Page 96 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

3.7 Postulates of quantum mechanics 87

For multi-particle systems with cartesian coordinates r 1 , r 2 , ... , the classical
function A(r 1 , r 2 , ... , p 1 , p 2 , ...) possesses the corresponding operator
^
A(r 1 , r 2 , ... ,("=i)= 1 ,("=i)= 2 , ...) where = k is the gradient with respect to
r k . For non-cartesian coordinates, the construction of the quantum-mechanical
^
operator A is more complex and is not presented here.
The classical function A is an observable, meaning that it is a physically
measurable property of the system. For example, for a one-particle system the
^
Hamiltonian operator H corresponding to the classical Hamiltonian function
p 2
H(r, p) ˆ ‡ V(r)
2m
:
2
2
2
2
where p ˆ p p ˆ p ‡ p ‡ p ,is
x
y
z
" 2
^ 2
H ˆÿ = ‡ V(r)
2m
^
The linear operator H is easily shown to be hermitian.
Measurement of observable properties
The third postulate relates to the measurement of observable properties. Every
individual measurement of a physical observable A yields an eigenvalue ë i of
^
the operator A. The eigenvalues are given by
^
Ajiiˆ ë i jii (3:45)
^
^
where jii are the orthonormal eigenkets of A. Since A is hermitian, the
^
eigenvalues are all real. It is essential for the theory that A is hermitian because
any measured quantity must, of course, be a real number. If the spectrum of A ^
is discrete, then the eigenvalues ë i are discrete and the measurements of A are
quantized. If, on the other hand, the eigenfunctions jii form a continuous,
in®nite set, then the eigenvalues ë i are continuous and the measured values of
^
A are not quantized. The set of eigenkets jii of the dynamical operator A are
assumed to be complete. In some cases it is possible to show explicitly that jii
forms a complete set, but in other cases we must assume that property.
The expectation value or mean value hAi of the physical observable A at
time t for a system in a normalized state Ø is given by
^
hAiˆhØjAjØi (3:46)
If Ø is not normalized, then the appropriate expression is
^
hØjAjØi
hAiˆ
hØjØi
Some examples of expectation values are as follows