I Determining Momentum from Wavefunction

[FermiDIrac19]

TL;DR Summary

Determining the momentum (2D) in a quantum system from the wavefunction values and the eigenergies.

The goal I am trying to achieve is to determine the momentum (2D) in a quantum system from the wavefunction values and the eigenergies. How would I go about this in a general manner? Any pointers to resources would be helpfull.

 

[PeroK]

Summary:: Determining the momentum (2D) in a quantum system from the wavefunction values and the eigenergies.

The goal I am trying to achieve is to determine the momentum (2D) in a quantum system from the wavefunction values and the eigenergies. How would I go about this in a general manner? Any pointers to resources would be helpfull.

How much QM do you know?

 

[FermiDIrac19]

How much QM do you know?

I took an introductory course on QM. But it seems I am missing some fundamental steps to understand this process.

 

[PeroK]

I took an introductory course on QM. But it seems I am missing some fundamental steps to understand this process.

When you say "determine the momentum" can you be mathematically precise about what you want to calculate?

 

[FermiDIrac19]

When you say "determine the momentum" can you be mathematically precise about what you want to calculate?

Well, specifically I want to determine what angle a state is in, in a quantum system. I thought this would be best determined from the x- and y- values of the momentum values of the system.

 

[PeroK]

Well, specifically I want to determine what angle a state is in, in a quantum system. I thought this would be best determined from the x- and y- values of the momentum values of the system.

If you know the position-space wavefunction, you can inverse Fourier transform it to get the momentum-space wavefunction. That tells you all there is to know about momentum for the given wavefunction.

Note that QM is probabilistic when it comes to position and momentum values that result from measurement.

 

[FermiDIrac19]

If you know the position-space wavefunction, you can inverse Fourier transform it to get the momentum-space wavefunction. That tells you all there is to know about momentum for the given wavefunction.

Note that QM is probabilistic when it comes to position and momentum values that result from measurement.

Thank you for the reply.

I should have specified that I am talking about wavefunction values in position space and determining the momentum computationally.

 

[PeroK]

Thank you for the reply.

I should have specified that I am talking about wavefunction values in position space and determining the momentum computationally.

There is no momentum to compute. That's why you have a wavefunction in QM instead of a definite position and momentum. The Heisenberg uncertainty principle and all that.

 

[FermiDIrac19]

There is no momentum to compute. That's why you have a wavefunction in QM instead of a definite position and momentum. The Heisenberg uncertainty principle and all that.

I'm confused. Could I not in a 2D eigenstate of a quantum system determine the momentum as an approxmation of a derivative in the 2D lattice of the wavefunction values?

 

[PeroK]

I'm confused. Could I not in a 2D eigenstate of a quantum system determine the momentum as an approxmation of a derivative in the 2D lattice of the wavefunction values?

No. This is not QM.

 

[FermiDIrac19]

No. This is not QM.

Can I not approximate this derivative to gain the expectaion value of momentum?

(sorry for the image, the latex didn't work)

 

[PeroK]

View attachment 287184

Can I not approximate this derivative to gain the expectaion value of momentum?

(sorry for the image, the latex didn't work)

It depends on the wavefunction. The alternative is to Fourier transform to momentum space.

Once you have a specific wavefunction then you use numerical methods.

 

[FermiDIrac19]

It depends on the wavefunction. The alternative is to Fourier transform to momentum space.

Once you have a specific wavefunction then you use numerical methods.

Thank you for your responses.

Do you have any resources I can use to read more about this?

I'm having trouble understanding the quantisation in my system.

To give more info: I have a system of graphene that I simulated in the python package kwant. This has open boundary condition in one direction and periodic boundary conditions in the other. From this I should be able to deduce the values of the momentum in each eigenstate using the momentum operator. How would I go about this?

 

[PeroK]

Thank you for your responses.

Do you have any resources I can use to read more about this?

I'm having trouble understanding the quantisation in my system.

To give more info: I have a system of graphene that I simulated in the python package kwant. This has open boundary condition in one direction and periodic boundary conditions in the other. From this I should be able to deduce the values of the momentum in each eigenstate using the momentum operator. How would I go about this?

A QM analysis of graphene sounds like quite an advanced specialist question!