Absolutely Small - Michael D. Fayer [31]
Figure 6.5 illustrates a distribution of momentum probability amplitude waves. p0 is the momentum of the wave at the center of the distribution of waves. It has a wavelength λ = h/p0. It is the wave with the biggest amplitude, that is, the biggest probability of finding it in the distribution. As the momentum is increased or decreased (λ is smaller or bigger) away from p0, the amount of a particular wave in the superposition (its probability) decreases. Δp is a measure of the width of the distribution. If Δp is large, then there is a large spread in p, and therefore, a large spread in wavelengths in the distribution. If Δp is small, the spread in wavelengths is small.
FIGURE 6.5. A plot of the probability of finding a particle in a particular momentum eigenstate with momentum p given that it is in a superposition of momentum probability amplitude waves. p0is the middle wave with the biggest amplitude in the distribution. Δp is a measure of the width of the distribution of eigenstates.
Momentum of a Free Particle in a Superposition State
What is the momentum of a free particle that is in a superposition of momentum eigenstates such as that shown in Figure 6.5? A superposition of momentum eigenstates just means that we add together (superimpose) a bunch of waves (probability amplitude waves) with each of the waves having a specific value of the momentum associated with it (an eigenstate). In any measurement of a property of a system, a particular value of that property will be measured. If we make a measurement of the momentum of a particle, we will measure a single value of the momentum. The nature of the disturbance that accompanies the measurement of an absolutely small object is to collapse the superposition state into a single eigenstate. Making a measurement changes a system by taking it from its initial superposition state to one particular eigenstate. This is what is meant by collapse.
In discussing the interference problem, we said that if we tried to find out if the photon was in the state T1 by placing a detector in leg 1 of the interferometer, we would destroy the superposition state necessary for the interference. The superposition state T would jump into either T1 or T2. Since the state T is a 50/50 superposition of T1 and T2, half the time a measurement will result in finding the system in T1 and half the time in T2. On any single measurement, it is impossible to know ahead of time which result will occur. Many measurements show that the superposition is 50/50 because half the time we find that the photon is in leg 1 of the apparatus (state T1) and half the time we find the system is in leg 2 of the apparatus (state T2).
The superposition of momentum eigenstates shown in Figure 6.5 is composed of a vast (infinite) number of states spread over a range of momenta characterized by the width of the distribution, Δp. Therefore, there is a wide range of momentum values that can be measured on any single measurement. If we make a single measurement, we will measure one of the many values. Let’s say we make a measurement and find a momentum a little bigger than p0. Call it p1 because it is our first measurement. In the process of making the measurement, we made a nonnegligible disturbance of the system. It was changed from being in the superposition state to a single eigenstate with momentum p1. So to make another measurement, we need to start over again and prepare the particle (the system) in the same way we did originally to get the same distribution of momenta. We make a second measurement. This time we measure a value that is quite a bit smaller than p0. Call this value p2. We prepare the system again, and make another measurement. We measure p3. Each time we make a measurement on an identically prepared system, we will measure a particular value of the momentum. In advance, we don’t know what the value will be. If we make many, many measurements,