For this example, we’ll explore 1 particle systems under the influence of an external force field with Potential
Where q1 is the charge of an electron, and q2 is the charge of a simulated atomic nucleus. Since we’re working in Atomic Units, the Potential reduces to
Where N is the number of protons in the nucleus (for Hydrogen N=1). The equation for the electron in a Hydrogen atom becomes:
The way to solve this is to express it in Spherical Coordinates, apply Separation of Variables (as usual) and lots of Algebra. Separable solutions wind up having the form:
There are many interesting avenues that we could explore here. For now, I’m just going to show a few of the possible states.
To start with, let’s see the ground state:
Next, let’s check out :
(7)Notice that this state does have nonzero fluid momentum, despite having a constant density (and being an energy eigenstate, where nothing is supposed to happen).
The actual swirling motion of the electron fluid explains why this state posses a magnetic moment. Here’s a plot of the momentum field.And here’s an animation showing the motion (all animations in this chapter take place over 500 Atomic Time units, or about 1 80 Trillionth of a second):
By combining Eigenstates, we can get some pretty impressive motion. Here are a few:
Below is . Spherically symmetric states bounce back and forth:
Below is . States without a magnetic moment do not spin, but tend to bounce up and down.:
Below is :
Below is . Note the similarity to the Classical system from Chapter 2. Not a coincidence.:
Below is . States with 2 different magnetic moments have sections of the fluid that spin at different rates.:
Below is .
For more animations, check out the Gallery.
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