One can learn a lot about a quantum system by inspecting a plot of its single-particle orbitals. In this video, you learn how single-particle orbitals (wave functions) and their corresponding densities can be visualized.
Here is an exercise about this plotting. The explicit expression for the wave function of the 2px orbital of the hydrogen atom is: ax*exp(-r/2) (‘a’ is a constant, take it to be 1). The expression for the density of this orbital is the square of it (you can look up the full expressions here).
Go here for an animated plot of the wave function and here for an animated plot of the density for this orbital. These plots are actually videos, which help you to see the wave function and density “in four dimensions”. Make sure you understand how this explicit expression gives rise to all the plots you see.
Once you understand that, it will be straightforward to make the following graphs yourself:
Finally, use your graph of the density to explain why you see on the animation for the density at that website two isolated spheres for sufficiently high density values.
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A graph with the 2px wave function and 2px density along the x-axis and along the y=x line, for the Hydrogen atom, is shown in this file.
How do you understand from this graph that for higher densities the objects in the animation are more spherical and more separated from each other?
The clue is to realize that these three-dimensional objects are isosurfaces: at any point in space where the density has a particular value (indicated by the sliding horizontal line), that point is coloured. For rather small values of the density, there is a point on the isosurface along the x-axis and along the y=x line, at similar distances from the origin (and there are other points further away from the origin when the density values come down again). If you would draw this for several more lines, you would obtain the squeezed spheres you see in the animation, almost touching each other near the origin. For higher density values, however, there are no points on the isosurface along the x-axis any longer. The two objects will be farther apart, and less squeezed — they look more like spheres now.
Do as explained hereabove, and make the two graphs with 2 wave functions and 2 densities. Then use these graphs two explain why you see on the animated plot of the density two separated spheres for sufficiently large eigenvalues. Type that answer hereunder and submit (keep your pictures for comparison, they do not need to be submitted).
Attention: You will receive feedback with the correct plots after submitting. Don’t get tempted to submit a dummy answer just to see the feedback quickly. You get the most out of your learning when you submit an honest attempt, after which you can compare and understand possible differences between the feedback and your pictures.
This response will be awarded full points automatically, but it will be reviewed and possibly adjusted after submission.
Anything unclear for you about this topic? Discuss it with your colleagues in the chapter forum.