(1) Write a subroutine integ.f that will use the trapezoidal

rule to evaluate the integral of a function.

Use it to confirm that the harmonic oscillator wavefunctions

you have in your subroutine are normalized and orthogonal

(2) Consider the quartic potential

V(x) = 0.5x**2 + beta*x**4

Use matrix methods to solve for the eigenvalues and eigenvectors

of this system

Plot out the lowest five wavefunctions!

In doing this, you can use the following steps as a guide:

(a) Do a number of beta values from ZERO to 0.25.

(b) Set up the Hamiltonian matrix elements as discussed in

class:

H_ij = Integral dx [ phi(i) H_{h.o.} phi(j) + phi(i) beta*x**4 phi(j) ]

where H_{h.o.} is the Harmonic oscillator Hamiltonian

made up of its kinetic energy part (second derivative) and the

potential part V(x)=0.5*x**2

(c) Work out the eigenvalues and eigenvectors for the

case beta=0.

Compare this to what we discussed should be the answer in class.

Come collect your $20 if you can

(d) Now go to non-zero beta values. Evaluate all the integrals,

and your Hamiltonian matrix should be symmetric (check!), but

non-diagonal. Use eigen.f to calculate eigenvalues and

eigenvectors. Use these eigenvectors to generate wavefunctions

for the QUARTIC OSCILLATOR.