How to graph Dirac equation, some complex numbers?

In summary, the conversation discusses the use of spin up plane wave solutions to the Dirac equation to represent the wave-function of a localized spin-up electron within a spherical coordinate system. It is noted that four complex numbers are needed at each spacetime point, and the function can be graphed at specific points within the coordinate system. The question of whether one or two rotations are needed to graph the function is also raised, along with the possibility of the function being a higher dimensional surface with potential topological twists. The conversation concludes with the clarification of the title question and a request for further insight on the topic.
  • #1
Spinnor
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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.
 
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  • #2
@Spinnor did you find any more insight on this topic?
 

FAQ: How to graph Dirac equation, some complex numbers?

1. How do I plot the Dirac equation on a graph?

The Dirac equation is a complex-valued differential equation, so it cannot be plotted directly on a traditional x-y graph. Instead, it is typically visualized using a four-dimensional graph with three spatial dimensions and one time dimension.

2. What is the significance of the complex numbers in the Dirac equation?

The complex numbers in the Dirac equation represent quantities that have both a magnitude and a direction in space. They are essential for describing the behavior of fundamental particles, such as electrons, in quantum mechanics.

3. Can I use a computer program to graph the Dirac equation?

Yes, there are many computer programs available that can plot the Dirac equation in four-dimensional space. These programs use numerical methods to solve the equation and generate a visual representation of the resulting wave function.

4. Are there any simplifications or approximations that can be made to graph the Dirac equation?

Some simplifications can be made to the Dirac equation, such as reducing it to a two-dimensional graph by assuming a constant value for time. However, these simplifications may not accurately reflect the behavior of the equation in its full form.

5. Can the Dirac equation be graphed in real-time?

Graphing the Dirac equation in real-time is possible with the use of specialized software and hardware. However, due to the complexity of the equation and the need for numerical calculations, real-time graphing may not be feasible for all systems.

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