Computational Physics - Department of Physics

460 14 Quantum Monte Carlo Methods

14.2.2.2 Postulate II

The only possible outcome of an ideal measurement of the physical quantityAare the eigen-
values of the corresponding quantum mechanical operatorÂ,

Âψν=aνψν,

resulting in the eigenvaluesa 1 ,a 2 ,a 3 ,···as the only outcomes of a measurement. The corre-
sponding eigenstatesψ 1 ,ψ 2 ,ψ 3 ···contain all relevant information about the system.

14.2.2.3 Postulate III

AssumeΦis a linear combination of the eigenfunctionsψνforÂ,

Φ=c 1 ψ 1 +c 2 ψ 2 +···=∑

ν

cνψν.

The eigenfunctions are orthogonal and we get

cν=

∫

(Φ)∗ψνdτ.

From this we can formulate the third postulate:

When the eigenfunction isΦ, the probability of obtaining the valueaνas the outcome of a
measurement of the physical quantityAis given by|cν|^2 andψνis an eigenfunction ofÂwith
eigenvalueaν.
As a consequence one can show that when a quantal system is in the stateΦ, the mean
value or expectation value of a physical quantityA(r,p)is given by

〈A〉=

∫

(Φ)∗̂A(r,−ih ̄▽)Φdτ.

We have assumed thatΦhas been normalized, viz.,
∫
(Φ)∗Φdτ= 1. Else

〈A〉=

∫

∫(Φ)∗ÂΦdτ

(Φ)∗Φdτ

14.2.2.4 Postulate IV

The time development of a quantal system is given by

ih ̄

∂Ψ

∂t

=ĤΨ,

withĤthe quantal Hamiltonian operator for the system.

14.3 First Encounter with the Variational Monte Carlo Method

The required Monte Carlo techniques for variational Monte Carlo are conceptually simple,
but the practical application may turn out to be rather tedious and complex, relying on a
good starting point for the variational wave functions. These wave functions should include