Question 1

The wave function for a particle in a one-dimensional box is   . Which statement is correct?

Accepted Answer

A)  This wavefunction gives the probability of finding the particle at x. 
B)  |Ψ(x)|2 gives the probability of finding the particle at x. 
C)  |Ψ(x)|2 Δx 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. 
E) 
  gives the probability of finding the particle between x and x + Δx. 
A)  This wavefunction gives the probability of finding the particle at x. 
B)  |Ψ(x)|2 gives the probability of finding the particle at x. 
C)  |Ψ(x)|2 Δx 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. 
E) 
  gives the probability of finding the particle between x and x + Δx. C

Question 2

A physically reasonable wave function, ψ(x), for a one-dimensional system must

Accepted Answer

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. 
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. D

Question 3

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

Accepted Answer

A)  a positive constant. 
B)  exponentially increasing. 
C)  oscillatory. 
D)  exponentially decreasing. 
E)  none of the above. 
A)  a positive constant. 
B)  exponentially increasing. 
C)  oscillatory. 
D)  exponentially decreasing. 
E)  none of the above. D

Question 4

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.

Accepted Answer

A)  1.5 × 1033 
B)  3.0 × 1033 
C)  4.5 × 1033 
D)  5.4 × 1033 
E)  1.0 × 1033 
A)  1.5 × 1033 
B)  3.0 × 1033 
C)  4.5 × 1033 
D)  5.4 × 1033 
E)  1.0 × 1033

Question 5

The graph below shows the value of the probability density |ψ(x)|2 in the region −3.00 m ≤ x ≤ +3.00 m. The value of the constant A is

Accepted Answer

A) 
  . 
B) 
  . 
C) 
  . 
D) 
  . 
E)  either
  or
  . 
A) 
  . 
B) 
  . 
C) 
  . 
D) 
  . 
E)  either
  or
  .

Question 6

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

Accepted Answer

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. 
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.

Question 7

When U(x) is infinitely large elsewhere, the wave function of a particle restricted to the region 0 < x < L where U(x) = 0, may have the form ψ(x) =

Accepted Answer

A) 
  . 
B) 
  . 
C)  Aenπx / L. 
D) 
  . 
E) 
  . 
A) 
  . 
B) 
  . 
C)  Aenπx / L. 
D) 
  . 
E) 
  .

Question 8

The expectations value of a function f(x) of x when the wave function depends only on x is given by  =

Accepted Answer

A) 
  . 
B) 
  . 
C) 
  . 
D) 
  . 
E) 
  . 
A) 
  . 
B) 
  . 
C) 
  . 
D) 
  . 
E) 
  .

Question 9

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.

Accepted Answer

A)  3 h2 
B)  2 h2 
C)  h2 
D)  4h2 
E)  0.5 h2 
A)  3 h2 
B)  2 h2 
C)  h2 
D)  4h2 
E)  0.5 h2

Question 10

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

Accepted Answer

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. 
E)  depends on the vector positions,
   
i, of each particle in 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. 
E)  depends on the vector positions,
   
i, of each particle in the system.

Question 11

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?

Accepted Answer

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. 
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.

Question 12

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?

Accepted Answer

A)  P  0. 
E)  P = ∞. 
A)  P  0. 
E)  P = ∞.

Question 13

The wave function for a particle in a box of length L is given by   . If the box extends from x = 0 to x = L, at which of the following positions is the highest probability for finding the particle?

Accepted Answer

A)  x = 0.33 L 
B)  x = 0.04 L 
C)  x = 0.60 L 
D)  Both (a) and (b). 
E)  Both (b) and (c). 
A)  x = 0.33 L 
B)  x = 0.04 L 
C)  x = 0.60 L 
D)  Both (a) and (b). 
E)  Both (b) and (c).

Question 14

A particle in a finite potential well has energy E, as shown below.   The wave function in region I where x I =

Accepted Answer

A)  Ae−Cx. 
B)  AeCx. 
C)  F sin kx. 
D)  G cos kx. 
E)  F sin kx + G cos kx. 
A)  Ae−Cx. 
B)  AeCx. 
C)  F sin kx. 
D)  G cos kx. 
E)  F sin kx + G cos kx.

Question 15

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?

Accepted Answer

A)  2 
B)  8 
C)  4 
D)  1 
E)  16 
A)  2 
B)  8 
C)  4 
D)  1 
E)  16

Question 16

A particle is in the ground state of a one-dimensional box of length 1.0 m. What is the minimum value of its momentum (in kg ⋅ m/s)?

Accepted Answer

A)  9.9 × 10−34 
B)  6.6 × 10−34 
C)  3.3 × 10−34 
D)  13 × 10−34 
E)  cannot be solved unless mass of particle is known. 
A)  9.9 × 10−34 
B)  6.6 × 10−34 
C)  3.3 × 10−34 
D)  13 × 10−34 
E)  cannot be solved unless mass of particle is known.

Question 17

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

Accepted Answer

A)  10 
B)  75 
C)  24 
D)  150 
E)  54 
A)  10 
B)  75 
C)  24 
D)  150 
E)  54

Question 18

Quantum tunneling occurs in

Accepted Answer

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. 
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.

Question 19

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

Accepted Answer

A) 
  . 
B) 
  . 
C) 
  . 
D) 
  . 
E)  either
  or
  . 
A) 
  . 
B) 
  . 
C) 
  . 
D) 
  . 
E)  either
  or
  .

Question 20

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

Accepted Answer

A) 
  . 
B) 
  . 
C) 
  . 
D) 
  . 
E) 
  . 
A) 
  . 
B) 
  . 
C) 
  . 
D) 
  . 
E) 
  .

Exam 41: Quantum Mechanics

The wave function for a particle in a one-dimensional box is . Which statement is correct?

A physically reasonable wave function, ψ(x), for a one-dimensional system must

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 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.

The graph below shows the value of the probability density |ψ(x)|2 in the region −3.00 m ≤ x ≤ +3.00 m. The value of the constant A is

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

When U(x) is infinitely large elsewhere, the wave function of a particle restricted to the region 0 < x < L where U(x) = 0, may have the form ψ(x) =

The expectations value of a function f(x) of x when the wave function depends only on x is given by < f(x) > =

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.

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

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?

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?

The wave function for a particle in a box of length L is given by . If the box extends from x = 0 to x = L, at which of the following positions is the highest probability for finding the particle?

A particle in a finite potential well has energy E, as shown below. The wave function in region I where x < 0 has the form ψI =

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 particle is in the ground state of a one-dimensional box of length 1.0 m. What is the minimum value of its momentum (in kg ⋅ m/s)?

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

Quantum tunneling occurs in

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

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

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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 E_n = nhf.

The graph below shows the value of the probability density |ψ(x)|² in the region −3.00 m ≤ x ≤ +3.00 m. The value of the constant A is

A particle in a finite potential well has energy E, as shown below. The wave function in region I where x < 0 has the form ψ_I =

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?