Absolutely Small - Michael D. Fayer [30]
FIGURE 6.4. The superimposition of 250 waves with equally spaced wavelengths spanning the wavelength range 0 to 4. Compared to Figure 6.3, which is the superposition of five waves, this superposition has a much larger peak at x = 0, the region of maximum constructive interference, and destructive interference reduces the other regions more. The amplitude of the superposition is dying out going toward +20.
THE SUPERPOSITION PRINCIPLE
In Chapter 5, the interference experiment was analyzed in terms of the superposition of two photon translation states, T1 and T2. A photon in the interferometer is described as being in the 50/50 superposition state, T = T1 + T2. The idea of superposition is central to the quantum theoretical description of nature; it is called the Superposition Principle and assumes that “Whenever a system is in one state, it can always be considered to be partly in each of two or more states.”
An original state can be regarded as a superposition of two or more states, as in the interference problem in which the translation state T of the photon was described as a superposition of T1 and T2. Conversely, two or more states can be superimposed to make a new state. It is this second statement that we will now use to understand the fundamental nature of particles. The fact that a photon can act like a particle in the photoelectric effect but act like a wave to give rise to the interference effect follows from the superposition principle and leads to the Heisenberg Uncertainty Principle.
Eigenstates
In connection with Figure 6.1, it was stated that a free particle with perfectly defined momentum p is a delocalized probability amplitude wave spread out over all space. If a particle exists in such a state, it is said to be in a momentum eigenstate. In discussing the interference problem, we called T1 and T2 pure states, but the correct name for them is eigenstates. Eigen is German for characteristic, so an eigenstate is a characteristic state. An eigenstate for a particular observable property, such as momentum, is a state with a perfectly defined value of that property. The momentum eigenstates of a free particle are delocalized over all space. One such eigenstate exists for each of the infinite number of possible values of the momentum. Each of these momentum eigenstates is associated with an exact value of momentum of the particle. The particle’s location is uniform over all space because the wavefunction associated with the eigenstate is spread out over all space. However, the Superposition Principle tells us that we can superimpose any number of momentum eigenstates to make a new state.
Superposition of Momentum Eigenstate Probability Amplitude Waves
To understand the nature of real particles, photons, electrons, and so on, we will superimpose a range of momentum eigenstate probability amplitude waves, such as the one shown in Figure 6.1. For each momentum p, the wave has a different wavelength, λ = h/p. In Figures 6.3 and 6.4, we saw that adding together waves of different wavelengths concentrated the amplitude of the wave in a particular region. As mentioned in both examples above, the amplitude of each wave in the superposition was the same. Now we will superimpose momentum probability amplitude waves with different amplitudes. There is one wave (a particular value of p) with the largest amplitude. The other waves with different wavelengths have amplitudes that decrease as the wavelength becomes greater than or less than the wavelength of the wave with the maximum amplitude. So, we have a distribution of wavelengths centered around the wavelength of the maximum amplitude wave. The wavelength with maximum amplitude is at the center of the distribution. By a distribution, we just mean that there is a range of wavelengths, in the same way that if you have a room full of people there will be a distribution of ages. There will be some people of average age, the center of the distribution, and some people older than the average and some