Absolutely Small - Michael D. Fayer [34]
you that the momentum is both 10 and 50 at the same time. In a single measurement, a single well-defined value will be measured. How do we get a single value when our wave packet has a spread in momenta? The wave packet is made up of a superposition of momentum eigenstates, that is, momentum probability amplitude waves with associated well-defined values of the momentum. When a measurement is made, the nonnegligible disturbance accompanying the measurement causes the system to “jump” from being in a superposition state to a particular eigenstate. The measurement gives the value of the momentum that goes with that eigenstate. Now, the measurement changed the system. To make another measurement, we need to start over and prepare the particle in the same way. By preparing it in the same way, a wave packet composed of the same superposition of momentum eigenstates will be generated. We now make the same measurement we did the first time. In general, we will measure a different value of the momentum because the wave packet is composed of many momentum waves each with a different observable value of the momentum associated with it. If we do this over and over again, each time preparing the wave packet in the same way and then making the measurement, we will get a spread in the measured values of the momentum. After a huge number of measurements, we might measure (neglecting units) the value 400 a thousand times, 390 eight hundred times, 410 eight hundred times, but 200 and 600 only twenty times. If we make a plot of all of these numbers we get a probability distribution like those shown for momentum on the left side of Figure 6.7. A probability distribution is an experimental determination of the composition of the wave packet. We now know how much (what is the probability) of each wave in the packet. The same description also applies to the position of our wave packet. On each measurement of the position of identically prepared wave packets, a single location for the particle will be found. After many measurements, a distribution of positions is determined like those illustrated on the right side of Figure 6.7.
THE HEISENBERG UNCERTAINTY PRINCIPLE
A very important point is that there is a relationship between the spread in momentum and the spread in position that is fundamental to the superposition state description of particles. When the spread in momentum (Δp) is large, there are many waves spread out along the x axis (see Figure 6.1) that combine to make the wave packet. These waves have different wavelengths (see Figure 6.2). When many waves with a wide range of wavelengths interfere, the region of constructive interference dies out very fast away from the maximum (see Figures 6.3 and 6.4). That means that the spread in position (Δx) is small. If there is only a small range of the momentum waves that make up the wave pack (Δp small), then the spatial region of constructive interference dies out slowly away from the maximum in the position distribution (see Figure 6.7). Therefore, the spread or uncertainty in position Δx is large. All of this occurs because of the probability amplitude wave nature of the wave functions that describe the momentum eigenstates. A wave packet is located, more or less, in the region of constructive interference, and there is little probability of finding the particle in the regions of substantial destructive interference.
The formal relationship between the spread in the momentum and the spread in the position, that is, between Δp and Δx, is called the Heisenberg Uncertainty Principle. Werner Karl Heisenberg (1901-1976) won the Nobel Prize in Physics in 1932 “for the creation of quantum mechanics, the application of which has, inter alia, led to the discovery of the allotropic forms of hydrogen.” The Heisenberg Uncertainty Principle is stated through a simple mathematical relationship, ΔxΔp ≥ h/4π, where h is Planck’s constant and the Δx and Δp define the widths of the distributions for position and momentum, as shown in Figure 6.7. (≥ means greater than or equal to.) Whether the equal