1. $v{}_A=v{}_B>v{}_C$. The de Broglie wavelength is $\text{λ}=h/mv$, so slower particles have longer wavelengths. The wave amplitude is not relevant.

2. c. The $n=2$ state has a node in the middle of the box. The antinodes are centered in the left and right halves of the box.

3. $n=5$. There are five antinodes and four nodes (excluding the ends).

4. d. The wave function reaches zero abruptly on the right, indicating an infinitely high potential-energy wall. The exponential decay on the left shows that the left wall of the potential energy is not infinitely high. The node spacing and the amplitude increase steadily in going from right to left, indicating a steadily decreasing kinetic energy and thus a steadily increasing potential energy.

5. c. $E=(n-\frac{1}{2})\hslash \omega \text{, so}\frac{5}{2}\hslash \omega$ is the energy of the $n=3$ state. An $n=3$ state has 3 antinodes.

6. b. The probability of tunneling through the barrier increases as the difference between $E$ and $U_0$ decreases. If the tunneling probability increases, the reflection probability must decrease.

1.

1. Infrared

2. 1.5 nm

3. 21 eV

5. 17 eV

9.

1. 0.159 nm

2. 0.195 nm

3. 0.275 nm

11. 2.3 pm

13.

15.

• a.

  

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• b.

  

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17.

1. 0.49 eV, 1.5 eV, 2.4 eV

2. 640 nm

19. 1.4 N/m

21. 519 nm

25. 0.0095%

27.

1. 250,000,000

29.

1. 37.7 eV, 151 eV, 339 eV, 603 eV

2. 11.0 nm, 4.12 nm, 6.59 nm, 2.20 nm, 2.75 nm, 4.71 nm

3. Ultraviolet

33.

35. 10%

37.

1. 0.15 nm

2. One atomic diameter

39.

1. $A_1={\displaystyle \frac{1}{{\left(\pi b^2\right)}^{1/4}}}$

2. $\text{Prob}(x<-b\text{or }x>b)={\displaystyle \frac{2}{\sqrt{\pi b^2}}}{\displaystyle {\int }_b^{\infty }}{e}^{-x^2/b^{2}dx}$

3. 15.7%

41.

• a. $P_{\text{class}}\left(y\right)=\left({\displaystyle \frac{1}{2h}}\right){\displaystyle \frac{1}{\sqrt{1-(y/h)}}}$

• b.

  

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43. $\frac{1}{4}\text{of the radius}$

45. 0.012

47. 9.7%

49. ${10}^{-1.17\times {10}^{32}}$