12
Quantum theory: techniques and
applications
Solutions to exercises
Discussion questions
E12.1(b)
The correspondence principle states that in the limit of very large quantum numbers quantum
mechanics merges with classical mechanics. An example is a molecule of a gas in a box. At room
temperature, the particle-in-a-box quantum numbers corresponding to the average energy of the gas
molecules (
1
2
kT per degree of freedom) are extremely large; consequently the separation between the
levels is relatively so small (
n
is always small compared to
n
2
, compare eqn 12.10 to eqn 12.4) that
the energy of the particle is effectively continuous, just as in classical mechanics. We may also look at
these equations from the point of view of the mass of the particle. As the mass of the particle increases
to macroscopic values, the separation between the energy levels approaches zero. The quantization
disappears as we know it must. Tennis balls do not show quantum mechanical effects. (Except those
served by Pete Sampras.) We can also see the correspondence principle operating when we examine
the wavefunctions for large values of the quantum numbers. The probability density becomes uniform
over the path of motion, which is again the classical result. This aspect is discussed in more detail in
Section 12.1(c).
The harmonic oscillator provides another example of the correspondence principle. The same
effects mentioned above are observed. We see from Fig. 12.22 of the text that probability distribution
for large values on
n
approaches the classical picture of the motion. (Look at the graph for
n
=
20.)
E12.2(b)
The physical origin of tunnelling is related to the probability density of the particle which according to
theBorninterpretationisthesquareofthewavefunctionthatrepresentstheparticle. Thisinterpretation
requires that the wavefunction of the system be everywhere continuous, event at barriers. Therefore,
if the wavefunction is non-zero on one side of a barrier it must be non-zero on the other side of the
barrier and this implies that the particle has tunnelled through the barrier. The transmission probability
depends upon the mass of the particle (specifically
m
1
/
2
, through eqns 12.24 and 12.28): the greater
the mass the smaller the probability of tunnelling. Electrons and protons have small masses, molecular
groups large masses; therefore, tunnelling effects are more observable in process involving electrons
and protons.
E12.3(b)
The essential features of the derivation are:
(1) The separation of the hamiltonian into large (unperturbed) and small (perturbed) parts which are
independent of each other.
(2) The expansion of the wavefunctions and energies as a power series in an unspecified parameters,
λ
, which in the end effectively cancels or is set equal to 1.

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