Physics 252 Pledged Homework
Due Friday April 23, Noon
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 x_{0}, x_{1}, x_{2}, x_{3}, ... 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 x_{1} + 0.5dx.
The reason for this procedure is as follows: if we know f'(x) at x_{1+1/2}, and f(x) at x_{1}, then to a very good approximation f(x_{2}) = f(x_{1}) + dx*f'(x_{1+1/2}), since this is using the slope of the function midway between x_{1} and x_{2} 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"(x_{2}). 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, extending last week's spreadsheet by two columns:
D |
E |
F |
G |
H |
(D15) position |
(E15) V(x)-E |
wavefn |
2nd deriv |
deriv at 1/2 |
x_{j} |
v(x _{j})-E |
f(x _{j}) |
f"(x_{j}) |
f'(x_{j+1/2}) |
The third row shown above is a "typical row".
The first two columns are filled in exactly as in the previous spreadsheet, with "position x" in D15, etc.
(Don't copy in the D15, that's just to show you where you are!)
So, D16 has 0, D17 is =D16+dx, etc.
Take V(x) = (1/2)Cx^{2}, since we shall first investigate a simple harmonic oscillator with spring constant C.
You should already have named cells for E and C, with some suitable values in them, say E = 1, C = 2, to give the spreadsheet something to do. (Note: C = 2 is a good choice for easy checking of accuracy, as you will see.)
In the third column, the entry in F16 is f(x_{0}) = f(0).
In the fourth column, G16 is =E16*F16 (using f" = (V-E)f).
In the fifth column H16 =f'(x_{0}) + 0.5*dx*G16. (Make sure you understand why this is!)
Going on to the fourth row, in F17 put =F16 + dx*H16. You can fill G17 using Schrödinger's equation.
What you put in H17 will test whether you are understanding what you are doing here.
HINT: don't just copy H16 -- that will give you the wrong answer! In fact, to figure out what to put in H17, it might be helpful to draw a horizontal line, mark off 0, 0.5dx, dx, 1.5dx, 2dx, ... then list below each point the cells from columns F, G, H corresponding to values at that point. Then reread the first four short paragraphs at the beginning of this question to be sure you understand the formulas you are entering in the cells.
Finding Simple Harmonic Oscillator Energy Levels More Accurately
1. Once you have set up this spreadsheet (for 600 rows), and got it to run, use it to find the first three energy levels of a harmonic oscillator.
This will be a lot quicker and easier if you automate the spreadsheet as described in the Finite Square Well Lecture!
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. Two 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.
Important hint: make sure you choose your origin so that the potential -- all of it -- is symmetrical about your origin!
(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.
3. A Deuteron Model
The deuteron, the nucleus of heavy hydrogen, is a bound state of a neutron and a proton. The orders of magnitude are correct if we take it that the proton sees the neutron as a square well (we'll make this one-dimensional for simplicity) of width 2X10^{-15} meters, and depth 10Mev.
(a)Use a spreadsheet to find how tightly bound the proton is in this well.
(b) Figure out a way to use your spreadsheet to find the total probability that in the lowest energy state the proton is actually outside the well.
(c) Determine whether or not there are any higher energy bound states in this model.
(d) Discuss whether the electrostatic force would be an important consideration if we replaced the neutron + proton by two protons.
4. A Quark Model
A popular theory of "elementary" particles, like the proton, is that they are actually made up of smaller objects, called quarks. In some important energy ranges, these quarks are bound together by a linear potential (so in the harmonic oscillator spreadsheet x^{2} would be replaced by just x). Also, some of these quarks are heavy enough that it's OK to treat the problem nonrelativistically. Again, we'll take the one-dimensional model, which has the right behavior.
(a) Find the first four energy levels.
(b) How does the energy level spacing vary with energy? Compare this with: the infinite square well, the simple harmonic oscillator, and the Bohr model for hydrogen.
(c) Show a wavefunction for a highly excited state, and discuss how the wavelength and average amplitude of the wavefunction vary with x.