How to graph Dirac equation, some complex numbers?

Click For Summary
SUMMARY

The discussion focuses on graphing solutions to the Dirac equation, specifically for a localized spin-up electron represented by a sum of plane wave solutions. It emphasizes the need for four complex numbers at each spacetime point and explores the implications of fixing time while graphing at specific spherical coordinates (θ=∏/2, r=R, and various φ values). The participants inquire about the necessity of graphing one or two rotations and whether the complex numbers, treated as 2D vectors, exhibit rotation when advancing in time. Additionally, the concept of the function as a higher-dimensional surface with potential topological twists is considered.

PREREQUISITES
  • Understanding of the Dirac equation and its solutions
  • Familiarity with complex numbers and their representation as vectors
  • Knowledge of spherical coordinates and their application in graphing
  • Basic concepts of topology related to higher-dimensional surfaces
NEXT STEPS
  • Research methods for visualizing solutions to the Dirac equation using software like Mathematica or MATLAB
  • Learn about the mathematical properties of complex numbers in quantum mechanics
  • Explore the implications of time evolution in quantum systems and its graphical representation
  • Investigate topological features in quantum field theories and their significance
USEFUL FOR

Physicists, mathematicians, and students interested in quantum mechanics, particularly those focusing on the graphical representation of wave functions and the Dirac equation.

Spinnor
Gold Member
Messages
2,231
Reaction score
419
Say we a have a sum of spin up plane wave solutions to the Dirac equation which represent the wave-function of a localized spin-up electron which is 90% likely to be found within a distance R of the origin of a spherical coordinate system. Four complex numbers at each spacetime point are needed? Fix time and graph this function at the points θ=∏/2, r=R, and phi=∏/2, ∏, 3∏/2, 2∏, ? 5∏/2, 3∏, 7∏/2, 4∏. Up is θ=0. Do we need to graph two rotations or one?

If we advance a very small "distance" forward (or backward) in time do the complex numbers considered as 2 dimensional vectors rotate a little?

Can the above function be considered some sort of higher dimensional surface? If so are there some sorts of topological "twists" to this function?

Thanks for any help!

Edit, the title of the thread should be "How to graph solutions to the Dirac equation, some complex numbers?" I reread the thread many times but not the title. Sorry.
 
Last edited:
Physics news on Phys.org

Similar threads

Undergrad Dirac equation in the hydrogen atom
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Dirac equation as one equation for one function
  • · Replies 40 ·
2
Replies
40
Views
5K
Graduate Getting particle/antiparticle solutions from the Dirac Equation
  • · Replies 6 ·
Replies
6
Views
2K
Graduate Graph sol. of Dirac eq., type "graph paper" needed?
  • · Replies 4 ·
Replies
4
Views
1K
Undergrad Spin conservation in the Dirac equation
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Born's rule, causality, and the Dirac equation
  • · Replies 17 ·
Replies
17
Views
3K
Graduate Dirac Equation computational complexity
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad Why does the Dirac equation lead to spin 1/2?
  • · Replies 8 ·
Replies
8
Views
7K
Graduate Solving the Radial Equation for the Dirac Hydrogen Atom Solution
  • · Replies 10 ·
Replies
10
Views
4K
Graduate Relation between Dirac's equation density matrix and current with spin
  • · Replies 1 ·
Replies
1
Views
2K