Quantum Theory Comes in Waves and Particles

Certainly there is no unique answer to Frank Wilczek’s question, “What Is Quantum Theory?” (Physics Today, June 2000, page 11 ). I was not certain that a pragmatic answer, which I proposed for a course in applied quantum mechanics, ¹ would be sustainable for the entire course, but it seems that it was. That answer was the assertion that everything is both a particle and a wave, and that everything else would follow from that one assertion. Planck’s constant arose when the velocity of a wave (the derivative of the frequency with respect to wavenumber) was equated to the velocity of a particle (the derivative of the energy with respect to momentum), so that energy is proportional to frequency and momentum to wavenumber, with the same proportionality constant ħ.

When Hamiltonian mechanics was introduced, “everything” became anything that can be described by such mechanics, or anything that satisfies a wave equation. Quantum theory does not say what nature gives, only how to predict the behavior of what is given. If nature serves up particles with half-integral angular momentum, the wavefunction that describes them must go into its negative when they are interchanged, and they therefore obey the Pauli principle.

Such principles, which historically arose as independent conjectures, now follow from the one assertion. It is a matter of seeing how this peculiar assertion can turn out to be absolutely true.

The relation between quantum theory and the physical world was seen as follows: Quantum theory tells what future scenarios are consistent with some initial information, and what the relative probability of each scenario is. Quantum theory cannot tell more than that, but neither can any other theory.

My pragmatic approach may not satisfy everyone’s philosophical needs. But the aim of an applied quantum course is to see how to use the theory and to introduce the many approximations that have made the theory accessible to physicists and engineers. If some basic consequences are difficult to accept, the problem must lie with the initial assertion from which the consequences follow.