Physics 252 Pledged Homework

 

 

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.

Solving f"(x) = (V(x) -E)f(x) by the leapfrog method.

 

Take a set of points x0, x1, x2, x3, ... as usual equally spaced dx apart, so xi+1 = xi + 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 x1+1/2 which is at x1 + 0.5dx.

 

The reason for this procedure is as follows: if we know f'(x) at x1+1/2, and f(x) at x1, then to a very good approximation f(x2) = f(x1) + dx*f'(x1+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'(x2+1/2) = f'(x1+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:

A

B

C

D

E

posn step

potl

wavefn

2nd_deriv

deriv at 1/2

x j

v(x j)-E

f(x j)

f"(xj)

f'(xj+1/2)

 

 

(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.

 

2. In the uncertainty principle lecture, we gave an estimate of the size of the hydrogen atom by minimizing total energy and using the uncertainty principle. Consider a vee-shaped potential, V(x) = -bx for x <0, V(x) = bx for x³ 0. Show that the energy of the lowest state is of order (h 2b2/m)1/3. (French, QM, 8-9)

 

Use your spreadsheet for this potential for a few values of b to see if this b dependence is correct.

 

3. (a) The three l = 1 angular wave functions for the hydrogen atom can be taken to be cosq , sinq sinj and sinq cosj . Explain with diagrams how the probability distribution for each of these wave functions varies over the sphere. Include a plot of cos2q as a function of q going around a circle. (This can be done with Excel, using the "radar" plot.)

 

(b) Prove that the sum of the probabilities for these three wave functions is constant over the sphere, and show qualitatively that this fits in with the distributions you have drawn.

 

 

4. (a) Develop a spreadsheet to solve the radial equation for hydrogen atom Schrödinger wave functions. On the top sheet, you need to be able to enter the energy, the angular momentum quantum number l, the initial value and derivative of the wave function u(r ) (see notes for notation), the step spacing dr , and it is useful to have an adjustable scale factor multiplying (V(r ) -E), so that it can be easily seen on the same graph as the wave function.

 

(b) Plot the first five l = 0 wave functions. Find the energies. Roughly estimate the average distance of the electron from the origin for each n.

 

(c) Plot the wave functions for n = 4 for all possible values of l. What degeneracies do you find? Estimate the average distance of the electron from the origin for each wave function in your set.