Page 47 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

38 Schro Èdinger wave mechanics

case of the free particle discussed in Chapter 1, we follow the formulation of
Born (1926).
The fundamental postulate relating the wave function Ø(x, t) to the proper-
2

ties of the associated particle is that the quantity jØ(x, t)j ˆ Ø (x, t)Ø(x, t)
gives the probability density for ®nding the particle at point x at time t. Thus,
the probability of ®nding the particle between x and x ‡ dx at time t is
2
jØ(x, t)j dx. The location of a particle, at least within an arbitrarily small
interval, can be determined through a physical measurement. If a series of
measurements are made on a number of particles, each of which has the exact
same wave function, then these particles will be found in many different
locations. Thus, the wave function does not indicate the actual location at
which the particle will be found, but rather provides the probability for ®nding
the particle in any given interval. More generally, quantum theory provides the
probabilities for the various possible results of an observation rather than a
precise prediction of the result. This feature of quantum theory is in sharp
contrast to the predictive character of classical mechanics.
According to Born's statistical interpretation, the wave function completely
describes the physical system it represents. There is no information about the
system that is not contained in Ø(x, t). Thus, the state of the system is
determined by its wave function. For this reason the wave function is also
called the state function and is sometimes referred to as the state Ø(x, t).
The product of a function and its complex conjugate is always real and is
positive everywhere. Accordingly, the wave function itself may be a real or a
complex function. At any point x or at any time t, the wave function may be
2
positive or negative. In order that jØ(x, t)j represents a unique probability
density for every point in space and at all times, the wave function must be
continuous, single-valued, and ®nite. Since Ø(x, t) satis®es a differential
equation that is second-order in x, its ®rst derivative is also continuous. The
iá
wave function may be multiplied by a phase factor e , where á is real, without
changing its physical signi®cance since
iá iá 2
[e Ø(x, t)] [e Ø(x, t)] ˆ Ø (x, t)Ø(x, t) ˆjØ(x, t)j

Normalization
The particle that is represented by the wave function must be found with
probability equal to unity somewhere in the range ÿ1 < x < 1, so that
Ø(x, t) must obey the relation

1
2
jØ(x, t)j dx ˆ 1 (2:9)
ÿ1