4(a) Write down the 7-dimensional TDSE for the hydrogen atom in pos...

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4(a) Write down the 7-dimensional TDSE for the hydrogen atom in position-space. Explain how the separation of the center-of-mass and relative position variables converts it into two 4-dimensional TDSE's. Explain why solving the center-of-mass TDSE leads to the 3d Gaussian Packet behavior of the center-of-mass. Explain why the radial variable r in the relative position TDSE can be separated from the angular variables 0 and 0. Explain how this separation of r from 0 and 0 leads to the effective potential for the hydrogen atom. (b) Sketch the energy levels of the hydrogen atom for n = 1,2,3, 4 and 5 versus n and l. Label the levels both spectroscopically (1s, 2s, 2p, ) and by their n and l values. Indicate the m degeneracy of each l level and add up the total degeneracy for each n level. (c) Make a composite sketch showing the l = 0,1,2, and 3 effective potentials for the hydrogen atom. Add the locations of the n = 1,2,3, and 4 energy levels to your effective potential sketch, and explain how the levels in the different wells are lined up. (d) Sketch just the l = 0 effective potential. Add the locations of the n = 1,2,3, and 4 energy levels to your effective potential sketch. Sketch the radial wavefunctions for each of these energy levels and label each wavefunction with the appropriate Rni designation. Sketch the corresponding probability distributions per unit volume I I dV. Sketch the corresponding radial probability distributions 4772 142/dr. (e) Sketch just the l = 1 effective potential. Add the locations of the n = 1,2,3, and 4 energy levels to your effective potential sketch. Sketch the radial wavefunctions for each of these energy levels and label each wavefunction with the appropriate Rni designation. Sketch the corresponding per volume and radial probability distributions. (f) Sketch just the l = 2 effective potential. Add the locations of the n = 1,2,3, and 4 energy levels to your effective potential sketch. Sketch the radial wavefunctions for each of these energy levels and label each wavefunction with the appropriate Rnl designation. Sketch the corresponding per volume and radial probability distributions. (g) Sketch the spherical harmonics for l = 0, 1 and 2. Be sure to show all of the allowed values of m. Explain the general rule for how many values of m there are for each value of 1. Explain how the spherical harmonics vary with 0. Explain the probability current for these spherical harmonics and explain the relationship between this probability current and the semiclassical vector model. Sketch the probability distributions for these spherical harmonics. Sketch the probability density versus 0 for the m = 0 and m = 1 cases above and explain in words how these probability densities vary with 0.

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