Absolutely Small - Michael D. Fayer [45]
We can immediately say that the lowest energy of a quantum particle in a 1-nm-length box cannot be zero. In the classical racquetball court, the velocity V could be zero, which means the momentum, p = mV is zero. In addition, the position x could have a perfectly well-defined value. For example, the ball could be standing still (V = 0) exactly in the middle of the court, which would be x = L/2. Then, for our classical racquet ball, Δp = 0 and Δx = 0. The product, ΔxΔp = 0, is not in accord with the Heisenberg Uncertainty Principle, which is okay because this is a classical system. However, the absolutely small particle in the nanometer size box is a quantum particle, and it must obey the Uncertainty Principle, that is, ΔxΔp ≥ h/4π. If V = 0 and x = L/2, we know both x and p. The result would be ΔxΔp = 0, the same as the classical racquetball. This is impossible for a quantum system. Therefore, V cannot be zero. The particle cannot be standing still at a specific point. If V cannot be zero, then Ek can never be zero. The Uncertainty Principle tells us that the lowest energy that a quantum racquetball can have cannot be zero. Our quantum racquetball can never stand still.
ENERGIES OF A QUANTUM PARTICLE IN A BOX
What energies can a quantum particle in a nanometer-size box have? This question can be answered without a great deal of math, but we need to think about waves again. In Chapter 6, we discussed the wavefunctions for free particles. The wavefunction for a free particle with a definite momentum p is a wave that extends throughout all space. So an electron with a perfectly well-defined momentum is a delocalized wave over all space. The probability of finding a free electron is equal everywhere. Such an electron has a well-defined kinetic energy, Ek = 1/2mV2, because it has a well-defined momentum, p = mV.
An electron in a nanometer-size box is something like our free particle in the sense that inside the box, Q = 0. Inside the box there is no potential, which means that there are no forces acting on the particle. This is just like the free particle; there are no forces acting on a free particle. However, there is a major difference between a particle in the box and a free particle, the walls of the box. An electron in a box is located only inside the box. Its wavefunction cannot be spread over all space because of the perfect nature of the box. The particle is inside the box and can’t ever be outside the box. The wavefunction gives the probability amplitude of finding the particle in some region of space. This is the Born interpretation of the wavefunction. If our electron can only be found inside the box and never outside of the box, there must be finite probability of finding the particle inside the box but zero probability of finding the particle outside the box. If the probability of finding the particle outside the box is zero, then the wavefunction must be zero for all locations outside the box.
The result of the reasoning just presented is that the wavefunction for a particle in a box is like a free particle wavefunction, but the wavefunction must be zero outside the box. In his interpretation of the nature of the quantum mechanical wavefunction, Born placed certain physical constraints on the form wavefunctions can have. One of these is that a good wavefunction must be continuous. This condition means that the change in the wavefunction with position must be smooth. An infinitesimal change in position cannot produce a sudden jump in the probability. This is really a simple idea. If the probability of finding a particle in some very small region of space is, for example, 1%, then moving over an unimaginably small amount can’t suddenly make the probability of finding the particle 50%. This is clear from the illustrations of wave packets in Figure 6.7. The probability changes smoothly with position. Therefore, we can say something else about the wavefunctions for a particle in the box in addition to the fact that they are waves with finite amplitudes inside the box and zero amplitude outside the box. Because