Mathematical Methods for Physics and Engineering : A Comprehensive Guide

22.8 Adjustment of parameters

It is easily verified that functions (b), (c) and (d) all satisfy (22.30) but, so far as mimicking
the correct solution is concerned, we would expect from the figure that (b) would be
superior to the other two. The three evaluations are straightforward, using (22.22) and
(22.23):

λb=

∫ 1

0 (2−^2 x)

(^2) dx
∫ 1
0 (2x−x
(^2) ) (^2) dx

=

4 / 3

8 / 15

=2. 50

λc=

∫ 1

0 (3x

(^2) − 6 x+3) (^2) dx
∫ 1
0 (x
(^3) − 3 x (^2) +3x) (^2) dx

=

9 / 5

9 / 14

=2. 80

λd=

∫ 1

0 (π

(^2) /4) sin (^2) (πx)dx
∫ 1
0 sin
(^4) (πx/2)dx =
π^2 / 8
3 / 8

=3. 29.

We expected all evaluations to yield estimates greater than the lowest eigenvalue, 2.47,
and this is indeed so. From these trials alone we are able to say (only) thatλ 0 ≤ 2 .50.
As expected, the best approximation (b) to the true eigenfunction yields the lowest, and
therefore the best, upper bound onλ 0 .

We may generalise the work of this section to other differential equations of

the formLy=λρy,whereL=L†. In particular, one finds

λmin≤

I

J

≤λmax,

whereIandJare now given by

I=

∫b

a

y∗(Ly)dx and J=

∫b

a

ρy∗ydx. (22.31)

It is straightforward to show that, for the special case of the Sturm–Liouville

equation, for which

Ly=−(py′)′−qy,

the expression forIin (22.31) leads to (22.22).

22.8 Adjustment of parameters

Instead of trying to estimateλ 0 by selecting a large number of different trial

functions, we may also use trial functions that include one or more parameters

which themselves may be adjusted to give the lowest value toλ=I/Jand

hence the best estimate ofλ 0. The justification for this method comes from the

knowledge that no matter what form of function is chosen, nor what values are

assigned to the parameters, provided the boundary conditions are satisfiedλcan

never be less than the requiredλ 0.

To illustrate this method an example from quantum mechanics will be used.

The time-independent Schr ̈odinger equation is formally written as the eigenvalue

equationHψ=Eψ,whereHis a linear operator,ψthe wavefunction describing

a quantum mechanical system andE the energy of the system. The energy