Chapter 41

Quantum Mechanics

Multiple Choice

1. A 15-kg mass, attached to a massless spring whose force constant is 2500 N/m, has an amplitude of 4 cm. Assuming the energy is quantized, find the quantum number of the system, n, if En = nhf.

a. 1.5 × 1033 b. 3.0 × 1033 c. 4.5 × 1033 d. 5.4 × 1033 e. 1.0 × 1033

2. Find the kinetic energy (in terms of Planck’s constant) of a baseball (m = 1 kg) confined to a one-dimensional box that is 25 cm wide if the baseball can be treated as a wave in the ground state.

a. 3 h2 b. 2 h2 c. h2 d. 4h2 e. 0.5 h2

3. Calculate the ground state energy (in eV) for an electron in a box (an infinite well) having a width of 0.050 nm.

a. 10 b. 75 c. 24 d. 150 e. 54

4. A particle is in the ground state of a one-dimensional box of length 1 m. What is the minimum value of its momentum (in kg · m/s)?

a. 9.9 × 10–34 b. 6.6 × 10–34 c. 3.3 × 10–34 d. 13.2 × 10–34 e. cannot be solved unless mass of particle is known.

333

334 CHAPTER 41

5. A particle is in the first excited state of a one-dimensional box of length 1 m. What is the minimum value of its momentum (in kg · m/s)?

a. 3.3 × 10–34 b. 6.6 × 10–34 c. 9.9 × 10–34 d. 13.2 × 10–34 e. 22.3 × 10–34

6. A particle is in the second excited state of a one-dimensional box of length 1 m. What is its momentum (in kg · m/s)?

a. 6.6 × 10–34 b. 3.3 × 10–34 c. 9.9 × 10–34 d. 13.2 × 10–34 e. cannot be solved unless mass of particle is known

7. What is the quantum number n of a particle of mass m confined to a one-dimensional box of length L when its momentum is 4h/L?

a. 1 b. 4 c. 2 d. 8 e. 16

8. What is the quantum number n of a particle of mass m confined to a one-dimensional box of length L when its energy is 2 h2/mL2?

a. 2 b. 8 c. 4 d. 1 e. 16

9. The wave function for a particle confined to a one-dimensional box located between x = 0 and x = L is given by Ψ(x) = A sin (nπ x/L) + B cos (nπ x/L) . The constants A and B are determined to be

L/2 , 0 a.

b. L/1 , L/1

c. 0, L/2

d. L/2 , L/2 e. 2/L, 0

Quantum Mechanics 335

10. Classically, the concept of “tunneling” is impossible. Why?

a. The kinetic energy of the particle would be negative. b. The velocity of the particle would be negative. c. The total energy of a particle is equal to the kinetic and potential energies. d. The kinetic energy must be equal to the potential energy. e. The total energy for the particle would be negative.

11. When a particle approaching a potential step has a total energy that is greater than the potential step, what is the probability that the particle will be reflected?

a. . 0P <b. . 0=P

=c. . 10>P∞=P

Pd. . e. .

12. A particle has a total energy that is less than that of a potential barrier. When the particle penetrates the barrier, its wave function is

a. a positive constant. b. exponentially increasing. c. oscillatory. d. exponentially decreasing. e. none of the above.

13. The ground state energy of a harmonic oscillator is

a. E = ω

b. E = ω/2

c. E = (2/3) ω d. E = 0 e. E = ω/4

14. Particles in degenerate energy levels all have the same

a. momentum. b. quantum numbers. c. energy. d. all of the above. e. velocity.

336 CHAPTER 41

15. The wave function for a particle in a one-dimensional box is ψ = A sin ⎟⎠⎞

⎜⎝⎛

Lxnπ

.

Which statement is correct?

a. This wavefunction gives the probability of finding the particle at x.

b. 2)(xΨ gives the probability of finding the particle at x.

c. xx ΔΨ 2)( gives the probability of finding the particle between x and x + Δx.

d. gives the probability of finding the particle at a particular value

of x.

∫ ΨL

dxx0

)(

e. ∫ ΨL

dxx0

2)( gives the probability of finding the particle between x and x + Δx.

16. The fact that we can only calculate probabilities for values of physical quantities in quantum measurements means that

a. radiation and matter are not described by mathematical relations between measurements.

b. the probabilities cannot be calculated from mathematical relationships. c. the results of physical measurements bear no relationship to theory. d. the average values of a large number of measurements correspond to the

calculated probabilities. e. the average of the values calculated in a large number of different theories

corresponds to the results of a measurement.

17. When the potential energy of a system is independent of time, the wave function of the system

a. is a constant. b. is directly proportional to the time. c. cannot be normalized. d. depends only on the center of mass, , of the system. Re. depends on the vector positions, ri

)(x

, of each particle in the system.

18. A physically reasonable wave function, ψ , for a one-dimensional system must

a. be defined at all points in space. b. be continuous at all points in space. c. be single-valued. d. obey all the constraints listed above. e. obey only (b) and (c) above.

Quantum Mechanics 337

19. The average position, or expectation value, of a particle whose wave function )(xψ depends only on the value of x, is given by =>< x

a. ∫∞+

∞−dxxx )(ψ .

b. . ∫∞+

∞−dxxx )(ψ

c. ∫∞+

∞−dxx

x)(

22ψ .

d. ∫∞+

∞−dxx

x)(

2

2

ψ .

e. . ∫∞+

∞−dxxxx )()(* ψψ

20. The expectations value of a function of x when the wave function depends

only on x is given by

)(xf

=>< )(xf

a. ∫ ∞−

∞+

∞+dxxxf )()( ψ .

b. . ∫ ∞−dxxxf )()( ψ

∞+dxxxf )()( *ψc. . ∫ ∞−

∞+dxxxfx )()()(* ψψd. . ∫ ∞−

e. ∫ ∞− 2

∞+dxx

xfx )(

)()(

2* ψψ .

21. If the interaction of a particle with its environment restricts the particle to a finite region of space, the result is the quantization of _____ of the particle.

a. the momentum b. the energy c. the velocity d. all of the above properties e. only properties (a) and (b)

338 CHAPTER 41

22. When U(x) is infinitely large elsewhere, the wave function of a particle rto the region Lx <<0 w

estricted here U(x) = 0 , e the form may hav ψ (x) =

a. ⎟⎠⎞

⎜⎝⎛

Lxn

Aπ

sin .

b. ⎟⎠⎞

⎜⎝⎛

Lxn

Bπ

cos .

c. . LxnAe /π

d. ⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

Lxn

BL

xnA

ππcossin .

e. ⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛

Lxn

BL

xnA

ππcossin .

23. A particle in a finite potential well has energy E, as shown below.

0 L

EU

I II III

The wave function in region I where 0<x has the form ψ = I

a. . CxAe−Cxb. . Ae

kxF sinc. .

d. . kxG cos

e. . kxGkxF cossin +

24. A particle in a finite potential well has energy E, as shown below.

0 L

EU

I II III

The wave function in region II where has the form 0>x =IIψ

a. . Cx−AeCxAeb. .

c. . kxF sin

kxG cos

kxGkxF cossin +

d. .

e. .

Quantum Mechanics 339

25. A particle in a finite potential well has energy E, as shown below.

0 L

EU

I II III

The wave function in region III where has the form Lx > =IIIψ

a. . CxAe−

b. . CxAec. . kxF sin

d. . kxG cos

e. . kxGkxF cossin +

26. Quantum tunneling occurs in

a. nuclear fusion. b. radioactive decay by emission of alpha particles. c. the scanning tunneling microscope. d. all of the above. e. only (b) and (c) above.

27. The graph below represents a wave function ψ (x) for a particle confined to

−2.0 ≤ x ≤ +2.00 m0 m . The value of the normalization constant A may be

ψ ( x)

A

–2 +2 x

−

14

. a.

−

12

. b.

c. +

12

.

d. +

14

.

e. either −

12

or +

12

.

340 CHAPTER 41

28. The graph below represents a wave function ψ (x) for a particle confined to

−2.00 m ≤ x ≤ +2.00 m . The value of the normalization constant A may be

ψ ( x)

–2 +2 x

–A

a. −

14

.

b. −

12

.

c. +

12

.

+

14

. d.

−

1e. either

2 or

+

12

.

29. The graph below represents a wave function ψ (x) for a particle confined to

−4.00 m ≤ x ≤ +4.00 m . The magnitude of the normalization constant A is

ψ (x)

+A

+4 x

–4

–A

1. a.

41

. b. 8

c.

12

.

d. 8 . e. 4.

Quantum Mechanics 341

30. Frank says that quantum mechanics does not apply to baseballs because they do not jump from quantum state to quantum state when being thrown. Francine agrees with him. She says that there is no uncertainty in a baseball’s position or momentum. Are they correct, or not, and why?

a. They are correct because the first excited state of a baseball is at a higher energy that any baseball ever receives. Therefore we cannot determine whether or not there is uncertainty in its position or momentum.

b. They are correct because the first excited state of a baseball is at a higher energy that any baseball ever receives. Therefore its position and momentum are completely uncertain until it is caught.

c. They are wrong because the baseball goes through so many quantum states in being thrown that we cannot observe the transitions. The uncertainties in its position and momentum are too small to observe.

d. They are wrong because the baseball goes through so many quantum states in being thrown that we cannot observe the transitions. Because of the number of transitions its position and momentum are completely uncertain until it is caught.

e. Quantum mechanics states that they are correct as long as they do not make any observations, but wrong as soon as they begin to make observations.

31. The graph below shows the value of the probability density | in the region ψ (x)|2

−3.00 m ≤ x ≤ +3.00 m . The value of the constant A is

|ψ (x)|2

A2

x

–3 +3

a. −

1.

3

−1

. b. 3

+1

. c. 3

d. +

13

.

e. either −

1

3 or

+

1

3.

342 CHAPTER 41

32. The wave function ψ (x) of a particle confined to 0 ≤ x ≤ L is given by ψ (x) = Ax .

ψ (x) = 0 for x < 0 and x > L . When the wave function is normalized, the probability density at coordinate x has the value

a.

2L2 x .

b.

2L2 x2 .

c.

2L3 x2 .

d.

3L3 x2 .

e.

3L3 x3 .

Quantum Mechanics 343

Chapter 41

Quantum Mechanics

1. a

2. b

3. d

4. c

5. b

6. c

7. d

8. c

9. a

10. a

11. d

12. d

13. b

14. c

15. c

16. d

17. e

18. d

19. e

20. d

21. d

22. a

23. b

24. e

25. a

26. d

27. c

28. b

29. b

30. c

31. e

32. d

344 CHAPTER 41