The Variational Method

This presentation shows a technique of how to solve for the approximate ground state energy using Schrodinger Equation in which the solution for wave function is not on hand

1. The Variational Method
James Salveo L. Olarve
PHYDSPHY, DLSU-M
January 29, 2010

2. This presentation shows a technique of how to solve for
the approximate ground state energy using Schrodinger
Equation in which the solution for wave function is not
on hand.
The intended reader of this presentation were physics
students. The author already assumed that the reader
knows Dirac Braket notation.
This presentation was made to facilitate learning in
quantum mechanics.
INTRODUCTION

3. Consider a quantum system described by a
Hamiltonian with a series of eigenstates with
corresponding energy eigenvalues
The Time-Independent Schrödinger Equation:
iii
EH yy=
Suppose you want to calculate the ground-state
energy for a system described by the Hamiltonian,
but you are unable to solve the Time-Independent
Schrödinger Equation.
TASK:

4. THEOREM:
HHE
g
º£ yy
That is , the expectation value of in the
state is certain to overestimate the
ground-state energy. Of course if just
happens to be one of the excited states,
then obviously exceeds ; but the
theorem says that the same holds for any
psi whatsoever
H
y
y
H
g
E

5. PROOF:
Since the (unknown) eigenfunctions of form a complete
set, we can express as linear combination of them:y
H
nnn
n
nntrial EHc yyyy ==å with
åå=
n
nn
m
mm cc yyyy
nmn
n
m
m
cc yyåå=
*
2
*
1 ååå ==
n
n
mn
mnnm cccd
Assuming the eigenfunctions have been orthonormalized
,normalized is Sincey

6. The expectation value for the Hamiltonian choosing
the trial function will be
åå=
n
nn
m
mm
cHcH yy
nmnn
n
m
m
cEc yyåå=
*
2
n
n
ncEå=
hence and , so
,eigenvaluesmallest the,definitionby is,energy state-ground But the
ngEE£
g
n
ng EcEH =³å
2

7. Reference
Introduction to Quantum Mechanics, David J.
Griffiths, 1994.