Deck 39: Quantum Mechanics

Question 1

The wave function for an electron that is confined to x ≥ 0 nm is
ψ(x) =
(a) What must be the value of A?
(b) What is the probability of finding the electron in the interval 1.15 nm ≤ x ≤ 1.84 nm?

Answer 1

(a) 0.93 (nm)⁻¹/²
(b) 0.17

Question 2

The square of the wave function of a particle, |ψ(x)|², gives the probability of finding the particle at the point x.

Answer 2

False

Question 3

An electron is in an infinite square well (a box) that is 2.0 nm wide. The electron makes a transition from the n = 8 to the n = 7 state, what is the wavelength of the emitted photon? (h = 6.626 × 10^-34 J ∙ s, mel = 9.11 × 10^-31 kg, 1 eV = 1.60 × 10^-19)

A) 880 nm
B) 750 nm
C) 610 nm
D) 1000 nm
E) 1100 nm

Answer 3

880 nm

Question 4

Find the value of A to normalize the wave function ψ(x) = .

A)
B)
C)
D)
E)

Question 5

An electron is in the ground state of an infinite well (a box) where its energy is 5.00 eV. In the next higher level, what would its energy be? (1 eV = 1.60 × 10^-19 J)

A) 1.25 eV
B) 2.50 eV
C) 10.0 eV
D) 15.0 eV
E) 20.0 eV

Question 6

An electron is bound in an infinite well (a box) of width 0.10 nm. If the electron is initially in the n = 8 state and falls to the n = 7 state, find the wavelength of the emitted photon. (c = 3.00 × 10⁸ m/s, h = 6.626 × 10^-34 J ∙ s, mel = 9.11 × 10^-31 kg)

Question 7

An electron in an infinite potential well (a box) makes a transition from the n = 3 level to the ground state and in so doing emits a photon of wavelength 20.9 nm. (c = 3.00 × 10⁸ m/s, h = 6.626 × 10⁻³⁴J ∙ s, ^mel = 9.11 × 10⁻³¹ kg)
(a) What is the width of this well?
(b) What wavelength photon would be required to excite the electron from its original level to the next higher one?

Question 8

A set of five possible wave functions is given below, where L is a positive real number.
Ψ1(x) = Ae^-x, for all x ψ2(x) = A cos x, for all x
Ψ3(x) =
Ψ4(x) =
Ψ5(x) =
Which of the five possible wave functions are normalizable? (There may be more than one correct choice.)

A) ψ1(x)
B) ψ2(x)
C) ψ3(x)
D) ψ4(x)
E) ψ5(x)

Question 9

An electron is bound in an infinite square-well potential (a box) on the x-axis. The width of the well is L and the well extends from x = 0.00 nm to x = 3.8 nm. In its present state, the normalized wave function of the electron is given by: ψ(x) = sin (2πx/L). What is the energy of the electron in this state? (h = 6.626 × 10^-34 J ∙ s, mel = 9.11 × 10^-31 kg, 1 eV = 1.60 × 10^-19)

A) 0.10 eV
B) 0.052 eV
C) 0.13 eV
D) 0.078 eV
E) 0.026 eV

Question 10

The probability density for an electron that has passed through an experimental apparatus is shown in the figure. If 4100 electrons pass through the apparatus, what is the expected number that will land in a 0.10 mm-wide strip centered at x = 0.00 mm?

A) 140
B) 1400
C) 450
D) 45

Question 11

A particle is confined to a one-dimensional box (an infinite well) on the x-axis between x = 0 and x = L. The potential height of the walls of the box is infinite. The normalized wave function of the particle, which is in the ground state, is given by ψ(x) = sin
, with 0 ≤ x ≤ L. What is the maximum probability per unit length of finding the particle?

A) 1/
B)
C) 2/
D) 1/L
E) 2/L

Question 12

The wave function for a particle must be normalizable because

A) the particle's momentum must be conserved.
B) the particle's angular momentum must be conserved.
C) the particle's charge must be conserved.
D) the particle must be somewhere.
E) the particle cannot be in two places at the same time.

Question 13

A particle is confined to a one-dimensional box (an infinite well) on the x-axis between x = 0 and x = L. The potential height of the walls of the box is infinite. The normalized wave function of the particle, which is in the ground state, is given by ψ(x) = sin
, with 0 ≤ x ≤ L. What is the probability of finding the particle between x = 0 and x = L/3?

A) 0.20
B) 0.22
C) 0.24
D) 0.26
E) 0.28

Question 14

An electron is in an infinite square well (a box) that is 8.9 nm wide. What is the ground state energy of the electron? (h = 6.626 × 10^-34 J ∙ s, ^mel = 9.11 × 10^-31 kg, 1 eV = 1.60 × 10^-19)

A) 0.0048 eV
B) 0.0057 eV
C) 0.0066 eV
D) 0.0076 eV
E) 0.0085 eV

Question 15

The wave function for an electron that is confined to x ≥ 0 nm is
ψ(x) =
(a) What must be the value of b?
(b) What is the probability of finding the electron in a 0.010 nm-wide region centered at x = 1.0 nm?

Question 16

The smallest kinetic energy that an electron in a box (an infinite well) can have is zero.

Question 17

The lowest energy level of a particle confined to a one-dimensional region of space (a box, or infinite well) with fixed length L is E₀. If an identical particle is confined to a similar region with fixed length L/6, what is the energy of the lowest energy level that the particles have in common? Express your answer in terms of E₀.

Question 18

Find the value of A to normalize the wave function ψ(x) = .

A) 1/L
B)
C) 1/L²
D) 1.
E) 1/

Question 19

An electron is in an infinite square well that is 2.6 nm wide. What is the smallest value of the state quantum number n for which the energy level exceeds 100 eV? (h = 6.626 × 10^-34 J ∙ s, mel = 9.11 × 10^-31 kg, 1 eV = 1.60 × 10^-19)

A) 43
B) 44
C) 45
D) 42
E) 41

Question 20

If an atom in a crystal is acted upon by a restoring force that is directly proportional to the distance of the atom from its equilibrium position in the crystal, then it is impossible for the atom to have zero kinetic energy.

Question 21

An electron with kinetic energy 2.80 eV encounters a potential barrier of height 4.70 eV. If the barrier width is 0.40 nm, what is the probability that the electron will tunnel through the barrier? (1 eV = 1.60 × 10^-19 J, mel = 9.11 × 10^-31 kg, h = 6.626 × 10^-34 J ∙ s)

A) 1.4 × 10^-2
B) 2.8 × 10^-2
C) 5.5 × 10^-2
D) 1.1 × 10^-2
E) 1.4 × 10^-1

Question 22

The energy of a particle in the second EXCITED state of a harmonic oscillator potential is What is the classical angular frequency of oscillation of this particle? (1 eV = 1.60 × 10^-19 J, = 1.055 × 10^-34 J ∙ s, h = 6.626 × 10^-34 J ∙ s)

A) 3.31 × 10¹⁵ rad/s
B) 2.08 × 10¹⁶ rad/s
C) 4.96 × 10¹⁵ rad/s
D) 6.95 × 10¹⁵ rad/s

Question 23

You want to confine an electron in a box (an infinite well) so that its ground state energy is 5.0 × 10^-18 J. What should be the length of the box? (h = 6.626 × 10^-34 J ∙ s, mel = 9.11 × 10^-31 kg)

A) 0.11 nm
B) 0.22 nm
C) 0.15 nm
D) 0.18 nm

Question 24

An 80-eV electron impinges upon a potential barrier 100 eV high and 0.20 nm thick. What is the probability the electron will tunnel through the barrier? (1 eV = 1.60 × 10^-19 J, mproton = 1.67 × 10^-27 kg, = 1.055 × 10^-34 J ∙ s, h = 6.626 × 10^-34 J ∙ s)

A) 0.027%
B) 2.7%
C) 0.27%
D) 2.8 × 10^-4 %
E) 2.0 × 10^-9 %

Question 25

A lithium atom, mass 1.17 × 10-⁻²⁶ kg, vibrates with simple harmonic motion in a crystal lattice, where the effective force constant of the forces on the atom is k = 49.0 N/m. (c = 3.00 × 10⁸ m/s, h = 6.626 × 10⁻³⁴ J ∙ s, = 1.055 × 10⁻³⁴ J ∙ s, 1 eV = 1.60 × 10⁻¹⁹ J)
(a) What is the ground state energy of this system, in eV?
(b) What is the wavelength of the photon that could excite this system from the ground state to the first excited state?

Question 26

An electron is confined in a harmonic oscillator potential well. What is the longest wavelength of light that the electron can absorb if the net force on the electron behaves as though it has a spring constant of 74 N/m? (^mel = 9.11 × 10^-31 kg, c = 3.00 × 10⁸ m/s, 1 eV = 1.60 × 10^-19 J, = 1.055 × 10^-34 J ∙ s, h = 6.626 × 10^-34 J ∙ s)

A) 210 nm
B) 200 nm
C) 220 nm
D) 230 nm

Question 27

A particle confined in a rigid one-dimensional box (an infinite well) of length 17.0 fm has an energy level E_n = 24.0 MeV and an adjacent energy level E_n+1= 37.5 MeV. What is the value of the ground state energy? (1 eV = 1.60 × 10^-19 J)

A) 1.50 MeV
B) 13.5 MeV
C) 0.500 MeV
D) 4.50 MeV

Question 28

The energy of a proton is 1.0 MeV below the top of a 1.2-MeV-high energy barrier that is 6.8 fm wide. What is the probability that the proton will tunnel through the barrier? (1 eV = 1.60 × 10^-19 J, ^mproton = 1.67 × 10^-27 kg, = 1.055 × 10^-34 J ∙ s, h = 6.626 × 10^-34 J ∙ s)

A) 11%
B) 9.1%
C) 14%
D) 7.5%

Question 29

The wave function of an electron in a rigid box (infinite well) is shown in the figure. If the electron energy 98.0 eV, what is the energy of the electron's ground state? ( ^mel = 9.11 × 10^-31 kg)

A) 6.13 eV
B) 3.92 eV
C) 10.9 eV
D) 24.5 eV

Question 30

An electron is trapped in an infinite square well (a box) of width 6.88 nm. Find the wavelength of photons emitted when the electron drops from the n = 5 state to the n = 1 state in this system. (c = 3.00 × 10⁸ m/s, h = 6.626 × 10^-34J ∙ s, ^mel = 9.11 × 10^-31 kg)

A) 6.49 μm
B) 5.45 μm
C) 5.91 μm
D) 7.07 μm

Question 31

The lowest energy level of a certain quantum harmonic oscillator is 5.00 eV. What is the energy of the next higher level?

A) 7.50 eV
B) 10.0 eV
C) 15.0 eV
D) 20.0 eV
E) 50.0 eV

Question 32

You want to have an electron in an energy level where its speed is no more than 66 m/s. What is the length of the smallest box (an infinite well) in which you can do this? (h = 6.626 × 10^-34 J ∙ s, mel = 9.11 × 10^-31 kg)

A) 5.5 µm
B) 11 µm
C) 2.8 µm
D) 1.4 µm

Question 33

Calculate the ground state energy of a harmonic oscillator with a classical frequency of 3.68 × 10¹⁵ Hz. (1 eV = 1.60 × 10^-19 J, h = 1.055 × 10^-34 J ∙ s, h = 6.626 × 10^-34 J ∙ s)

A) 7.62 eV
B) 15.2 eV
C) 11.4 eV
D) 5.71 eV

Question 34

A 3.10-eV electron is incident on a 0.40-nm barrier that is 5.67 eV high. What is the probability that this electron will tunnel through the barrier? (1 eV = 1.60 × 10^-19 J, ^mel = 9.11 × 10^-31 kg, h = 1.055 × 10^-34 J ∙ s, h = 6.626 × 10^-34 J ∙ s)

A) 0.56%
B) 0.35%
C) 0.40%
D) 0.25%
E) 0.48%

Question 35

A 10.0-g bouncy ball is confined in a 8.3-cm-long box (an infinite well). If we model the ball as a point particle, what is the minimum kinetic energy of the ball? (h = 6.626 × 10^-34 J ∙ s)

A) 8.0 × 10^-64 J
B) 3.2 × 10^-46 J
C) 1.3 × 10^-20 J
D) zero

Question 36

An electron is confined in a one-dimensional box (an infinite well). Two adjacent allowed energies of the electron are 1.068 × 10^-18 J and 1.352 × 10^-18 J. What is the length of the box? (h = 6.626 × 10-³⁴ J ∙ s, ^mel = 9.11 × 10^-31 kg)

A) 1.9 nm
B) 0.93 nm
C) 1.1 nm
D) 2.3 nm

Question 37

An electron is confined in a harmonic oscillator potential well. A photon is emitted when the electron undergoes a 3→1 quantum jump. What is the wavelength of the emission if the net force on the electron behaves as though it has a spring constant of 9.6 N/m? (^mel = 9.11 × 10^-31 kg, c = 3.00 × 10⁸ m/s, 1 eV = 1.60 × 10^-19 J, = 1.055 × 10^-34 J ∙ s, h = 6.626 × 10^-34 J ∙ s)

A) 290 nm
B) 150 nm
C) 190 nm
D) 580 nm

Question 38

The atoms in a nickel crystal vibrate as harmonic oscillators with an angular frequency of 2.3 × 10¹³ rad/s. The mass of a nickel atom is 9.75 × 10^-26 kg. What is the difference in energy between adjacent vibrational energy levels of nickel? (h = 6.626 × 10-³⁴ J ∙ s, = 1.055 × 10^-34 J ∙ s, 1 eV = 1.60 × 10^-19 J)

A) 0.015 eV
B) 0.019 eV
C) 0.023 eV
D) 0.027 eV
E) 0.031 eV

Question 39

Find the wavelength of the photon emitted during the transition from the second EXCITED state to the ground state in a harmonic oscillator with a classical frequency of 3.72 × 10¹³ Hz. (c = 3.00 × 10⁸ m/s, 1 eV = 1.60 × 10^-19 J, = 1.055 × 10^-34 J ∙ s, h = 6.626 × 10^-34 J ∙ s)

A) 4.03 μm
B) 2.26 μm
C) 2.98 μm
D) 5.24 μm

Question 40

One fairly crude method of determining the size of a molecule is to treat the molecule as an infinite square well (a box) with an electron trapped inside, and to measure the wavelengths of emitted photons. If the photon emitted during the n = 2 to n = 1 transition has wavelength 1940 nm, what is the width of the molecule? (c = 3.00 × 10⁸ m/s, h = 6.626 × 10^-34J ∙ s, ^mel = 9.11 × 10^-31 kg)

A) 1.33 nm
B) 1.12 nm
C) 1.21 nm
D) 1.45 nm