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Topics: Wave functions, probability amplitude,
probability distribution, energy eigenvalues, hydrogen atom, quantum number, energy
eigenstate, principle quantum number. Pre-requisite skills: An understanding of basic quantum physics and the meaning of the wave function and eigenvalue problem. Approximate completion time: Under an hour. |
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Provide sufficient detail to verify that the assignment was completed in a meaningful manner. |
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Applet by Wolfgang Christian |
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1. This applet plots the probability amplitude, not the probability density, of the elecron of a hydrogenic atom as a function of the radial distance from the nucleus. Notice that for l = 0 the probability amplitude is nonzero at the origin. Because this system has spherical symmetry, the probability-amplitude-squared must be multipled by r, the radial coordinate, in order to obtain the probability density.
4. For a given value of l, the quantum number m can only take on certain values. What are those values? Again, use the applet to test your answer. 5. For the hydrogenic atom, the energy levels only depend on the principal quantum number n. Since a wave function is defined according to all three quantum numbers, and not just n, this means that there are often a large number of different wave functions that correspond to the same energy level. We call this number the degeneracy of the energy level.
6. A node is a point in the wave function that is 0, not counting the ends.
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