### Physics 214 Sample Final Exam(36 questions)

The grading button and a description of the scoring criteria are at the bottom of this page.

True-false questions are worth 2 points each, three-choice multiple choice questions are worth 3 points each, five-choice multiple choice questions are worth 6 points each. The maximum possible score is 177.

### QUESTION 1

A beam of photons with wavelength 150 nm and beam of electrons having the same energy as the photons go through the same slit of width 355 nm. You observe the diffraction pattern on a distant screen. All angles are measured from the centerline. The photons produce their first dark band at an angle α. Is the magnitude of α bigger than, equal to or smaller than the magnitude the angle β where the electrons produce their first dark band?

(a)   | α | > | β |
(b)   | α | = | β |
(c)   | α | < | β |

### QUESTION 2

It takes 3.0 eV of energy to excite an electron in a 1-dimensional infinite well from the ground state to the first excited state. What is the width L of the box?

(a)   L = 0.25 nm
(b)   L = 0.61 nm
(c)   L = 2.10 nm
(d)   L = 10.6 nm
(e)   L = 109 nm

### QUESTION 3

This and the following question refer to the same situation

A particle of unknown mass is in a 1-dimensional box of width L = 3.0 × 10-10 m with infinitely high potential walls at x = 0 and at x = L and zero potential for 0 < x < L. The particle is in the second excited state of the box.

What is the de Broglie wavelength λ of the particle?

(a)   λ = 1.0 × 10-10 m
(b)   λ = 2.0 × 10-10 m
(c)   λ = 3.0 × 10-10 m
(d)   λ = 6.0 × 10-10 m
(e)   Not enough information is given.

### QUESTION 4

For which values of x is the probability of finding the particle largest?

(a)   only at x = L/6 and x = 5L/6 (two values of x)
(b)   only at x = L/4 (one value of x)
(c)   only at x = L/6 and at x = L/2 and at x = 5L/6 (three values of x)
(d)   The probability is everywhere the same.
(e)   Not enough information is given.

### QUESTION 5

This and the following question are related.

How many distinct (n,l,m) states of the hydrogen atom with n = 3 are there? Neglect electron spin.

(a)   1 distinct state
(b)   5 distinct states
(c)   9 distinct states

### QUESTION 6

Which one of these statements about the energies of the (n,l,m) states of the hydrogen atom with n = 3 is correct?

(a)   They are all equal to -1.51 eV.
(b)   The highest (most positive) energy is equal to -1.51 eV, the others are all smaller.
(c)   At least one energy is equal to -2.44 eV.
(d)   At least one energy is equal to +3.55 eV.
(e)   States with larger l have higher energies.

### QUESTION 7

This and the following question are related.

A particle is in a bound energy state of the finite depth potential well shown.

If we measure the particle's position in the well, which is the more likely result?

(a)   The particle has a larger wavelength in the left half of the well.
(b)   The particle has a larger wavelength in the right half of the well.
(c)   The particle wavelength is the same on both sides of the well.

### QUESTION 8

At which values of x is the total energy of the particle the largest?

(a)   at x = 0
(b)   at x = L
(c)   The energy is the same everywhere.

### QUESTION 9

The longest wavelength of light that can be absorbed by a particular harmonic oscillator is λ = 1000 nm. What is the second longest wavelength that can be absorbed?

(a)   λ = 188 nm
(b)   λ = 300 nm
(c)   λ = 500 nm
(d)   Every λ ≤ 1000 nm can be absorbed.
(e)   No shorter wavelengths can be absorbed.

### QUESTION 10

Consider a well that has an adjustable shape, so that we can vary its two lowest energy levels, E1 (the ground state) and E2. We will put a particle into a superposition of the two energy states (i.e., Ψ = 1 + 2). Which of these manipulations will increase the frequency of oscillation of the particle's spatial probability density?

(a)   Increase E1, keeping E2 constant.
(b)   Increase |E2 - E1|.
(c)   Increase a, keeping b constant.
(d)   Increase |b - a|.
(e)   none of the above

### QUESTION 11

This and the following two questions refer to the same situation

An electron is confined to an infinite 1-dimensional well of width L = 1.5 nm. At t = 0 it is in a superposition of the ground state and second excited state: Ψ(x,t=0) = 1 + 3 , where Ψ1 and Ψ3 are individually normalized.

What is the frequency of oscillation f of the spatial probability density?

(a)   f = 1.27 × 1014 Hz
(b)   f = 2.54 × 1014 Hz
(c)   f = 3.24 × 1014 Hz
(d)   f = 3.55 × 1015 Hz
(e)   f = 2.29 × 1015 Hz

### QUESTION 12

Which one of these pairs of values of a and b correctly normalizes the wave function and can result, at some instant of time, in a zero probability density at the middle of the well (i.e., at x = L/2)?

(a)   a = b = 1/√2
(b)   a = b = 1/2
(c)   a = √(2/3), b = √(1/3)
(d)   a = √(2/3), b = -√(1/3)
(e)   No values of a and b will ever result in a zero probability density at the middle of the well.

### QUESTION 13

Suppose now that a = 0.399 and b = 0.917. If we measure the energy, what is the probability P3 of obtaining E3?

(a)   P3 = 1.00
(b)   P3 = 0.399
(c)   P3 = 0.518
(d)   P3 = 0.841
(e)   P3 = 0.917

### QUESTION 14

This and the following question refer to the same situation

The work function (energy needed to remove an electron) of gold is 5.1 eV. Two pieces of gold (at the same potential) are separated by a distance L.

For what value of L will the transmission probability for an electron to cross from one to the other be T ≈ 10-3? Assume that G = 1 in the formula for the tunneling probability.

(a)   L = 0.001 nm
(b)   L = 0.02 nm
(c)   L = 0.1 nm
(d)   L = 0.3 nm
(e)   L = 4 nm

### QUESTION 15

Suppose we increase L by a factor of two from the value required for the transmission probability T to be ≈ 10-3. What is the new value of T?

(a)   T ≈ 10-6
(b)   T ≈ 0.5 × 10-3
(c)   T ≈ 1 × 10-3
(d)   T ≈ 2 × 10-3
(e)   T ≈ 0.03

### QUESTION 16

This and the following question refer to the same situation

An electron is confined in a 3-dimensional rectangular box (V = 0 inside and V = outside) with sides L = 4 nm, 4 nm and 5 nm. What minimum energy E must a photon have in order to excite the electron out of its ground state. (The electron absorbs the photon.)

(a)   E = 0.015 eV
(b)   E = 0.045 eV
(c)   E = 0.071 eV
(d)   E = 0.125 eV
(e)   E = 0.241 eV

### QUESTION 17

Now we put 10 electrons into the box. Assuming the total energy of the system is as low as possible, and including the effects of spin, how many electrons have the energy of the second excited state?

(a)   none
(b)   1
(c)   2
(d)   4
(e)   all 10

### QUESTION 18

Consider a wave function that is a spherical blob with diameter D as shown. The average value of its kinetic energy (if one makes a large number of measurements of identical systems) is <KE0>. Now, shrink the wave function in all three dimensions, so that it is a similar spherical blob with diameter half D. What will be the new average kinetic energy <KE>?

(a)   <KE> = <KE0>
(b)   <KE> = 2 <KE0>
(c)   <KE> = 4 <KE0>

### QUESTION 19

A hydrogen atom is initially at rest (approximately), in an excited state of unknown quantum number n. The electron drops to the next lower energy level emitting a photon. Estimate the largest possible recoil velocity of the atom.

(a)   0 (the atom does not recoil when it emits a photon)
(b)   3.26 m/s
(c)   121.6 m/s

### QUESTION 20

This and the following question refer to the same situation.

You are sitting 10 meters from a musician (left-most dot) playing an instrument with a steady note at some unknown frequency f. You are facing in a direction perpendicular to the direction of the musician, as shown in the figure below (not to scale).

The speed of sound in air is 346 m/sec, while the speed in your head depends on whether the sound propagates directly across (i.e., through your brain, with diameter 15 cm), at vbrain = 1540 m/s, or propagates around the periphery through the skull bone (assumed to be spherical for this problem) at vskull = 4080 m/s.  Assuming the individual intensities from these two paths are the same, for what frequency will there be destructive interference at your left eardrum?

(a)   f = 12.6 kHz
(b)   f = 8.3 kHz
(c)   f = 2.3 kHz

### QUESTION 21

A second musician ("X") now joins in and they both play a steady middle-C note (f = 278 Hz) in phase. The second musician sits 2 meters away from the first musician in the perpendicular direction from the line between you and the first musician, as shown in this figure.

The intensity that each musician produces at your right eardrum when they play alone is 2 W/m2. What is the net intensity, I, at your right eardrum? The intensity that each musician produces at your right eardrum when they play alone is 2 W/m2. What is the net intensity I at your right eardrum?

(a)   I = 0 W/m2
(b)   I = 0.8 W/m2
(c)   I = 1.5 W/m2
(d)   I = 6.2 W/m2
(e)   I = 7.0 W/m2

### QUESTION 22

The Apollo 11 mission landed on the Moon in 1969. Moon hoax enthusiasts often cite a lack of teloscopic evidence for hardware they left behind. The Hubble Space Telescope (HST) is the largest orbiting telescope, with a 2.4-meter aperture mirror. At closest approach, the HST-moon distance is 376,000 km.

By approximately what factor would we have to increase the HST mirror aperture to resolve the lunar rover left by the Apollo 11 astronauts under the most optimistic circumstances? Use a wavelength of 700 nm for the light being detected by the HST and require that we need to resolve 15 cm features in order to identify it.

(a)   factor = 10
(b)   factor = 100
(c)   factor = 1000
(d)   factor = 10000
(e)   factor = 100000

### QUESTION 23

Light of unknown wavelength falls on a tiny pinhole. The pattern created by the pinhole has its first diffraction minimum at an angle 2° from the center line. In a second experiment light falls on a pinhole with a diameter that is half of that in the first experiment. What is the angle θmin of the first diffraction minimum for the second (smaller) pinhole?

(a)   θmin = 1°
(b)   θmin = 2°
(c)   θmin = 4°
(d)   θmin = 8°
(e)   θmin = 16°

### QUESTION 24

This and the following question refer to the same situation.

In a two-slit interference experiment, a viewing screen is placed 5 meters directly behind two slits separated by 4 μm. Coherent, monochromatic light of wavelength λ = 700 nm emerges (in phase) from the slits. (Assume the slit width is very small compared to the wavelength λ.)

At what value of x on the screen does the largest-order intensity maximum occur (i.e., the one that is the farthest from the center line)?

(a)   x = 1.2 m
(b)   x = 4.4 m
(c)   x = 5.0 m
(d)   x = 9.0 m
(e)   x = 15.7 m

### QUESTION 25

What happens to the separation between the existing intensity maxima on the screen as we increase the slit spacing?

(a)   The separation decreases.
(b)   The separation increases.
(c)   There is no change.

### QUESTION 26

This and the following question refer to the same situation.

Consider doubly ionized Lithium (Li++), which has 1 electron orbiting a charge +3 nucleus.

Compare ao,Li the most likely distance for the electron in a Li++ atom to be from the nucleus, with ao,H the most likely distance for the electron in a hydrogen atom to be from the nucleus, assuming the electron is in the ground state in both cases. Which of the following is true?

(a)   ao,Li  <  ao,H
(b)   ao,Li  =  ao,H
(c)   ao,Li  >  ao,H

### QUESTION 27

What is the maximum wavelength λmax of light that would completely ionize the Li++ (i.e., free the electron from the nucleus)?

(a)   λmax = 0.665 nm
(b)   λmax = 10.1 nm
(c)   λmax = 21 nm
(d)   λmax = 137 nm
(e)   λmax = 487 nm

### QUESTION 28

Which one of the following statements is true?

(a)   An ideal metal conducts because each electron 'belongs' to every nucleus in the crystal lattice, while in an ideal insulator, each electron is localized to a single nucleus.

(b)   Metals conduct much better than insulators because, per gram of material, metals have orders of magnitude more electrons.

(c)   At sufficiently low temperatures, (intrinsic) semi-conductors are effectively insulators.

### QUESTION 29

A Ca atom (atomic number = 20) is in the ground state. Light is shined on the atom, exciting the most energetic electron. Which of the following are possible quantum numbers (n,l,m) of this excited electron? We omit the electron spin quantum number.

(a)   (4, 0, 0)
(b)   (4, 1, -1), (4, 1, 0) and (4, 1, 1)
(c)   (5, 2, +1) and (5, 2, -1)
(d)   all of the above
(e)   none of the above

### QUESTION 30

You are given a collection of quantum dots (treat as 1-D infinite square wells) and a collection of diatomic molecules (treat as simple harmonic oscillators). You find that the same wavelength of light will excite both the quantum dots and the molecules from their ground states to their first excited states. What is the ratio of the photon energies required to further excite each system from the first excited state to the second excited state?

(a)   Equantum dot / Emolecule  =  1
(b)   Equantum dot / Emolecule  =  6/5
(c)   Equantum dot / Emolecule  =  5/3
(d)   Equantum dot / Emolecule  =  15/2
(e)   The information given is not sufficient to answer this question.

### QUESTION 31

Electron spin resonance (ESR) is a technique much like nuclear magnetic resonance (NMR), except that it relies on using radio-frequency photons to flip the spin of an electron in a magnetic field. If the magnetic field applied to the electrons is increased from 0.010 tesla to 0.011 tesla, how much, Δf, will the transition frequency change?

(a)   Δf = 7 MHz
(b)   Δf = 14 MHz
(c)   Δf = 28 MHz

### QUESTION 32

This question and the following two refer to the same situation

An electron is moving in the potential U(x) shown to the right. This potential is an even function of x. The energies, E1 = -1.5 eV and E2 = +0.5 eV, of the two lowest energy states are indicated by the dashed lines. Five x values are labeled A-F in the figure. Ψ1(x) is the wave function of the ground state, and Ψ2(x) is the wave function of the first excited state.

Which one of the following statements is not true?

(a)   Ψ1(A) = Ψ1(F)
(b)   Ψ2(A) = -Ψ2(F)
(c)   Both Ψ1 and Ψ2 are proportional to sin(kx) in the regions A < x < B and D < x < F (though 'k' might be different for Ψ1 and Ψ2)
(d)   1/dx = 0 at x = C
(e)   d|Ψ2|2/dx > 0 for x < A

### QUESTION 33

Which wave function penetrates farther into the region x > F ? That is, for which wave function is Ψ(x) / Ψ(F) larger when x > F ?

(a)   Ψ1
(b)   Ψ2
(c)   They are the same.

### QUESTION 34

If the system is prepared in a superposition of Ψ1 and Ψ2, at what frequency does the probability density oscillate?

(a)   f = 4.8 × 1014 Hz
(b)   f = 3.6 × 1014 Hz
(c)   f = 1.24 × 1014 Hz

### QUESTION 35

This question and the following one refer to the same situation

The graph at the right shows the results of a photoelectric effect experiment in which the stopping voltage was measured for several frequencies of light. The slope of the line is 4.0 × 10-15 volts/Hz, and the x-intercept is f0 = 6.0 × 1014 Hz.

What is the work function Φ of the material?

(a)   Φ = -6.0 eV
(b)   Φ = -4.0 eV
(c)   Φ = +2.4 eV
(d)   Φ = +4.0 eV
(e)   Φ = +6.0 eV

### QUESTION 36

What value of Planck's constant, h, does this data yield?

(a)   h = 6.0 × 10-34J.s
(b)   h = 6.2 × 10-34J.s
(c)   h = 6.4 × 10-34J.s
(d)   h = 6.6 × 10-34J.s
(e)   h = 6.8 × 10-34J.s

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