \psi_0 \right> + c_1 \left< \psi_k | \psi_1 \right> + c_2 \left< \psi_k | \psi_2 \right> . \ . \ .[/tex]
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, [itex] \left< \psi_k | \psi_i \right> = 1[/itex]. Thus,
[tex] \left< \psi_i | \psi \right> = c_i. [/tex]
But of course, you can't really solve for the [itex] c_i [/itex]s without knowing the general solution to the particular potential at hand, since you'll need to know that to find the [itex] \psi_i [/itex]s. Good luck! :wink:

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

1. What is meant by "scattering" in the context of gaussian wave packets?

Scattering refers to the interaction between a wave packet and a potential, causing the wave packet to change direction or spread out as it passes through the potential.

2. How is a gaussian wave packet typically described in scattering experiments?

A gaussian wave packet is often described by its amplitude, width, and central momentum, which determine its shape and behavior as it moves through a potential.

3. What role does the potential play in the scattering of a gaussian wave packet?

The potential acts as an obstacle or barrier for the wave packet, causing it to change direction or spread out as it passes through. Different potentials can result in different scattering patterns.

4. Can a gaussian wave packet be scattered without encountering a potential?

Yes, a gaussian wave packet can also be scattered by other particles or by imperfections in the medium it is traveling through, without encountering a potential barrier.

5. What factors can affect the scattering of a gaussian wave packet at a potential?

The shape and strength of the potential, the initial properties of the wave packet (such as its width and momentum), and the angle of incidence can all affect the scattering behavior of a gaussian wave packet at a potential.

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