# Chapter 40 - Quantum Mechanics I: Wave Functions - Problems - Exercises - Page 1354: 40.17

See explanation.

#### Work Step by Step

Calculate the probability density from the given wavefunction. $$|\psi(x)|^2=\frac{2}{L}\sin^2 \frac{2 \pi x}{L}$$ a. The probability is zero where the probability density is zero. $\sin \frac{2 \pi x}{L}=0$ $\frac{2 \pi x}{L}=n \pi$, where n = 0, 1, 2,… $x=0, \frac{L}{2}, L$ are the only solutions for x inside the box. b. The probability is highest where the probability density is maximum. A sine function’s maximum magnitude is 1. $\sin \frac{2 \pi x}{L}=\pm 1$ $\frac{2 \pi x}{L}=\frac{n \pi}{2}$, where n = 1, 3, 5,… $x=\frac{L}{4}, \frac{3L}{4}$ are the only solutions for x inside the box. c. Yes, this matches the picture.

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