Chem. 540, Fall 2010
Instructor: Nancy Makri
Computer Assignment 3
Eigenstates of General Hamiltonians in 1d
In this assignment you will calculate energy eigenvalues using a symbolic algebra program, but
this time the Hamiltonian matrix is not readily available.
You will have to choose a basis set and
evaluate the matrix elements by performing the integrals.
The system is a particle of mass
1
m
=
in a potential
( )
V x
(to be specified below).
Thus, the
Hamiltonian has the form
2
ˆ
ˆ
ˆ
( )
2
p
H
V x
m
=
+
.
Your goal is to calculate the five lowest energy eigenfunctions and eigenvalues with accuracy of
0.1% in the energies.
For convenience I suggest using particleinabox basis functions.
You
should do this for two potentials, (i)
2
1
2
( )
V x
x
=
and (ii)
4
1
2
( )
V x
x
=
.
I recommend you write the
program with the harmonic potential in mind, in which case you know the answers, so you can
check things.
Once everything is ok, you could copy the program and replace the potential
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 Fall '08
 Mccall
 Matrices, Energy, Eigenvalue, eigenvector and eigenspace, Eigenfunction, basis functions

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