Once Before Time - Martin Bojowald [27]
ATOMS: STABILITY FROM UNSTEADINESS
An electron of a given energy, bound in an atom by electric attraction, has a spherical wave function of fixed wavelength surrounding the nucleus. How quantum mechanics leads to the realization of only a discrete set of energies can be seen in the simplified context of a wave function on a circle surrounding a central point. The wavelength, the distance between neighboring hills of the wave, is related by quantum mechanics to the electron’s velocity or momentum: The faster the electron, the more rapidly the wave changes, and so the wavelength is shorter. If we imagine following such a wave once around the center, laying wave crests next to wave troughs at the required distance, the form will generally change when we return to the starting point after a single orbit. A peak no longer falls on its initial position, unless an integer number of wavelength intervals can be fitted into the circumference of the circle (see figure 5). If there is a mismatch, it causes a spatial shift of the new crests, repeated every time we orbit around. At some point, a new crest falls on an old trough, and the negative heights in the trough exactly cancel out the positive heights in the crest: The whole wave disappears in destructive interference. Only in exceptional cases, for those finely tuned circles whose circumference is an exact multiple of the wavelength, can the wave be in such harmony that each orbit will lay crests exactly on crests and no destruction will occur.
In this way, an electron on a circle of fixed radius can move only with velocities in a discrete set; otherwise its wave function, and thus the electron, could not exist. At this stage, we have only considered the wavelike nature of quantum mechanics, not properties of the forces confining the electron to the circle. Forces provide additional conditions, for they tell us how the electron is being accelerated or slowed down. For motion along a circle, moreover, a balance must be realized between the force exerted by the center, attracting the electron, and the electron’s normal tendency to follow a straight line, moving outward from the circle. A further relationship arises, telling us how fast the electron must move when it has a certain distance from the center. Now, there are two conditions for the electron, both tying the velocity to the radius: one relationship that also exists classically, since it refers only to the force, and the new observation in quantum mechanics, requiring the wave to fit wholly on the circle. Each condition takes a different form, and can thus be realized only for special radii. Changing an allowed radius slightly will render the wavelength out of tune with the circumference, and it is impossible to satisfy both conditions. In particular, the requirement of integer multiples of the wavelength to fit on a closed circle, a condition arising only due to the wavelike nature of quantum mechanics, implies that only discrete, isolated radii can be occupied.
5. Drawing a wave of a given wavelength along a circle is possible only for a discrete set of specific radii. If the radius and wavelength do not match (as shown by the dashed wave), the wave after one orbit is displaced from its original position to the degree that its variations away from the circle begin to cancel each other out.
For an electron orbiting the nucleus of an atom, these are the orbits around the nucleus the electron is “allowed” to take, for only then can a nonzero wave function result. Since the velocity has already been tied to the radius, and both determine the energy, energies must also lie in a discrete set, isolated from each other. While motion in an atom is more complex than