FIGURE Q40.1 shows the de Broglie waves of three equal-mass particles. Which one is moving most slowly? Explain.

FIGURE Q40.1

The correspondence principle says that the average behavior of a quantum system should begin to look like the Newtonian solution in the limit that the quantum number becomes very large. What is meant by “the average behavior” of a quantum system?

A particle in a potential well is in the $n=5$ quantum state. How many peaks are in the probability density $P\left(x\right)=\left|\psi \left(x\right)\right|{}^2?$

What is the quantum number of the particle in FIGURE Q40.4? How can you tell?

FIGURE Q40.4

Rank in order, from largest to smallest, the penetration distances ${\eta }_{\text{A}}$ to ${\eta }_{\text{C}}$ of the wave functions corresponding to the three energy levels in FIGURE Q40.5.

FIGURE Q40.5

Consider a quantum harmonic oscillator.

1. What happens to the spacing between the nodes of the wave function as $\left|x\right|$ increases? Why?

2. What happens to the heights of the antinodes of the wave function as $\left|x\right|$ increases? Why?

3. Sketch a reasonably accurate graph of the $n=8$ wave function of a quantum harmonic oscillator.

FIGURE Q40.7 shows two possible wave functions for an electron in a linear triatomic molecule. Which of these is a bonding orbital and which is an antibonding orbital? Explain how you can distinguish them.

FIGURE Q40.7

Four quantum particles, each with energy $E$, approach the potential-energy barriers seen in FIGURE Q40.8 from the left. Rank in order, from largest to smallest, the tunneling probabilities $(P_{\text{tunnel}}){}_A$ to $(P_{\text{tunnel}}){}_D$.

FIGURE Q40.8

An electron has a 0.0100 probability (a 1.00% chance) of tunneling through a potential barrier. If the width of the barrier is doubled, will the tunneling probability be 0.0050, 0.0025, or 0.0001? Explain.