Page 39 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

30 The wave function

information that can be known about the particle that it represents. The wave
function is a complete description of the quantum behavior of the particle. For
this reason, the wave function is often also called the state of the system.
In the double-slit experiment, the patterns observed on the detection screen
are slowly built up from many individual particle impacts, whether these
particles are photons or electrons. The position of the impact of any single
particle cannot be predicted; only the cumulative effect of many impacts is
predetermined. Accordingly, a theoretical interpretation of the experiment must
involve probability distributions rather than speci®c particle trajectories. The
probability that a particle will strike the detection screen between some point x
and a neighboring point x ‡ dx is P(x)dx and is proportional to the range dx.
The larger the range dx, the greater the probability for a given particle to strike
the detection screen in that range. The proportionality factor P(x) is called the
probability density and is a function of the position x. For example, the
probability density P(x) for the curve I A in Figure 1.9(a) has a maximum at
the point A and decreases symmetrically on each side of A.
If the motion of a particle in the double-slit experiment is to be represented
by a wave function, then that wave function must determine the probability
density P(x). For mechanical waves in matter and for electromagnetic waves,
the intensity of a wave is proportional to the square of its amplitude. By
analogy, the probability density P(x) is postulated to be the square of the
absolute value of the wave function Ø(x)
2

P(x) ˆjØ(x)j ˆ Ø (x)Ø(x)
On the basis of this postulate, the interference pattern observed in the double-
slit experiment can be explained in terms of quantum particle behavior.
A particle, photon or electron, passing through slit A and striking the
detection screen at point x has wave function Ø A (x), while a similar particle
passing through slit B has wave function Ø B (x). Since a particle is observed to
retain its identity and not divide into smaller units, its wave function Ø(x)is
postulated to be the sum of the two possibilities
Ø(x) ˆ Ø A (x) ‡ Ø B (x) (1:48)

When only slit A is open, the particle emitted by the source S passes through
slit A, thereby causing the wave function Ø(x) in equation (1.48) to change or
collapse suddenly to Ø A (x). The probability density P A (x) that the particle
strikes point x on the detection screen is, then
2
P A (x) ˆjØ A (x)j
and the intensity distribution I A in Figure 1.9(a) is obtained. When only slit B