Warped Passages - Lisa Randall [65]
De Broglie assumed that a particle with momentum p was associated with a wave whose wavelength was inversely proportional to momentum—that is, the smaller the momentum, the longer the wavelength. The wavelength was also proportional to Planck’s constant, h.*The idea behind de Broglie’s proposal was that a wave that oscillated frenetically (that is, one with small wavelength) carried more momentum than one that oscillated lethargically (with large wavelength). Smaller wavelengths mean more rapid oscillations, which de Broglie associated with larger momentum.
If you find the existence of this particle-wave perplexing, that’s because it is. When de Broglie first suggested his waves, no one knew what they were supposed to be. Max Born proposed a surprising interpretation: that the wave was a function of position whose square gives the probability for finding a particle at any location in space.† He named this a wavefunction. Max Born’s insight was that particles cannot be pinned down and can be described only in terms of probabilities. This is a big a departure from classical assumptions. It means that you cannot know the particle’s exact location. You can only specify the probability of finding it somewhere.
But even though a quantum mechanical wave describes only probabilities, quantum mechanics predicts this wave’s precise evolution through time. Given the values at any one time, you can determine the values at any later time. Schrödinger developed the wave equation that shows the evolution of the wave associated with a quantum mechanical particle.
But what does this probability of finding a particle mean? It’s a puzzling idea—after all, there’s no such thing as a fraction of a particle. That a particle can be described by a wave was (and in some ways still is) one of the most surprising aspects of quantum mechanics, particularly as it is known that particles often behave like billiard balls, and not like waves. A particle interpretation and a wave interpretation seem incompatible.
The resolution to this apparent paradox hinges on the fact that you never detect the wave nature of a particle with just one particle. When you detect an individual electron, you see it in some definite location. In order to map out the entire wave, you need a set of identical electrons, or an experiment that is repeated many times. Even though each electron is associated with a wave, with a single electron you will measure only one number. But if you could prepare a large set of identical electrons, you would find that the fraction of electrons in each location is proportional to the probability wave assigned to an electron by quantum mechanics.
The wavefunction of an individual electron tells you about the likely behavior of many identical electrons with this same wavefunction. Any individual electron will be found only in a single place. But if there were many identical electrons, they would exhibit a wave-like distribution of locations. The wavefunction tells you the probability of the electron ending up in those locations.
This is analogous to the distribution of height in a population. Any individual has their own height, but the distribution tells us the likelihood that an individual will have any particular height. Similarly, even if one electron behaves like a particle, many electrons together will have a distribution of positions delineated by a wave. The distinction is that an individual electron is nonetheless associated with this wave.
In Figure 43 I’ve plotted an example of a probability function for an electron. This wave gives the relative probability of finding the electron at a particular location. The curve I have drawn takes a definite value for every point in space (or rather, every point along a line, since the flatness of the paper forces me to draw only one dimension of space). If I could make many copies of this same electron, I could take a series of measurements of the electron