Computational Physics - Department of Physics

11.4 Improved Monte Carlo Integration 365

Why care at all and not be content with just a change of variables in cases where that
is needed? Below we show several examples of how to improve a Monte Carlo integration
through smarter choices of PDFs which render the integrand smoother. However one classic
example from quantum mechanics illustrates the need for a good sampling function.
In quantum mechanics, the probability distribution function is given byp(x) =Ψ(x)∗Ψ(x),
whereΨ(x)is the eigenfunction arising from the solution of e.g., the time-independent
Schrödinger equation. IfΨ(x)is an eigenfunction, the corresponding energy eigenvalue is
given by
H(x)Ψ(x) =EΨ(x),

whereH(x)is the hamiltonian under consideration. The expectation value ofH, assuming that
the quantum mechanical PDF is normalized, is given by

〈H〉=

∫

dxΨ(x)∗H(x)Ψ(x).

We could insertΨ(x)/Ψ(x)right to the left ofHand rewrite the last equation as

〈H〉=

∫

dxΨ(x)∗Ψ(x)H(x)

Ψ(x)

Ψ(x), (11.22)

or
〈H〉=

∫

dxp(x)H ̃(x),

which is on the form of an expectation value with

H ̃(x) =H(x)

Ψ(x)

Ψ(x).

The crucial point to note is that ifΨ(x)is the exact eigenfunction itself with eigenvalueE,
thenH ̃(x)reduces just to the constantEand we have

〈H〉=

∫

dxp(x)E=E,

sincep(x)is normalized.
However,in most cases of interest we do not have the exactΨ. But if we have made a
clever choice forΨ(x), the expressionH ̃(x)exhibits a smooth behavior in the neighbourhood
of the exact solution. The above example encompasses the main essence of the Monte Carlo
philosophy. It is a trial approach, where intelligent guesses lead to hopefully better results.

11.4.1Change of Variables

The starting point is always the uniform distribution

p(x)dx=

{

dx 0 ≤x≤ 1

0 else

withp(x) = 1 and satisfying ∫∞

−∞

p(x)dx= 1.

All random number generators provided in the program library generate numbers in this
domain.