Absolutely Small - Michael D. Fayer [46]
Figure 8.3 illustrates a discontinuous wavefunction (not allowed) inside a box. The wavefunction is called ϕ (Greek letter phi). The vertical axis gives the amplitude of the wavefunction. The dashed line shows where zero is. Wavefunctions, which are probability amplitude waves, can oscillate positive and negative. The wave-function shown in Figure 8.3 has values at the walls that are not 0. However, the wavefunction must be zero outside the box, that is, for values of x less than 0 and greater than L it must be zero. As drawn, the wavefunction jumps suddenly from nonzero values at the walls to zero values immediately beyond the walls outside the box. Therefore, the wavefunction as drawn in Figure 8.3 is not a good wavefunction because it is not continuous. This function cannot represent a quantum particle in the box.
FIGURE 8.3. A wavefunction inside the box that is discontinuous. The wavefunction is called ϕ. The vertical axis is the amplitude of the wavefunction. The dashed line shows where the wavefunction is zero, which must be outside the box. The wavefunction has a nonzero value at the walls and then must drop discontinuously (not smoothly) to zero outside the box.
Wave Function Must Be Zero at the Walls
For the wavefunctions representing the particle in the box to be physically acceptable functions, their values at the walls must be zero so that there is no discontinuity at the walls. This is not a difficult condition to meet. Figure 3.1 illustrates a wave in free space. It oscillates positive and negative. Every time it goes from positive to negative or negative to positive, it crosses through zero. In fact, the zero points are separated by one-half of a wavelength. So what we need to do to get good particle in a box wavefunctions is pick waves with wavelengths such that they fit in the box with their zero points right at the walls. Figure 8.4 shows three examples of waves that are acceptable particle in the box wavefunctions. The one on the bottom, labeled n = 1, is composed of a single half wavelength. It starts on the left with an amplitude of 0, goes through a maximum, and then is zero again at the wall at position L. The next wave up, labeled n = 2, is one full wavelength. Again, it starts at the left wall with amplitude zero, goes through a positive peak, back through zero, a negative peak, and is zero at the wall at position L. The wave labeled n = 3 is one and one-half wavelengths. Any wave that is an integer number of half wavelengths, that is 1, 2, 3, 4, 5, etc. half wavelengths, and has a wavelength so that it starts at zero on the right and ends at zero on the left is okay.
The label n is the number of half wavelengths in the particular wavefunction. For n = 1, the wavelength, λ, is 2L because the box has length L, and n = 1 corresponds to a half wavelength. For n = 2, the wavelength is L because exactly one wavelength fits between the walls. For n = 3, 3 half wavelengths = L. That means 1.5λ = L. Then λ = L/1.5, so λ = 2L/3. Notice that there is a general rule here. λ = 2L/n, where n is an integer. For n = 1, λ = 2L. For n = 2, λ = 2L/2 = L. For n = 3, λ = 2L/3, and so forth.
FIGURE 8.4. Three examples of wavefunctions, ϕ, inside the box that are continuous. They have been shifted upward for clarity of presentation. The vertical axis is the amplitude of the wavefunction. The dashed line shows where the wave function is zero, which it must be outside the box. The wavefunctions, which have zero values at the walls, are continuous across the walls.
Nodes Are Points Where the Wavefunction Crosses Zero
Nodes are another important feature of the wavefunctions. Nodes are points where the wavefunction crosses zero, going from positive to negative, or negative to positive. The n = 1 wavefunction has no nodes. The n = 2 wavefunction has one node right in the middle of the box. The