Chapter 4, Problem 37P

(a) Using Equation 4.36, make a graph of the $n=3$ wave function and the square of this wave function for a particle in a box with the edges at $x=0$ and $x=7\text{Å}$ . Estimate the probability of finding the particle between $x_1=3\text{Å}$ and $x_2=4\text{Å}$ . You may either do an integral or find the area under the curve by counting the number of rectangular boxes (roughly estimating the fractional boxes), and then multiply by the area of each box, watching your units. Show your calculation with units for credit.

(b) Verify that the $n=3$ wave function in (a) is an allowed quantum state (sometimes called an eigenfunction) of $\wp$ by substituting it for $\psi$ in the left side of the equation $\wp \psi =E\psi$ immediately following Eq. 4.32. Calculate the derivatives to see if your result $E\psi$ is a constant multiplied by the original $n=3$ wavefunction. What does the constant you get tell you about the energy of thiseigenstate? Is it in agreement with the energy for the $n=3$ level you would calculate from Equation 4.37?