Determining Momentum from Wavefunction

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SUMMARY

This discussion focuses on determining the momentum in a two-dimensional quantum system using wavefunction values and eigenergies. The primary method suggested is to perform an inverse Fourier transform on the position-space wavefunction to obtain the momentum-space wavefunction, which encapsulates all necessary momentum information. The conversation highlights the probabilistic nature of quantum mechanics, specifically referencing the Heisenberg uncertainty principle, which complicates direct momentum calculations. Additionally, the user mentions utilizing the Python package Kwant for simulating a graphene system with specific boundary conditions.

PREREQUISITES
  • Understanding of quantum mechanics principles, particularly wavefunctions and eigenstates.
  • Familiarity with Fourier transforms in the context of quantum mechanics.
  • Basic knowledge of the Heisenberg uncertainty principle.
  • Experience with the Python package Kwant for quantum simulations.
NEXT STEPS
  • Study the process of inverse Fourier transforming wavefunctions to obtain momentum-space representations.
  • Explore the Heisenberg uncertainty principle and its implications for momentum calculations in quantum systems.
  • Learn about numerical methods for analyzing specific wavefunctions in quantum mechanics.
  • Investigate advanced quantum simulations using the Kwant package, focusing on boundary conditions and their effects on momentum states.
USEFUL FOR

Quantum physicists, computational physicists, and researchers working on quantum simulations, particularly those interested in graphene and advanced quantum systems.

FermiDIrac19
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TL;DR
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.
 
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FermiDIrac19 said:
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?
 
PeroK said:
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.
 
FermiDIrac19 said:
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?
 
PeroK said:
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.
 
FermiDIrac19 said:
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.
 
PeroK said:
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.
 
FermiDIrac19 said:
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.
 
PeroK said:
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?
 
  • #10
FermiDIrac19 said:
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.
 
  • #11
PeroK said:
No. This is not QM.
Screenshot 2021-08-06 215307.png


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

(sorry for the image, the latex didn't work)
 
  • #12
FermiDIrac19 said:
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.
 
  • #13
PeroK said:
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?
 
  • #14
FermiDIrac19 said:
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!
 

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