Quantum Mechanics Physics 215

Homework 5. Due in Class Friday March 10

1. One might be tempted to conclude from the lack of angular momentum in the ground state of the hydrogen atom that the electron is stationary. Show that this is not so by calculating the probability that the electron's momentum, if measured, would be found to line in a momentum element centered at momentum . What are electron's mean kinetic and potential energy? Show that the Virial Theorem is satisfied.
2. The electron current density is defined as:

   

   where is the electron mass. (Note: e > 0 and the charge of the electron is -e.)

   (a) Evaluate the current density for the n=2, l=1, m=-1 state of the hydrogen atom as a function of position. (It is particularly convenient to express the current in spherical components). Sketch a picture of the flow of current.

   (b) Calculate the current flowing in a ring of cross section dA and the magnetic moment it produces (using classical electromagnetic theory). Integrate to find the entire magnetic moment produced by the current distribution.

   (c) How do your answers above change for the n=2, l=1, m=1 state of hydrogen? Interpret the difference physically.

   (d) Obtain the general result for the current density and total (integrated) magnetic moment of hydrogen with arbitrary n, l and m.

3. Consider the following set of expectation values for the electron radius in the hydrogen atom:

   

   (a) Derive the following recurrence relation:

   

   where is the Bohr radius. This result is valid when k > -(2l + 1).

   Hint: First, show that the radial equation can be written in the following form:

   

   where is a suitably rescaled radial variable. Multiply this equation by and also by and partially integrate the two results. One can then obtain a recurrence relation for

   (b) Evaluate and .

   Hint: In evaluating for k=-1 use the Virial Theorem. For k=1 and k=2, use the results of part (a).

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