a. State the general normalization condition. Customize the general normalization condition specifcally for the infnite square well, paying particular attention to the limits of integration.

b. Integrate the normalization condition to determine the values of A_n. You may use an integration table. What is very important about the amplitude of these wave functions?

c. Write the full, normalized solution to the wave function describing a particle in a one-dimensional infnite square well.

d. Sketch the three lowest stationary states of the particle in the infnite square well.

e. Sketch the probability densities of the three lowest stationary states of the particle in the infnite square well.

f. Consider what the probability density would look like for the n = 10 state, the n = 100 state, the n = 1, 000 state. What do you notice about the probability density as n increases?

- Classically, there is an equal probability of nding the particle anywhere in the box, and thus the probability density (x) = c where c is a constant.

g. Integrate the classical probability density to determine the value of c. Does this make sense? How would this value change if the well were twice as wide? Explain why this is reasonable.

- There are several very important properties of the wave functions describing a particle in an innite square well. 1. As we saw when we normalized the wave functions, the amplitude is independent of the state n. 2. Solutions have nodes like classical standing waves. It is important to note that quantum mechanics does not count the boundaries as nodes. Thus in quantum mechanics, the number of nodes is equal to n 1, where as in classical mechanics, the number of nodes is equal to n + 1. 3. Solutions alternate between even and odd functions.

h. Return to your sketches of the first three wave functions and prove this to yourself.

4. As you increase the energy, you add one more node. 5. Each set of wave functions is orthogonal, meaning that _^m (x)_n (x) dx = 0 for all m =\ n.

i. Prove this to yourself by integrating the general form of the wave functions over the limits 0 to a. You may use an integral table.