40.7 Wave-Function Shapes

Bound-state wave functions are standing de Broglie waves. In addition to boundary conditions, two other factors govern the shapes of wave functions:

  1. The de Broglie wavelength is inversely dependent on the particle’s speed. Consequently, the node spacing is smaller (shorter wavelength) where the kinetic energy is larger, and the spacing is larger (longer wavelength) where the kinetic energy is smaller.

  2. A classical particle is more likely to be found where it is moving more slowly. In quantum mechanics, the probability of finding the particle increases as the wave-function amplitude increases. Consequently, the wave-function amplitude is larger where the kinetic energy is smaller, and it is smaller where the kinetic energy is larger.

We can use this information to draw reasonably accurate wave functions for the different allowed energies in a potential-energy well.

TACTICS BOX 40.1 Drawing wave functions

1 Draw a graph of the potential energy U(x). Show the allowed energy E as a horizontal line. Locate the classical turning points.

2 Draw the wave function as a continuous, oscillatory function between the turning points. The wave function for quantum state n has n antinodes and (n−1) nodes (excluding the ends).

3 Make the wavelength longer (larger node spacing) and the amplitude higher in regions where the kinetic energy is smaller. Make the wavelength shorter and the amplitude lower in regions where the kinetic energy is larger.

4 Bring the wave function to zero at the edge of an infinitely high potential-energy “wall.”

5 Let the wave function decay exponentially inside a classically forbidden region where E<U. The penetration distance η increases as E gets closer to the top of the potential-energy well.

Workbook Exercises 10–13

EXAMPLE 40.8 Sketching wave functions

FIGURE 40.18 shows a potential-energy well and the allowed energies for the n=1 and n=4 quantum states. Sketch the n=1 and n=4 wave functions.

FIGURE 40.18 A potential-energy well.

The figure illustrates a potential energy well of varying widths.

VISUALIZE The steps of Tactics Box 40.1 have been followed to sketch the wave functions shown in FIGURE 40.19.

FIGURE 40.19 The n=1 and n=4 wave functions.

The figure illustrates two wave functions of a potential energy well of varying widths.