Download QHarmonic_Osc

Physics 139A Introduction to Quantum Mechanics Spring 2007
From http://scipp.ucsc.edu/~johnson/phys139a/homework.html and http://scipp.ucsc.edu/~johnson/
Numerical solution of the harmonic oscillator
This problem illustrates a simple method of numerical integration, called “wag the dog” by
Griffiths, that can be used to find energy eigenvalues and their eigenfunctions (in graphical form)
for 1-D problems. You will use a spreadsheet (Excel or Open-Office Calc) to do the calculation.
The goal is to solve equation 2.72 in the textbook:
d 2
  2  K *
2
d
2E
K




m
x

You must do for this problem just one eigenvalue, corresponding to the second excited state (n =2;
see Figure 2.6 for the n = 0 case). The answer, of course, is K =5, but let’s pretend not to know that.
Since this is an even state, it is enough to solve for just ξ ≥ 0, starting with initial values at ξ = 0,
ψ(0) = 1 and ψ '(0) = 0.
The starting value for the wave function is arbitrary, and we will not worry about normalization in
this problem. The starting point for the derivative is exactly right for an even function. The idea is
to start at ξ = 0 and take small steps of size ε in the positive direction. From the first terms in the
Taylor series expansion, you can predict new values of ψ and ψ' at the end of a step according to
           *     *  2
           * 
1
2
One just has to substitute into these equations     ( 2  K ) * and then they can be iterated to
graph the wave function.
This sort of numerical integration can be used in mechanics to predict a trajectory of a projectile,
for example, given the initial position and velocity, but here the goal is a bit different. This is a
boundary-value problem. You have to make sure that the wave function goes to zero as ξ→∞. That
will only happen for special values of K, i.e. the eigenvalues. So you step through trial values of K
until you find one that looks good. You’ll never get it exact. Due to numerical imprecision,
eventually your solution will blow up if you go out far enough in ξ. It is enough to make it look
right out to some reasonable distance beyond the classical limit (ξ= √5 in this case), so please make
your graphs cover just the range 0 ≤ξ ≤ 5.
Here is a recipe (to be understood and remembered for future assignments): use the first row to
label the first 3 columns in your spread sheet Xi, Psi, and Psi-prime. Place values for K =4.9 and ε=
0.01 somewhere off to the side, say in cells H1 and H2 respectively. Enter the starting values for ξ,
ψ, ψ' in the second row (cells A2, B2, and C2) just under your labels. Then, in the 3rd row enter the
formulas:
=A2+H$2 (in cell A3)
=B2+(C2+0.5*(A2^2-H$1)*B2*H$2)*H$2 (in cell B3)
=C2+(A2^2-H$1)*B2*H$2 (in cell C3)
The first of these formulas simply increments the value of ξ by the amount 0.01 for each successive
row. The second and third formulas calculate the new values of ψ and ψ', according to the equations
given above. The $ signs are needed to keep the references to K and ε from changing when you
copy these formulas. Now, select the 3 cells in the third row. Then hold the cursor on the lower
right hand corner of the selection box and drag downward to copy the formulas in the remaining
rows all the way down to at least row 502.
Then, select the first two columns and make an “x,y scatter” chart, which will be a plot of the wave
function. Go to the Insert menu and click on Chart. Then select the type “XY (Scatter)”. Place the
chart somewhat below the value for K. Then, start adjusting the value of K bit by bit, watching how
the wave function responds. Find a value that makes the wave function approach the horizontal axis
and stay there. Print just the first page of your spread sheet, showing the value of K, the plot, and
the 3 columns. Also, print it twice more, once with K=4.98 and again with K=5.00.
Note that your best eigenvalue will disagree somewhat with the analytic solution, because of the
numerical approximations used here, plus rounding errors. Better integration algorithms exist (such
as Runge-Kutta), but that’s not important for our purposes here.