40.3.5 Solve III: Normalize the Wave Functions

We can determine the constant $A$ by requiring the wave functions to be normalized. The normalization condition, which we found in Chapter 39, is

\int_{- \infty}^{\infty} {| ψ (x) |}^{2} d x = 1

This is the mathematical statement that the particle must be somewhere on the $x$ -axis. The integration limits extend to $\pm \infty$ , but here we need to integrate only from 0 to $L$ because the wave function is zero outside the box. Thus

\int_{0}^{L} {| ψ_{n} (x) |}^{2} d x = A_{n}^{2} \int_{0}^{L} {sin}^{2} (\frac{n π x}{L}) d x = 1

(40.24)

A_{n} = {[\int_{0}^{L} {sin}^{2} (\frac{n π x}{L}) d x]}^{- 1 / 2}

(40.25)

We placed a subscript $n$ on $A_{n}$ because it is possible that the normalization constant is different for each wave function in the family. This is a standard integral. We will leave it as a homework problem for you to show that its value, for any $n$ , is

A_{n} = \sqrt{\frac{2}{L}} n = 1, 2, 3, \dots

(40.26)

We now have a complete solution to the problem. The normalized wave function for the particle in quantum state $n$ is

ψ_{n} (x) = {\begin{array}{l} \sqrt{\frac{2}{L}} sin (\frac{n π x}{L}) & 0 \leq x \leq L \\ 0 & x < 0 and x > L \end{array}

(40.27)