We have now seen persuasive evidence that, on an atomic or subatomic scale, a particle such as an electron can’t be described simply as a point that has three position coordinates and three velocity components. In some situations, such a particle behaves like a wave, and we have spoken a few times about using a wave function to describe the state of a particle. Let’s now describe more specifically the quantum-mechanical language that we use to replace the classical scheme of coordinates and velocity components. Our new scheme for describing the state of a particle has a lot in common with the language of classical wave motion. In Chapter 19 we described transverse waves on a string by specifying the position of each point in the string at each instant of time by means of a wave function (Section 19-4). If Y represents the displacement from equilibrium of a point on the string, then the function y(x, t) represents that displacement at any distance x from the origin and at any time t. Once we know the wave function for a particular wave motion, we know everything there is to know about the motion. We can find the position and velocity of any point on the string at any time, and so on. We worked out specific forms for these functions for sinusoidal waves, in which each particle undergoes simple harmonic motion.
We followed a similar pattern for sound waves in Chapter 21. The wave function p(x, t) for a wave traveling along the x-direction represented the pressure variation at any point x at any time t. We used this language once more in Section 33-4, in which we used two wave functions to describe the electric and magnetic fields of electromagnetic waves at any point in space at any time.
Thus it is natural to use a wave function as the central element of our new quantum mechanical language. The symbol that is sually used for this wave function is ‘I’ or ‘1’, the Greek letters “psi” (pronounced “sigh”). In general, we will use ‘I’ for a function of the space coordinates and time, and will use ‘I’ for a function of the space coordinates only, not of time. Just as we can use the wave function y(x, t) for mechanical waves on a string to provide a complete description of the motion of the particles of the string, we can use the quantum-mechanical wave function ‘I'(x, y, z, t) for a particle to give us information about that particle. CAUTION Keep in mind that a quantum-mechanical wave function is unlike any wave you’ve yet encountered. Mechanical waves require a medium to travel through; the stretched string is the medium for transverse waves on a string, and air is the medium for sound waves. But the wave function for a particle is not a mechanical wave that needs some material medium in order to propagate. The wave function describes the particle, but we can’t define the function itself in terms of anything material. We can only describe how it is related to physically observable effects.