Gaussian Wave packet

[gvetaw]

Hello guys,

@andrewdurkiewicz @ion201 @orbeckst So our problem is that we have a PDE to solve which is second ordered in space and first ordered in time. I was thinking that we should use the technique that we learned on Tuesday for for solving the diffusion eqn since Schrodinger eqn is parabolic as well. That would mean that we would use a lattice as demonstrated from the lecture notes of PDEs. As far as building the Gaussian wave packet goes I think we should take baby steps and first do a fourier transform that is one-dimensional and time independent. Once that works we can build a three-dimensional wave packet and then let it evolve with time by using a finite difference scheme. After those steps are complete we would then just simply add the results with an if statement to two basis for spin up or spin down as defined in the literature from the resource directory. Let me know if it sounds like a plan.

[orbeckst]

There are specialized techniques for solving the Schrödinger equation, and just doing what we did in class is not enough. This is one of the cases where the project would lead you to things beyond what we did in class (which was one of the intentions of doing projects).

The Computational Physics book has some solutions (essentially, Crank-Nicholson with a split across real and imaginary parts of the wave function and tricks to keep the solution finite/normalized).

[andrewdurkiewicz]

What about solving in momentum space?

[orbeckst]

You'll have to go and read – I don't know the answer.

[andrewdurkiewicz]

Professor, I am confused as to one thing. Ive been doing a lot of reading up on this and here is where we are at with the hamiltonian relationship to the wave packet. The hamiltonian matrix is defined for the regions (taking the reference frame to be moving along with the beam. This makes the momentum part of the hamiltonian go to zero):

from which the corresponding energy eigenstates are as follows:

which, gives us the state:

where,

what confuses me about this type of notation, is whether or not we are choosing our Psi(up) and Psi(down)? My guess is that this is completely dependent on what we choose initially. Am I correct in this?

[gvetaw]

If you go the script I have been working on all day you will notice that it is the exact same psi I already have defined.

[andrewdurkiewicz]

sorry, I have not git pulled on this computer. But if so, can you answer my question? Do we define Psi(up) and Psi(down)?

[gvetaw]

you need to superimpose the states before the electron goes through the magnet. Before it goes through the magnet it can have BOTH states. When it leaves the magnet it is in one of the states. It is best two separate the states out and use an if state.

[andrewdurkiewicz]

but before it goes in, it is a free particle. It does not matter what state it is in. It will go straight down the y axis until it hits the magnetic field.

[gvetaw]

Not quite sure what you mean

[andrewdurkiewicz]

Basically, we don't really need to worry about the spin until its measured at t = 0. (magnet goes from t = 0 to t = T) The wave accelerates in the x and z direction due to the gradient of the potential of the hamiltonian. This only exist in a specific domain. So, we only really need to measure the spin immediately before it goes into the magnet. The result of this collapses the superposition into one state. Only one. We both did this. Nate and I have no clue how to do anything in momentum space. We are not familiar with it at all, and thus...I dont understand any of your code.

[andrewdurkiewicz]

I feel like we already have it all worked out. Find the expectation values for x and z with what we have right now after we normalize it. Then, in the program we just vary the x and z values, t = T (leaves the magnet). These values are what we want. I dont see any problem doing it this way. I dont see any with your way. All I know is that I wont be of much help in momentum space. I dont understand it.

[gvetaw]

The gaussian is in momentum space the spinor function and the psi is in position space. The code is full of comments and is clear what I am trying to do. I am going to proceed on with what I started and use the literature that has been in the resource directory from day one. The code follows the literature step by step. I am next going to use Oliver's suggestion for solving the Schordinger eqn

[gvetaw]

I just read the resource you provided so I don't see why there is an issue here.

[gvetaw]

I feel that we had this stated two weeks ago that we were going to follow the uploaded literature and agreed to use the Crank-Nicholson yesterday. Out of the blue with one week to finish you guys want to go in a different direction.

[andrewdurkiewicz]

We have to meet up tomorrow and talk about it. Because I don't think I understand.

[andrewdurkiewicz]

So, when we graph this , how are we supposed to go from momentum to our a position graph. We should get a probability density that matches an oval. I you have graphed a gaussian in terms of momentum. We want position (to match the original experiment).

[gvetaw]

Could you explain the probability density that matches an oval? The gaussian which i think you are asking about is the spinor function. The spinor function is the inverse Fourier transform of the gaussian, g, in momentum space. Once you do the transform,which numpy provides a module for, gives a gaussian in position space with an complex exponential factor. This doesn't matter because the probability density is real i.e absolute value squared. So if the width of the gaussian is very wide in momentum space the width of the gaussian is narrow in position space. You can play around with beta and see this.

[gvetaw]

Sure we can meet tomorrow. What time?

[andrewdurkiewicz]

The spin is either 1/2 or -1/2 in both the s_x and s_z so we should see the probability density for x split into two peaks. This is what i expect at least. One peak for 1/2 another for -1/2.

[andrewdurkiewicz]

before 10:30 if you can

[gvetaw]

I do think it really matters as long as you stay consistent. The literature used 1/(2pi) but if you like we can use 1/sqrt(2pi). We are not using the pauli matrix for the x component from what I read we are using only sigma z and it's basis. Thus the electron first has a superimposition of both spin up and spin down state from the z basis, goes through an entanglement process in the magnetic field, and comes out in one state. The gaussian only states that the electron is localized in space and highly peaked around some position. But if there were two electrons you would see what you are thinking of because you have two electrons with each with a psi

[gvetaw]

Also the electron can only be in two states either spin up or spin down and the average or expectation is zero.

[ion201]

Before we get into momentum space, etc; I think we should approach the simpler problem of only this Hamiltonian perturbation. This is a much more straightforward approach to the problem and will save us some effort. I don't want to spend too much time brooding over momentum evolution in the Hamiltonian when it will not necessarily affect our results.

[andrewdurkiewicz]

Guys I am sorry, I am sick. Can we discuss and figure out today what road we are going down? Nate, did you discuss with Greg about the project path today in class?

[andrewdurkiewicz]

Greg, I am working all day tomorrow so that I don't work Friday. Ill be free to meet anytime Thursday-Sunday. Lets meet Thursday, figure out which path to take and break responsibilities up.