This applet shows the superposition of eigenfunctions of the two-dimensional Coulomb potential for energy quantum numbers n = 1,2,3,4 and the corresponding values of m. For a given n the sliders define the coefficients for m in the order -(n-1), -(n-2), ..., -1, 0, 1, ..., (n-1). All coefficients are scaled automatically to give a normalised wave function.

Start the simulation with the default values and compare the results for the real or imaginary part and the absolute value. Explain your findings.

The next experiment shows the superposition of eigenfunctions of different energies, but with the same angular momentum: First set all c4 sliders to 0, then set the sliders for n = 3, m = 2 and n = 4, m = 2 to 1. Start the simulation and again compare the results for the real or imaginary part and the absolute value.

Using a proper combination of eigenfunctions one should be able to get any start configuration. The following coefficients

m=-3 | m=-2 | m=-1 | m=0 | m=1 | m=2 | m=3 | |
---|---|---|---|---|---|---|---|

n=1 | 0.000403 | ||||||

n=2 | 0.0 | -0.022160 | 0.0 | ||||

n=3 | 0.027054 | -0.170473 | 0.304744 | -0.170473 | 0.027054 | ||

n=4 | 0.007128 | -0.089421 | 0.406419 | -0.661812 | 0.406419 | -0.089421 | 0.007128 |

are computed to approximate a polar Gaussian with the parameters

r0 | dr | φ0 | dφ | Lz |
---|---|---|---|---|

1.0e-9 | 3e-10 | 0 | 1 | 0 |

Enter these values, start the simulation and compare the result with the corresponding polar Gaussian.