Page 43 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

34 The wave function

ment reinforces the postulate that the wave function for a particle is the sum of
indistinguishable paths and is modi®ed when the paths become distinguishable
by means of a measurement. The nature of the modi®cation is the collapse of
the wave function to one of its components in the sum. Moreover, this new
collapsed wave function may be expressed as the sum of subsequent indis-
tinguishable paths, but remains unchanged if no further interactions with
measuring devices occur.
This statistical interpretation of the signi®cance of the wave function was
postulated by M. Born (1926), although his ideas were based on some
experiments other than the double-slit and Stern±Gerlach experiments. The
concepts that the wave function contains all the information known about the
system it represents and that it collapses to a different state in an experimental
observation were originated by W. Heisenberg (1927). These postulates regard-
ing the meaning of the wave function are part of what has become known as
the Copenhagen interpretation of quantum mechanics. While the Copenhagen
interpretation is disputed by some scientists and philosophers, it is accepted by
the majority of scientists and it provides a consistent theory which agrees with
all experimental observations to date. We adopt the Copenhagen interpretation
of quantum mechanics in this book. 3

Problems
1.1 The law of dispersion for surface waves on a sheet of water of uniform depth d is 4

ù(k) ˆ (gk tanh dk) 1=2

where g is the acceleration due to gravity. What is the group velocity of the
resultant composite wave? What is the limit for deep water (dk > 4)?
1.2 The phase velocity for a particular wave is v ph ˆ A=ë, where A is a constant. What
is the dispersion relation? What is the group velocity?
1.3 Show that

1
A(k)dk ˆ 1
ÿ1
for the gaussian function A(k) in equation (1.19).

3 The historical and philosophical aspects of the Copenhagen interpretation are more extensively discussed
in J. Baggott (1992) The Meaning of Quantum Theory (Oxford University Press, Oxford).
4 For a derivation, see H. Lamb (1932) Hydrodynamics, pp. 363±81 (Cambridge University Press, Cam-
bridge).