, which remains constant. As the ball whirls, its height above or below the centre of the circle changes continuously. The height is an image of the red mist, which is sometimes positive (above the centre), sometimes negative (below it). The distance sideways – to the right (positive) or the left (negative) – is an image of the green mist. The square of is the image of the constant intensity of the blue mist. A stationary state is like having children swinging such balls at the same rate everywhere in Q, all perfectly in phase – they all reach the top of the circle together. The only thing not perfectly uniform is the string length, , which can change from point to point in Q. In a momentum eigenstate, is the same everywhere. It is a very special state, but in a more general stationary state does vary over Q. The stationary Schrödinger equation governs its variation.
It does this by imposing a condition at each point of Q. The sum of two numbers, calculated in definite ways, must equal a third. The first number is the most interesting but the most difficult to find. Take a quantum system of three bodies. Its configuration space Q has nine dimensions. Each point in Q corresponds to a position of the three bodies in absolute space. Imagine holding two bodies fixed, and moving the third along a line in absolute space. This will move you along a line in Q. Suppose that along it you plot , the string length, as a curve above the line. At each point, this curve will have a certain curvature. At some places it will curve strongly, towards or away from the line, at others weakly. In the calculus, the curvature is the second derivative.
At each point of Q there are nine such curvatures because Q has nine dimensions, one for each of the three directions in absolute space in which each particle can move. The first number in the Schrödinger condition is the sum of these nine curvatures after each has been multiplied by the mass of the particle for which it has been calculated. I shall call this the curvature number.
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.
In Chapter 14 I described how molecules appear in the Schrödinger picture: as immense collections of all the configurations they could conceivably have, with the blue mist of the quantum probability strongly concentrated on the most probable configurations. These most probable configurations, generally clustered around a single point in Q, are the ones represented by the ball-and-strut models. I can now begin to make good my claim that Schrödinger found the laws of creation. His stationary equation determines the structures – indeed, creates the structures – of all these amazing atoms and molecules that constitute so much of the matter in the universe, our own bodies included. The equation does it by determining which structures are probable. But I mean creation not only in this sense of the structure of atoms and molecules, but in an even deeper one. The full explanation is still to come, but we are getting closer to our quarry.