Absolutely Small - Michael D. Fayer [27]
For a photon, the momentum is given as p = h/λ, where h is Planck’s constant and λ is the wavelength of the light. Therefore, the momentum is related to the wavelength (the color) of the light. Prince Louis-Victor Pierre Raymond de Broglie (1892-1987) won the Nobel Prize in Physics in 1929 “for his discovery of the wave nature of electrons.” De Broglie showed theoretically that particles, such as electrons or baseballs, also have a wave description. As discussed below, the wave description of electrons—or any type of particle—is in terms of the same types of waves as a photon, probability amplitude waves, as introduced in Chapter 5.
The wavelength associated with a particle is λ = h/p. This is a simple rearrangement of the formula for the photon momentum given above. If both sides of the photon momentum formula are multiplied by λ and divided by p, then the expression for the wavelength associated with a particle is obtained. De Broglie’s important result is that the relationship between the momentum and the wavelength is the same for photons (light) as it is for material particles, such as electrons and baseballs. Therefore, the properties of photons are described in fundamentally the same way as the properties of electrons, as well as baseballs. The wavelength associated with a particle is called the de Broglie wavelength. (We will see with physical examples in the next chapter why baseballs don’t seem to have wavelike properties, but photons and electrons do.)
WHAT A FREE PARTICLE WAVEFUNCTION LOOKS LIKE
For a free particle with some particular value of its momentum, p, what does the wavefunction look like? Recall that the wavefunction is related to the probability of finding the particle someplace in space. Figure 6.1 shows a graph of the wavefunction for a free particle with the momentum, p. As discussed above, the wavelength of the wavefunction associated with the particle is λ = h/p. As can be seen in the figure, the wavefunction for a free particle is represented by two waves called the real and the imaginary parts of the wavefunction. These components are equivalent. The term imaginary is a mathematical term. It does not imply that in some sense the imaginary component is less important than the part referred to as real. It is just jargon to identify the two components, although they do have differences in the way they are represented mathematically. The real and the imaginary components of the wavefunction have the same wavelength, but are shifted by one-fourth of the wavelength. That means that one wave is shifted in phase relative to the other by 90°. The two components of the wavefunction do not interfere with each other, either constructively or destructively, because in a mathematical sense and in essence they are perpendicular to each other.
FIGURE 6.1. The wavefunction for a free particle with momentum p, which has wavelength, λ = h/p. A quantum mechanical wavefunction can have two parts, called real and imaginary. Both waves have the same wavelength. They are just shifted by one-fourth of a wavelength, which is the same as a 90° shift in the phase. These two components are separate from each other. They do not interfere either constructively or destructively. For a free particle with the well-defined value of the momentum, p, the wave function extends from positive infinity to negative infinity, +∞ to - ∞.
A PARTICLE WITH WELL-DEFINED MOMENTUM IS SPREAD OVER ALL SPACE
The important feature of the wavefunction shown in Figure 6.1 is that it extends from positive infinity to negative infinity, that is, from +∞ to -∞. In Figure 6.1, only a small section of the wavefunction in a small region of space is shown because we cannot plot +∞ to -∞ on a finite piece of paper. The wave shown in the figure just keeps going to the right and to the left. It is uniform across all space. This means that for a quantum mechanical particle with a definite value of the momentum, p, we are equally likely to find the particle anywhere along the x axis, the horizontal axis in the graph. The vertical axis tells