Page 243 - Instant notes

The wave nature of matter 229

(The value n=0, corresponding to a wavefunction of constant amplitude, is allowable for
a particle on a ring, in contrast to the boundary conditions for the particle in a box which
require nodes in the wavefunctions at the walls of the box.) The allowed energy quanta
for the particle on the ring are therefore:

Both positive and negative values of the quantum number are permitted, corresponding to
circular motion with the same kinetic energy in either a clockwise or anticlockwise
direction. The corresponding allowed quantized values for the angular momentum are:

The existence of an n=0 quantum number means that a rotating particle has no
irremovable zero point energy. This conclusion is consistent with the uncertainty
principle. Although the particle is confined to a circle, nothing is known about the
particle’s position within the whole range of possible angular positions from 0 to 360° so
zero angular momentum is possible.

Degeneracy

The existence of different states of motion with the same energy is known as
degeneracy. For the rotating particle all states with |n|>0 are doubly degenerate. The state
with n=0 is non-degenerate because in this state the particle is stationary and there is no
possibility of different directions of travel.

Quantum tunneling

When a particle of energy E is confined by a non-infinite potential barrier V, quantum
mechanics shows there is still some probability of finding the particle in the region of
space on the other side of the barrier, even when V>E. In the classical mechanics
description the particle has insufficient energy to surmount the barrier and zero
probability of existence on the other side. The probability of this quantum tunneling
decreases as both the height and width of the potential barrier increase (Fig. 5).
Tunneling arises because the wavefunction does not fall abruptly to zero at