The End of Time: The Next Revolution in Physics

The second number is much easier to find. Recall that any configuration of bodies has an associated potential energy. The configuration (and the nature of the bodies, their masses, etc.) determines it uniquely. For gravity, this was explained in Figure 17. The second number, which I shall call the potential number, is found simply by multiplying the potential by

The third number is also easy to find. If ω is the frequency of the state (the number of ‘rotations of the balls’ in a second), then, by the quantum rules, the energy of the state is E = hω, where h is Planck’s constant. This is the relationship Einstein found between the energy and frequency of a photon. The third number, which I shall call the energy number, is then found by multiplying the energy E by .

The condition imposed by the stationary Schrödinger equation is then

Curvature number + Potential number = Energy number

(Planck’s constant also occurs in the first number, to ensure that all three numbers have the same physical nature.)

However, finding this condition, which must hold everywhere in Q, was only half the story. Schrödinger thought that an atom in a stationary state was like a violin string vibrating in resonance. Because its two ends are fixed, the amplitude at the ends is zero. He therefore imposed on not only the above condition, but also the condition that it should tend to zero at large distances. It was this requirement that enabled him to make the huge discovery that convinced him – and very soon everyone else – that he had found the secret of Bohr’s quantum prescriptions.

This hinges on an extremely interesting property of the stationary Schrödinger equation. As yet E is a fixed but unknown number. It may be smaller or greater than the potential V, which varies over Q. The interesting thing is that the above condition forces to do very different things depending on the value of E – V. Where it is greater than zero, oscillates. As Schrödinger said rather quaintly, ‘it does not get out of control’. However, where E – V is less than zero, the condition forces an entirely different behaviour on . It must either tend rapidly to zero or else grow rapidly – exponentially in fact – to infinity. The latter would be a disaster. Schrödinger therefore commented that things become tricky and must be handled delicately. Indeed, he showed that it is only in exceptional cases, for special values of E, that does not ‘explode’ but instead subsides to zero at infinity. These are the cases he was looking for. Well-behaved solutions exist for only certain values of E, which are discrete (separated from each other) if E is less than zero.

The well-behaved solutions are called eigenfunctions, and the corresponding values of E are called (energy) eigenvalues. It is a fundamental property of quantum mechanics that any system always has at least one eigenfunction. The eigenfunction of any system that has the lowest value of its energy eigenvalue (there is often only one such eigenfunction) is called the ground state. In general, there are also eigenfunctions with higher energies, called excited states. Finally, if E is large enough for E – V to be positive everywhere, the eigenfunctions oscillate everywhere, though more rapidly where the potential is lowest. The negative eigenvalues E form the discrete spectrum, and the corresponding states are called bound states because for them has an appreciable value only over a finite region. The remaining states, with E greater than zero, are called unbound states, and their energy eigenvalues form the continuum spectrum.

Schrödinger won the 1933 Nobel Prize for Physics mainly for his wave-mechanical calculation for the hydrogen atom. He found that the energy eigenvalues of its stationary states are precisely the energies of the allowed states in Bohr’s model. This was a huge advance, since Schrödinger’s formalism had an inner unity and consistency to it completely lacking in the older model. Brilliant successes of the new wave mechanics, many achieved by Schrödinger himself, soon came flooding in, leaving no doubt about the great fruitfulness of the new scheme.