Page 46 - Science at the nanoscale

9:2
RPS: PSP0007 - Science-at-Nanoscale
June 9, 2009
Brief Review of Quantum Mechanics
36
2
volume element dV is given by |ψ(r, t)| dV. Since the probability
of finding the particle in all space is 1, the wavefunction satisfies
the following normalisation condition:
∞
Z
2
(3.6)
|ψ(r, t)| dV = 1
−∞
Once the wavefunction that describes the system is known, how
does one obtain the various physical observables of the system?
For example, if we are interested in the energy E of the particle,
how can we determine E if we know ψ(r, t)? With each physical
observable, there is an associated mathematical operator that can
be used to “operate” on the wavefunction. The action of the oper-
ator is to carry out a mathematical operation on the wavefunction
and extract the value of the observable. Mathematically it can be
represented as
(3.7)
Qψ = qψ
where Q denotes the operator while q denotes the observable
value.
For example, if the operator Q chosen is the energy
operator, then the value q corresponds to the energy value. For
the operator Q, there may exist a special set of functions which
are known as the eigenfunctions ψ of the operator
j
(3.8)
Qψ = q ψ
j j
j
with the corresponding eigenvalues q . This set of functions form
j
a complete and basic set of linearly independent functions. Any
wavefunction representing a physical system can be expressed as ch03
a linear combination of the eigenfunctions of any physical observ-
able of the system.
a (3.9)
ψ = ∑ j ψ j
where a j represents a coefficient related to the probability of the
particular eigenfunction. Hence the operator Q can be used to
extract a linear combination of eigenvalues multiplied by coeffi-
cients related to the probability of their being observed.
Once the wavefunction ψ that describes a physical system is
known, the expectation value of the physical observable, q, can
be expressed in terms of the wavefunction and the operator, Q,