Absolutely Small - Michael D. Fayer [47]
Figure 8.4 shows the probability amplitude waves. As discussed, the probability of finding the particle in a certain region of space is proportional to the square of the wavefunction (actually the absolute value squared, but for our purposes, there is no difference). Figure 8.5 shows the square of the wavefunctions that are displayed in Figure 8.4. The square of the wavefunctions are always positive because the probability of finding a particle in some region of space cannot be negative. Where the amplitude is large, there is a large probability of finding the particle. As n increases, the number of nodes increases. As we will discuss in the next and later chapters, atomic and molecular wavefunctions also have nodes.
A question that is frequently asked is, how does a particle get through a node? For example, for n = 2, there is a node exactly in the middle of the box. In a classical system if we had a ball on the left side of the box and it was traveling to the right, but we said it could never be in the center of the box, we would be confident that the ball could never get to the right side of the box. However, we cannot think classically about an absolutely small particle, such as an electron in a molecular-sized box. It does not have a simultaneous definite position and momentum that can be described by an observable trajectory. A quantum particle, an electron, is described as a probability amplitude wave. Waves have nodes. Even classical waves have nodes. A quantum particle does not have to “pass through” a node because it is a delocalized probability amplitude wave. The idea of a trajectory in which to get from point A to point B a particle must pass through all the points in between just doesn’t apply to the proper wave description of electrons and other absolutely small particles.
FIGURE 8.5. The squares of the first three wavefunctions, ϕ2, for the particle in a box. They have been shifted upward for clarity of presentation. The vertical axis is the amplitude of the wavefunction squared. The dashed line shows where the wavefunction is zero. The square of the wavefunctions are always positive because they represent probabilities. The wavefunctions shown in Figure 8.4 can be positive or negative.
The Energies Are Quantized
Now we will determine the possible energies that an absolutely small particle in a box can have. The classical ball in a racquetball court can have any energy, and the energy is continuous. We can determine what energies a particle, such as an electron, can have in a tiny box by using the rule for the possible wavelengths, λ = 2L/n, that allowed probability amplitude waves can have inside the box (see Figure 8.4). Here tiny means a box that is small in the absolute sense, that is, the wavelength is comparable to the size of the box. We will also need several other physical relationships that we have met previously. The other relations we need are the de Broglie wavelength, p = h/λ, where p is the momentum and h is Planck’s constant; the fact that the momentum is p = mV, where m is the mass and V is the particle velocity; and the kinetic energy of the particle, . Now let’s combine these formulas.
First, square p. Then,
p2 = m2V2.
If we now divide both sides of the equation by 2m, we see that the right side gives the kinetic energy, and the left side gives . So we have the following expression for the kinetic energy,
Using the de Broglie relation, we can replace p2 with p2 =h2/λ2. Putting this into the expression for the energy gives,
Finally, we will use our rule, λ = 2L/n, for the possible wavelengths. Then λ2=4L2/n2. Substituting