Particle-in-a-Box Wave Functions

This image shows the four lowest-energy wave functions for a particle in a box, or infinite well (Figure 4.9, page 156). Their variation with position x along the horizontal axis is given by sin(npx/L). The lowest-energy/ground state, for which n = 1 and the position-dependence is thus sin(1px/L), is zero at x = 0 and x = L, but nowhere in between--it has one antinode; the next-lowest, sin(2px/L), is zero at x = 0, x = L/2, and x = L--it has two antinodes; etc.

However, also represented are the functions' variation with time t (noted in footnote 8, page 156). All stationary-state wave functions have a "temporal part" given by exp(-iwt), whose real part is cos(wt) and imaginary part is -sin(wt). Thus, both real and imaginary parts are oscillatory, of angular frequency w, and they are out of phase by one-quarter cycle (since one is a cosine while the other is a sine). When one is large the other is small, and vice versa. In the image shown the "real axis" is vertical and the "imaginary axis" is perpendicular to the plane of the well.

Note that the higher the energy the faster the wave function revolves about the x axis through the real-imaginary directions. This happens because of the relationship between energy and frequency: E = (h/2p) w. The higher the energy, the higher will be the frequency. The energy levels are given by E = n²h²/(8p²mL²), so we see that the the next-to-lowest energy, n = 2, is four times the lowest energy (2² vs 1²), and its frequency should therefore be four times as high. By similar logic the n = 3 wave function should have a frequency nine times that of the ground state, and the n = 4 should have a frequency sixteen times as high. You should check to see that this is indeed the case in the image.