University of the Philippines College of Science

PHYSICS 73

Long Problem Set 3

1st semester AY 2010 - 2011

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INSTRUCTIONS: Choose the best answer and shade the corresponding circle in your answer sheet. To change your answer, cross-out and sign your original

answer and then shade your new answer.

USEFUL CONSTANTS: Mass of electron: 9.11 x 10-31 kg Charge of electron: 1.6 x 10-19 C 1eV = 1.6 x 10-19 J h = 6.626 x 10-34 J s ħ = 1.05 x 10-34 J s

Meeting 29

1. In the photoelectric effect, which of the following quantities will increase as the intensity

of the light incident on the metal increases?

a. Photocurrent

b. Stopping potential

c. Wavelength of the incident light

d. Frequency of the incident light

e. Maximum kinetic energy of the photoelectrons

2. Silver has a work function of 4.3 eV. What would be the frequency of the light incident on

this metal surface for the stopping potential to be equal to 3.0 V?

a. 2.32 x 1015

Hz

b. 1.76 x 1015

Hz

c. 3.45 x 1015

Hz

d. 1.29 x 1015

Hz

e. 4.39 x 1015

Hz

3. What is the momentum of a photon of blue light (wavelength 470 nm)?

a. 2.45 x 10-27

kg m/s

b. 3.74 x 10-27

kg m/s

c. 1.41 x 10-27

kg m/s

d. 4.23 x 10-27

kg m/s

e. 5.38 x 10-27

kg m/s

Meeting 30

4. In an x-ray tube, electrons are accelerated up to a voltage of 2.48 kV. What is the

wavelength of the x-rays produced by this tube?

a. 0.05 nm

b. 0.5 nm

c. 0.005 nm

d. 5 nm

e. 0.0005 nm

5. An x-ray is scattered over an angle of 45 degrees. What is the difference between the

wavelengths of the incident and scattered x – ray?

a. 0.711 pm

b. 4.14 pm

c. 0.113 pm

d. 0.659 pm

e. 1.71 pm

6. The difference between the wavelengths of an incident and a scattered x-ray is equal to

one-fourth the value of the ratio h/mc. What is the value of the scattering angle?

a. 20.70 b. 10.4

0 c. 41.4

0

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d. 75.50 e. 5.2

0

Meeting 31

7. In the Bohr model of the hydrogen atom, what is the lowest possible energy that can be

occupied by an electron orbiting the nucleus?

a. 13.6 eV

b. -3.40 eV

c. 1.51 eV

d. -13.6 eV

e. -1.51 eV

8. An electron orbiting a hydrogen nucleus makes a transition from the sixth excited state to

the first excited state. What is the wavelength of the photon emitted by the atom when the

electron makes this transition?

a. 294 nm

b. 307 nm

c. 0.938 nm

d. 0.585 nm

e. 397 nm

9. The radius of the orbit of an electron in a hydrogen atom is equal to 1.32 nm. What is the

angular momentum of an electron occupying this orbit?

a. 5.275 x 10-34

J s

b. 4.220 x 10-34

J s

c. 3.165 x 10-34

J s

d. 2.110 x 10-34

J s

e. 1.055 x 10-34

J s

Meeting 32

10. An American Major League Baseball pitcher can throw pitches that can reach speeds of

up to 100 miles per hour, or 44.7 meters per second. If a baseball whose mass is 0.145 kg is

thrown at this speed, what is its de Broglie wavelength?

a. 6.63 x 10-34

m

b. 3.31 x 10-34

m

c. 1.02 x 10-34

m

d. 2.25 x 10-34

m

e. 4.48 x 10-34

m

11. A particle with mass 1.5 x 10-25

kg is moving with speed b = 0.90. What is its de Broglie

wavelength?

a. 5.7 nm

b. 3.3 nm

c. 4.9 nm

d. 2.1 nm

e. 1.0 nm

12. A consequence of the wave-particle duality is that electrons can be:

a. Multiplied

b. Divided

c. Transmitted

d. Diffracted

e. Insulated

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Meeting 33

13. According to the Heisenberg Uncertainty Principle, which of the following situations

cannot physically happen?

a. An electron's position can be determined with certainty.

b. An electron's momentum can be determined with certainty.

c. An electron's momentum cannot be determined with certainty.

d. a and b simultaneously

e. a and c simultaneously

14. A particle's momentum is measured to be 2.55 kg m/s, with an error in measurement of

0.638 kg m/s. If the position of the particle is measured at the same instant that this

momentum measurement was made, what would be the smallest error in the measurement of

the particle's position?

a. 6.63 x 10-34

m

b. 5.28 x 10-35

m

c. 2.11 x 10-34

m

d. 0.638 m

e. 8.27 x 10-35

m

15. In a single-slit diffraction experiment, the width of the slit is measured to be 0.100 nm.

What is the minimum value of the uncertainty in the momentum of a photon that enters this

slit?

a. 5.28 x 10-25

kg m/s

b. 1.04 x 10-24

kg m/s

c. 2.11 x 10-44

kg m/s

d. 3.25 x 10-24

kg m/s

e. 4.48 x 10-30

kg m/s

Meeting 34

16. Consider a free particle with definite momentum. Which of the following solutions to the

time-independent Schrodinger equation for this system describes this state?

a. Aeikx

+ Aeik'x

b. Aeikx

c. Ae-kx

d. Ax2

e. Ax+B

17. What would be the probability density function for a solution to the time-independent

Schrodinger equation for a given system whose explicit form is

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18. If the wave function is normalized, what is the probability of finding the state described

by the wave function somewhere in space?

a. 0

b. 0.5

c. 0.25

d. 1

e. p/2

Meeting 35

For the next four questions: A particle is constrained to move at constant speed between two

walls x=0.00 cm and x=7.00 cm. It has a constant probability density inside the walls and

zero outside it.

19. What is the probability density?

a. (¼)(cm-1

)

b. 0.125 cm-1

c. 0 cm-1

d. eikx

cm-1

e. 0.143 cm-1

20. What is the probability that the particle is between 3.00 cm and 7.00 cm?

a. ¼

b. 1/7

c. 0

d. 1

e. 0.57

21. A particle is described by the wavefunction

Normalize the wavefunction. That is, find A in terms of a and b.

a. A = (3/b)1/2

b. A = (3/a)1/2

c. A = (3ab)1/2

d. A = (3a/b)1/2

e. A = 3/ab

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Meeting 36

22. Determine the mass of a free particle whose wave function is in the plane wave

(x,t) = A exp(i( 2.5 x 1011

m-1

x – (2.1 x 1013

Hz) t)

A. 1.56 x 10-25

kg

B. 9.80 x 10-25

kg

C. 2.48 x 10-26

kg

D. 5.28 x 10-35

kg

E. 28 x 10-37

kg

23. The state of a particle is given by ψ(x) = Aψ1 + Bψ2, where ψ1 and ψ2 are solutions to the

time-independent Schrodinger equation with energies E1 and E2 respectively. Which of the

following statements is true?

a. Measuring the energy of the particle will yield E1 + E2

b. Measuring the energy of the particle will always give a value E1.

c. Measuring the energy of the particle will always give a value E2.

d. Measuring the energy will give a value of both E1 and E2.

e. The particle has no definite energy.

24. The time-independent Schrodinger equation is actually an eigenvalue equation where the

solution wavefunction ψ is the eigenfunction. Which of the following observables correspond

to the eigenvalue of ψ?

a. position

b. momentum

c. energy

d. frequency

e. Planck’s constant

Meeting 37

25. Consider an electron trapped in a one-dimensional infinite square well with boundaries at

x = -L/2 and x = L/2. What is the value of the energy at the 2nd

excited state?

a.

b.

c.

d.

e.

For the next two questions: Let Ψn(x) be the energy eigenfunctions of the infinite square well

of width L, and energy En. Suppose that a particle in the square well was prepared in the state

(t= 0.00 s)

ψ(x, t=0.00 s) = cos(45°) Ψ1(x) + sin(45°)cos(30°)Ψ2(x) + sin(45°)sin(30°) Ψ3(x)

26. What is the probability amplitude that the particle is at the 2nd

excited state?

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A. 0.3536

B. 0.7071

C. 0.6124

D. 0.1250

E. 0.3750

27. What is the probability of finding the particle in the first excited state?

A. 0.3536

B. 0.7071

C. 0.6124

D. 0.1250

E. 0.3750

Meeting 38

The following three problems will be about the finite square well of width L, height U0. If the

height of the well U0= 6E∞, where E∞ = (h2)/(8mL

2), then the first three eigenenergies are

known to be E1=0.625 E∞, E2= 2.43 E∞, E3= 5.09 E∞.

28. What would happen to the number of bound states as the depth of a finite square well

increases?

a. The number of bound states increases.

b. The number of bound states remains the same.

c. The number of bound states decreases.

d. The number of bound states are reduced by half.

e. The number of bound states would fluctuate between two values.

29. Electron in a finite square well. An electron is assumed to be modeled by a particle in a

1D finite square well of width 1.50 nm. If the electron is in the ground state, what maximum

wavelength can a photon have and still liberate the electron?

A. 345 nm

B.1259 nm

C. 520 nm

D. 2974 nm

E. 1.36 x 10-5

m

30. Proton in a finite square well. A proton is bound in a square well of width 4.0 fm = 4.0 x

10-15

m. The depth of the well is six times the ground level energy E∞ of the corresponding

infinite well. The proton makes a transition from the level with energy E1 to the level with

energy E3, by absorbing a photon. What is the energy of this photon?

A. 12.8 MeV

B. 1.33 MeV

C. 76.8 MeV

D. 10.9 MeV

E. 5.79 MeV

Meeting 39

Tunneling formulae:

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31. An electron is moving past the square barrier shown on the right, but with E greater than

the barrier height U0. If E=2U0, what is the ratio of the DeBroglie wavelength of the electron

in the region x> L to the wavelength for 0<x<L?

A. 1: 2

B. 1: (2)1/2

C. 2:1

D. (2)1/2

:1

E. none of the above

32. An electron encounters a barrier with length L and depth that is the half the electron’s

kinetic energy 2U0=E0. What is the corresponding transmission probability?

A. (1/16)E0

B. 1

C. 16 E0

D. E0

E. 2E0

33. which of the following will increase the tunnelling probability?

a. increasing the energy of the particle

b. increasing the width of the barrier

c. increasing the mass of the particle

d. increasing the height of the barrier

e. increasing the charge of the particle

Meeting 40

34. The 3rd

excited state of a particle in a 1D harmonic oscillator potential is

a. 3/2 ħω

b. 3ħω

c. 5/2 ħω

d. 7/2 ħω

e. 2 ħω

35. The ground state energy of a harmonic oscillator is 5.60 eV. If it undergoes a transition

from its n = 3 to its n = 2 state by emitting a photon, what is the wavelength of the photon?

a. 111 nm

b. 222 nm

c. 333 nm

d. 444 nm

e. 1.77 x 10-26

m

36. Which of the following statements is false?

a. the ground state energy for the classical harmonic oscillator is zero.

b. the ground state energy for the quantum harmonic oscillator is nonzero.

c. the energies of both Newtonian and Quantum harmonic oscillators are quantized.

d. there is a nonzero probability for the quantum harmonic oscillator to exist at |x| > A where

A is its classical amplitude.

e. it is impossible to find a classical oscillator outside its classical amplitude A.

Meeting 41

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37. Consider an electron in a 3D particle in a box with equal lengths along x, y and z.

Describe the degeneracy for the 3rd

excited state.

a. non degenerate

b. doubly degenerate

c. triply degenerate

d. four-fold degeneracy

e. six-fold degeneracy

38. What would be the ground state energy for a three-dimensional harmonic oscillator?

A. 2

1

B.

C. 2

3

D. 0

E. 2

39. Degenerate? The energy eigenfunctions of the three-dimensional harmonic oscillator

may be labeled by three quantum numbers, n1, n2, and n3 . The allowed values of each

quantum number are the set of positive integers and 0. If the energy is given by

E= (n1+n2+n3+3/2) hω/( 2π)

What is the degree of degeneracy of the 1st excited state?

A. nondegenerate

B. 2

C. 3

D. 4

E. 5

Meeting 42

40. Consider a hydrogen atom whose state at t=0 is described by the following wave function

What is the probability that a measurement of the square of the total angular momentum

yields 2.22 x 10-68

kg2 m

4/s

2?

A. 0.867

B. 0.750

C. 1.183

D. 0.500

E. 3.34 x 10-4

41. Degeneracy. How many states correspond to the 3rd

excited energy level of a Hydrogen

atom if there is no external magnetic field present and spin is neglected?

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A. 1

B. 4

C. 16

D. 3

E. 9

42. Orbital angular momentum. What is the minimum angle between L and Lz if l =

4?

A. 116.550

B. 63.440

C. 26.560

D. 47.870

E. 42.130

Meeting 43

43. The Zeeman effect in hydrogen may be regarded as evidence of quantization of ______:

A. the square of the angular momentum,

B. the angular momentum in the z direction

C. energy

D. kinetic energy

E. potential energy

44. Consider a hydrogen atom exposed in a uniform external magnetic field. Which set of

quantum numbers describe the 4th

excited state if spin is neglected?

a. n = 4, l = 0, ml = 0

b. n = 3, l = 0, ml = 0

c. n = 2, l = 1, ml = 0

d. n = 2, l = 1, ml = -1

e. n = 2, l = 1, ml = 1

Meeting 44

45. Consider a two electron system in 1 dimension. Let Ψ(x,+;y,+) be the probability density

amplitude of the finding the first electron at x with spin up, and the second electron at y with

spin up. Which of the following functional forms (choices A to D) for this probability

amplitude is impossible?

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A. Ψ(x,+;y,+) =A exp(-x2-y

2)

B. Ψ(x,+;y,+) =A exp(-x2+2x-y

2)- A exp(-x

2+2y-y

2)

C. Ψ(x,+;y,+)= Aexp(-x3-y

2)- Aexp(-y

3-x

2)

D. Ψ(x,+;y,+)= Asin(3x)sin(2x)-Asin(3y)sin(2x)

E. all of the above are allowed

END OF EXAM