Computational Physics

40 The variational method for the Schrödinger equation

(Note that, in general, the lowest-but-one variational eigenstate 

1 is not

perpendicular toφ 0 so this result does not guarantee 1 ≥λ 1 .)

(c) Consider a vector ′ 1 =α

 0 +β

 1 which is perpendicular toφ 0. From (b) it is

clear that〈 ′ 1 |H| ′ 1 〉/〈 ′ 1 | ′ 1 〉≥λ 1. Show that

〈 ′ 1 |H| ′ 1 〉

〈 ′ 1 | ′ 1 〉

=

|α|^20 +|β|^21

|α|^2 +|β|^2

and that from this it follows that 1 ≥λ 1. This result can be generalised for

higher states.

3.2 The overlap matrixSis defined as

Spq=〈χp|χq〉.

Consider a vectorψthat can be expanded in the basisχpas:

ψ=

∑

p

Cpχp.

(a) Supposeψis normalised. Show thatCthen satisfies:

∑

pq

Cp∗SpqCq=1.

(b) Show that the eigenvalues ofSare positive.

3.3 [C] In this problem, it is assumed that a routine for diagonalising a real, symmetric
matrix is available.
(a) [C] Using a library routine for diagonalising a real, symmetric matrix, write a
routine which, given the overlap matrixS, generates a matrixVwhich bringsS
to unit form:
V†SV=I.
(b) [C] Write a routine which uses the matrixVto produce the solutions
(eigenvectors and eigenvalues) to the generalised eigenvalue problem:
HC=ESC.
The resulting routines can be used in the programs ofSections 3.2.1and3.2.2.

3.4 [C] The potential for a finite well is given by

V(x)=

{

0 for|x|>|a|

−V 0 for|x|≤|a|

In this problem, we determine the bound solutions to the Schrödinger equation using

plane waves on the interval(−L,+L)as basis functions:

ψn(x)= 1 /

√

2 Leiknx

with

kn=±

nπ

L

, n=0, 1,...