Chem. 540, Fall 2008
Instructor: Nancy Makri
Computer Assignment 3
The Variational Principle
In this assignment you will apply the variational principle to calculate approximations to the ground
state energy of one-dimensional systems described by the Hamiltonian
2
ˆ
ˆˆ
( )
2
p
H V x
m
.
Use the (un-normalized) trial function
2
( ) x
x e
For each of the potentials
(a)
2 2
1
( ) 2
V x m x
(b)
4
1
( ) 2
V x x
use a symbolic algebra program to calculate the expectation value of the Hamiltonian (the energy) with
respect to this trial function, evaluate the first derivative of this expectation value with respect to the
nonlinear variational parameter
, and set this derivative equal to zero to find the optimal value of
. To
confirm that this is indeed a minimum, plot the energy as a function of
. How does your energy compare
to the exact result (in the case of the harmonic potential) or (in the case of the quartic potential) the result
you obtained from the basis set expansion in the last assignment? What can you say about the
wavefunction that minimizes the energy?