K. M. Jinks Durham University Durham City, England
A Particle in a Chemical Box
The conventional quantum mechanical problem of a particle in a one-dimensional box can be made more interesting chemically without introducing any more difficult mathematics by employing the potential shown in the figure rather than the usual infinite well. The diagram shows a potential which can be recognized as heing similar to those encountered in a study of intermolecular forces, so this simplified mathematical model may he used to illustrate some of the essential quantum mechanical features of a molecular interaction. If we take the inert gases as a specific example, the potential shown in the diagram can represent the mutual interaction of two Ne atoms due to van der Waals forces. Classical mechanics predicts that if a Ne atom has a kinetic energy less than W i t can never get out of the "box" of length a, so we have a Ne?: molecule and bound states for the system. Since the kinetic energy of a gas depends on its temperature, it would be possible to form molecules of any inert gas simply by lowering the temperature; e.g. the potential in N~~ to a temperatureof -33°K. so UD to -33°K there should he a simificant proportion of N& molecules in the vapor (the normal boiling point of neon is -2IPK). The quantum mechanical situation is very different, and we will show that not only does the existence of hound states depend on the potential w a n d the mass of the particle, it also depends strongly on the width a of the well. i.e. on the "size" of the atoms. Moreover a DOtential df the form shown need not give rise to any bound states a t all, so it is important to investigate the conditions for the existence of hound states. The mathematical problem is to solve the Schrodinger equation
,,
i i z dZ$ 2", d$
+ (E - V)+
=
subject to the restrictions: (1) V = =, x
312 / Journal of Chemical Education
0
< 0; (2) v = - W,
+a v(x) X
-
0
-W
0 a Potenlial diagram used to illustrate some of the quantum mechanical tea-
'
a
interaction.
O < x a. Following the general method, the equation is solved for the separate regions (1) to (3), and the resulting wave functions combined to give the wave function for the whole system by applying the usual continuity conditions. For case (1)the Schrodinger equation reduces to z;;E-@ fix d2+
- 4=0
The only function iI. which is consistent with this equation forallx
