1. Read the notes below on constructing a "leapfrog" spreadsheet to solve Schrödinger's equation for a potential, and test it out by finding the first few eigenvalues (bound state energies) for the simple harmonic oscillator potential.

Take a set of points x₀, x₁, x₂, x₃, ... as usual equally spaced dx apart, so x_i+1 = x_i + dx.

We shall find f(x) and its second derivative f"(x) at this series of points, using the value of the first derivative f(x) at the points midway between this set of points, for example at the point x_1+1/2 which is at x₁ + 0.5dx.

The reason for this procedure is as follows: if we know f'(x) at x_1+1/2, and f(x) at x₁, then to a very good approximation f(x₂) = f(x₁) + dx*f'(x_1+1/2), since this is using the slope of the function midway between x₁ and x₂ as a straight line approximation to the function in that interval.

We actually find f'(x) at these midpoints by using the value of f"(x) at the original set of points. It's really the same trick: f'(x_2+1/2) = f'(x_1+1/2) + dx* f"(x₂). Again, this approximates a function with a straight line having the slope in the middle of the interval.

The method is implemented as follows:

Take five columns of the spreadsheet and label them as below:

(You don't need to use A, B, C, D, E -- I just did that for convenience here)

The first two columns are filled in as in the previous spreadsheet.

The initial entry (which we take to be row 3) in the third column is f(x₀), in D3 (say) =B3*C3 (using f" = (V-E)f), in E3 =f'(x₀) + 0.5*dx*D3.

Going on to the fourth row, in C4 put =C3 + dx*E3 ... and you figure out the rest.

Once you have set up this spreadsheet, and got it to run, use it to find the first three energy levels of a harmonic oscillator. Compare your results with:
(a) the exact answer,
(b) the results using the simpler spreadsheet. State approximately what level of accuracy the two spreadsheets give for these problems. How is this improved if you double the number of rows used in the spreadsheet?

2. Square Wells

(a)Use the new spreadsheet, or the one from last week's homework if you prefer, to find the lowest four energy levels in a one-dimensional potential that consists of two square wells, each of depth 20, and width 3.14159, a distance S apart. Find how the energy difference between the two lowest energy levels varies with S: find it for S = 0.1, 1, 5, 10. How does the energy in these lowest states relate to the energy levels of a particle in one of these wells by itself? Plot some of the wavefunctions.

(b)An ionized hydrogen molecule looks to the single electron orbiting the two protons like a double well. Do you think the wavefunction in the lowest state is odd or even about the central point? Explain.

(c)Now consider a potential in one dimension which has three such wells. Take the separation S=1. Find the lowest energy wavefunction, and discuss the energy difference to the next wavefunctions. What happens as S is varied? This model is a first attempt at understanding electronic energy levels in metals.

3. Muonic atoms

Recall that the muon is a particle having the same charge as an electron, but with a mass 207 times greater. The muon has a lifetime of only two microseconds, but that's long enough, if trapped in matter, for it to drop through a succession of Bohr-type orbits around a nucleus, emitting photons corresponding to the energy differences between the orbits.

(a) Suppose a muon is orbiting a hydrogen nucleus, and drops from n = 3 to n = 2 then to n = 1. What would be the frequency of the photons emitted? What kind of photons are these? (Take the nucleus to have infinite mass.)

(b) Describe how your answer would change if you took into account the fact that the nucleus has a finite mass. Is this more important for this muonic atom than for a regular hydrogen atom?

(c) Will the presence of the muon affect the atom's electron significantly?

(d) Now consider a muon captured by a Uranium nucleus, of charge 92 and nuclear radius 7.10^-15 meters. Find the radii of the lowest three Bohr orbits. Don't just use the formula in this part - derive from first principles!