Absolutely Small - Michael D. Fayer [32]
Momentum Not Perfectly Defined for a Particle in a Superposition of States
A particle in a superposition of momentum eigenstates, such as that shown in Figure 6.5, does not have a perfectly well-defined value of the momentum. In a single measurement, we cannot say what value of the momentum will be measured. We can say we are most likely to measure a value near p0. With many measurements, we can determine the probability distribution. A classical particle, like the one illustrated in Figure 2.5, has a perfectly well-defined momentum. We can measure it without changing it. If it is a free particle, we can make many measurements of the momentum at different times, and we will always measure the same value of p. This is not the case for an absolutely small quantum particle in a momentum superposition state. We will measure a single value of p on a single measurement, but the act of making the measurement fundamentally changes the nature of the particle. The particle goes from being in a superposition state to being in an eigenstate (a single wave with a single value of the momentum). It goes from being in a state in which there is a probability distribution of momenta to a single value of the momentum that is observed. To recover the distribution, the particle needs to be prepared again.
WHERE IS A PARTICLE WHEN IT IS IN A MOMENTUM SUPERPOSITION STATE?
In connection with Figure 6.1, we said that a particle in a single momentum eigenstate is delocalized over all space. This doesn’t go along well with the description of the photoelectric effect. Now the question is where is a particle that is in a momentum superposition state? We have already hinted at the answer with the discussions surrounding Figures 6.2 through 6.4. In Figures 6.3 and 6.4, we saw that a superposition of waves with different wavelengths produced a spatial distribution that was concentrated in a region of space. In Figure 6.3, the wavelengths went from 0.8 to 1.2, and the pattern was not as concentrated as the one in Figure 6.4, which was formed from wavelengths that went from 0 to 4. Figure 6.6 shows the spatial distribution associated with distribution of waves (momentum eigenstates) shown in Figure 6.5. There is a position where the value is maximum, which is also the average value. For larger and smaller values of x relative to x0, the amplitudes (probabilities) become smaller.
FIGURE 6.6. A plot of the probability of finding the particle at a location x given that it is in the superposition of momentum eigenstates shown in Figure 6.5. x0is the middle position with the greatest probability. Δx is a measure of the width of the spatial distribution.
What does the probability distribution of positions (x values) mean? A particle with the momentum probability distribution shown in Figure 6.5 gives rise to the spatial probability distribution shown in Figure 6.6. A single measurement of the position will measure a particular value of the position. Call it x1. When the position measurement is made on the absolutely small quantum particle, it causes a nonnegligible disturbance that collapses the position probability distribution into a position eigenstate with a perfectly defined value of the position. To make another measurement, the system (a particle) must be prepared again in the identical manner so that it has the same momentum probability distribution and, therefore, the same spatial probability distribution. The second measurement of the particle’s position will give a value, x2,