Deck 41: Quantum Mechanics

Question 1

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) P < 0.
B) P = 0.
C) P = 1.
D) P > 0.
E) P = ∞.

Answer 1

P > 0.

Question 2

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.
E) depends on the vector positions,

_i, of each particle in the system.

Answer 2

depends on the vector positions,

_i, of each particle in the system.

Question 3

A particle is in the first excited state of a one-dimensional box of length 1.0 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 × 10−³⁴
E) 22 × 10−³⁴

Answer 3

6.6 × 10−³⁴

Question 4

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.

Question 5

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)?

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 6

The ground state energy of a harmonic oscillator is

A)

B)

C)

D) E = 0
E)

Question 7

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.

Question 8

A particle is in the second excited state of a one-dimensional box of length 1.0 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 × 10−³⁴
E) cannot be solved unless mass of particle is known

Question 9

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 > =

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

Question 10

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.

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

Question 11

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)

Question 12

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.

Question 13

A physically reasonable wave function, ψ(x), 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.

Question 14

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

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?

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

Question 16

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 (nπx/L) + B cos (nπx/L) . The constants A and B are determined to be

A)
, 0
B)
,

C) 0,

D)
,

E) 2/L, 0

Question 17

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

Question 18

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

Question 19

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

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

Question 20

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

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.

Question 21

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) 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 22

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.

Question 23

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

Question 24

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.

Question 25

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 =

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

Question 26

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) =

A)
.
B)
.
C) Aenπx ^{/ L}.
D)
.
E)
.

Question 27

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

Question 28

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, What is the probability of finding the particle between x = 0.60 L and x = 0.70 L?

A) 0.10
B) 0.20
C) 0.25
D) 0.05
E) The probability is not given.

Question 29

A particle in a finite potential well has energy E, as shown below. The wave function in region III where x > L has the form ψ_III =

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

Question 30

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

A)
.
B)
.
C)
.
D)
.
E) 4.

Question 31

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

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

Question 32

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

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

Question 33

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

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