Chapter 5 and 39

A particle of mass m is in the ground state of an infinite potential energy well ofwidth L. The energy of the particle is 2.0 eV (c) What would be the ground-state energy of this particle if the width of the well were changed to 2L? (1) 0.5 eV (2) 1.0 eV (3) 2.0 eV (4) 4.0 eV

(1) 0.5 eV

(c) What are the continuity conditions at x = 0? (1) Both ψ and dψ/dx are continuous at x = 0. (2) ψ is continuous at x = 0, but dψ/dx is not. (3) dψ/dx is continuous at x = 0, but ψ is not. (4) Neither ψ nor dψ/dx are continuous at x = 0.

(1) Both ψ and dψ/dx are continuous at x = 0.

(b) How would the wavelength of a particle change as it moves from the x < 0 region to the x >0 region? (1) Increases (2) Decreases (3) Remains the same

(2) Decreases

A particle of mass m is in the ground state of an infinite potential energy well ofwidth L. The energy of the particle is 2.0 eV (b) Suppose the particle is in the second excited state (n = 3) from which it can jump to any lower state by emitting a photon whose energy is equal to the difference in the energies of the two states. Considering all possible jumps that lead eventually to the ground state, which photon energy would NOT be observed? (1) 6.0 eV (2) 10.0 eV (3) 12.0 eV (4) 16.0 eV

(3) 12.0 eV

The ground-state energy of a simple harmonic oscillator is 1.0 eV. (b) How much energy must be added to move the oscillator from the first excited state to the second excited state? (1) 0.5 eV (2) 1.0 eV (3) 2.0 eV (4) 3.0 eV (5) 4.0 eV

(3) 2.0 eV

The ground-state energy of a simple harmonic oscillator is 1.0 eV. (a) If the oscillator is in its ground state, how much energy must be added for it to reach the first excited state? (1) 0.5 eV (2) 1.0 eV (3) 2.0 eV (4) 3.0 eV (5) 4.0 eV

(3) 2.0 eV

The ground-state energy of a simple harmonic oscillator is 1.0 eV. (c) The oscillator is in the third excited state (n = 3). It can jump to any lower state, in the process emitting a photon whose energy is equal to the difference in energy between the states. How many different photon energies can be emitted if these oscillators can take any possible path from the excited state to the ground state? (1) 1 (2) 2 (3) 3 (4) 4 (5) 5 (6) 6

(3) 3

A particle of mass m is in the ground state of an infinite potential energy well ofwidth L. The energy of the particle is 2.0 eV (a) How much energy must be added to the particle to cause it to jump to the first excited state? (1) 2.0 eV (2) 4.0 eV (3) 6.0 eV (4) 8.0 eV

(3) 6.0 eV

Particles are incident from the negative x axis onto a potential energy step at x = 0.At the step the potential energy drops from the positive value U0 for all x < 0 to the value 0 for all x > 0. The energy of the particles is greater than U0. (a) Which statement best describes the behavior of the particles? (1) All particles are transmitted from x < 0 to x > 0. (2) All particles are reflected back to x < 0 at the step. (3) Some particles are reflected and some are transmitted. (4) Some particles are reflected and some are absorbed

(3) Some particles are reflected and some are transmitted.

In the one-dimensional infinite well, how does the energy spacing between the excited states change as the energy of the states increases? (1) The spacing is constant. (2) The spacing decreases. (3) The spacing increases. (4) The spacing changes randomly.

(3) The spacing increases.

The probability to find a particle at any specific location in space: (1) is directly proportional to the amplitude of the wave function. (2) can never be zero. (3) depends on the squared amplitude of the wave function. (4) can sometimes be infinite.

(3) depends on the squared amplitude of the wave function.

A beam of particles is incident from the negative x axis onto a positive potential energy step located at x = 0. The kinetic energy of the particles is less than the potential energy of the step. Which is the best description of the behavior of the particles? (1) All particles are reflected precisely at x = 0. (2) Some particles are reflected at the step and some are transmitted into the x > 0 region. (3) Some particles are reflected and some are absorbed. (4) All particles are reflected, but they can penetrate a short distance into the x > 0 region. (5) All particles are absorbed at the step.

(4) All particles are reflected, but they can penetrate a short distance into the x > 0 region.

The Schrödinger equation is (a) a second-order differential equation. (b) an equation based on conservation of energy. (c) an equation whose solution gives the wave function that describes a particle. How many of the above statements are true? (1) Zero (2) One (3) Two (4) All three

(4) All three

The ground state of a particle in simple harmonic motion has energy 0.5 eV and the first excited state has energy 1.5 eV. What is the energy of the next excited state? (a) 2.5 eV (b) 3.5 eV (c) 3.0 eV (d) 2.0 eV

(a) 2.5 eV

In a certain infinite potential energy well, the particle has a ground-state energy of 2.0 eV. Which of the following is NOT a possible value for the energy of one of the excited states of this particle in the well? (a) 36 eV (b) 50 eV (c) 18 eV (d) 8 eV

(a) 36 eV

A particle in the first excited state of a one-dimensional infinite potential energy well (with U = 0 inside the well) has an energy of 6.0 eV. What is the energy of this particle in the ground state? (a) 1.0 eV (b) 1.5 eV (c) 2.0 eV (d) 3.0 eV

(b) 1.5 eV

An electron in the ground state of an infinite potential energy well has an energy of 8.0 eV. How much additional energy must be supplied for the electron to jump from the ground state to the first excited state? (a) 8.0 eV (b) 16.0 eV (c) 24.0 eV (d) 32.0 eV

(c) 24.0 eV

A particle is moving in an infinite potential energy well in one dimension. In the ground state, the energy of the particle is 5.0 eV. Which one of the following is a possible energy for an excited state? (a) 25.0 eV (b) 40.0 eV (c) 80.0 eV (d) 100.0 eV

(c) 80.0 eV

The ground-state energy of a particle in an infinite one-dimensional potential energy well is 6.0 eV. What is the energy of the first excited state? (a) 12.0 eV (b) 18.0 eV (c) 7.5 eV (d) 24.0 eV

(d) 24.0 eV

Consider the following two possible solutions to the Schrödinger equation in the entire interval from x = 0 to x = +∞: (i) ψ(x) = Ae-kx/x (ii) ψ (x) = Ae+kx (A and k are real positive constants.) Which are allowable wave functions? (a) Only i (b) Only ii (c) Both i and ii (d) Neither i nor ii

(d) Neither i nor ii

A beam of particles is incident from the negative x direction on a potential energy step at x = 0. When x < 0, the potential energy of the particles is zero, and for x > 0 the potential energy has the constant positive value U0. In the region x < 0, the particles have a kinetic energy K that is smaller than U0. What is the form of the wave function in the region x > 0? (a) kx kx Ae Be− + (b) ikx ikx Ae Be− + (c) kx Ae (d) kx Ae− (e) Acos sin kx B kx

(d) kx Ae−

The probability density for a particle in the ground state of a one-dimensional infinite potential energy well: (1) has a single maximum at the center of the well. (2) has a minimum at the center of the well and maxima at the sides of the well. (3) has several maxima and minima in the well. (4) is constant throughout the well.

1 has a single maximum at the center of the well.

An electron in an atom initially has an energy 5.5eV above the ground state energy. It drops to a state with energy 3.2eV above the ground state energy and emits a photon in the process. The wave associated with the photon has a wavelength of: A. 5.4×10^−7 m B. 3.0×10^−7 m C. 1.7×10^−7 m D. 1.15×10^−7 m E. 1.0×10^−7 m

A

If P(r) is the radial probability density then the probability that the separation of the electron and proton is between r and r + dr is: A. P dr B. (|P|^2) dr C. 4π(r^2)P dr D. 4π(r^2)|P|dr E. 4π(|P|^2) dr

A

The binding energy of an electron in the ground state in a hydrogen atom is about: A. 13.6 eV B. 3.4 eV C. 10.2 eV D. 1.0 eV E. 27.2 eV

A

The ground state energy of an electron in a one-dimensional trap with zero potential energy in the interior and infinite potential energy at the walls is 2.0eV. If the width of the well is doubled, the ground state energy will be: A. 0.5eV B. 1.0eV C. 2.0eV D. 4.0eV E. 8.0eV

A

The quantum number n is most closely associated with what property of the electron in a hydrogen atom? A. Energy B. Orbital angular momentum C. Spin angular momentum D. Magnetic moment E. z component of angular momentum

A

Which of the following sets of quantum numbers is possible for an electron in a hydrogen atom? A. n = 4, l= 3,m_l = −3 B. n = 4, l= 4,m_l = −2 C. n = 5, l= −1, m_l =2 D. n = 3, l= 1,m_l = −2 E. n = 2, l= 3,m_l = −2

A

A particle is confined to a one-dimensional trap by infinite potential energy walls. Of the following states, designed by the quantum number n, for which one is the probability density greatest near the center of the well? A. n =2 B. n =3 C. n =4 D. n =5 E. n =6

B

An electron in an atom drops from an energy level at −1.1 × 10^−18 J to an energy level at −2.4×10^−18 J. The wave associated with the emitted photon has a frequency of: A. 2.0×10^17 Hz B. 2.0×10^15 Hz C. 2.0×10^13 Hz D. 2.0×10^11 Hz E. 2.0×10^9 Hz

B

An electron in an atom initially has an energy 7.5eV above the ground state energy. It drops to a state with an energy of 3.2eV above the ground state energy and emits a photon in the process. The momentum of the photon is: A. 1.7×10^−27 kg·m/s B. 2.3×10^−27 kg·m/s C. 4.0×10^−27 kg·m/s D. 5.7×10^−27 kg·m/s E. 8.0×10^−27 kg·m/s

B

Four different particles are trapped in one-dimensional wells with infinite potential energy at their walls. The masses of the particles and the width of the wells are 1. mass = 4m0, width = 2Lo 2. mass = 2m0, width = 2Lo 3. mass = 4m0, width = Lo 4. mass = m0, width = 2Lo Rank them according to the kinetic energies of the particles when they are in their ground states. A. 1, 2, 3, 4 B. 1, 2, 3 and 4 tied C. 1 and 2 tied, then 3, 4 D. 4, 3, 2, 1 E. 3, 1, 2, 4

B

Take the potential energy of a hydrogen atom to be zero for infinite separation of the electron and proton. Then the ground state energy is −13.6eV. The energy of the first excited state is: A. 0 B. −3.4 eV C. −6.8 eV D. −9.6 eV E. −27 eV

B

Take the potential energy of a hydrogen atom to be zero for infinite separation of the electron and proton. Then the ground state energy is−13.6eV. When the electron is in the first excited state the ionization energy is: A. 0 B. 3.4 eV C. 6.8 eV D. 10.2 eV E. 13.6 eV

B

The wave function for an electron in a state with zero angular momentum: A. is zero everywhere B. is spherically symmetric C. depends on the angle from the z axis D. depends on the angle from the x axis E. is spherically symmetric for some shells and depends on the angle from the z axis for others

B

Two one-dimensional traps have infinite potential energy at their walls Trap A has width L and trap B has width 2L. For which value of the quantum number n does a particle in trap B have the same energy as a particle in the ground state of trap A? A. n =1 B. n =2 C. n =3 D. n =4 E. n =5

B

A particle is trapped in a one-dimensional well with infinite potential energy at the walls. Three possible pairs of energy levels are 1. n = 3 and n =1 2. n = 3 and n =2 3. n = 4 and n =3 Order these pairs according to the difference in energy, least to greatest. A. 1, 2, 3 B. 3, 2, 1 C. 2, 3, 1 D. 1, 3, 2 E. 3, 1, 2

C

A particle is trapped in a finite potential energy well that is deep enough so that the electron can be in the state with n = 4. For this state how many nodes does the probability density have? A. none B. 1 C. 3 D. 5 E. 7

C

The Balmer series of hydrogen is important because it: A. is the only one for which the quantum theory can be used B. is the only series that occurs for hydrogen C. is in the visible region D. involves the lowest possible quantum number n E. involves the highest possible quantum number n

C

The radial probability density for the electron in the ground state of a hydrogen atom has a peak at about: A. 0.5 pm B. 5 pm C. 50 pm D. 500 pm E. 5000 pm

C

A particle in a certain finite potential energy well can have any of five quantized energy values and no more. Which of the following would allow it to have any of six quantized energy levels? A. Increase the energy of the particle B. Decrease the energy of the particle C. Make the well shallower D. Make the well deeper E. Make the well narrower

D

A particle in a certain finite potential energy well can have any of five quantized energy values and no more. Which of the following would allow it to have any of six quantized energy levels? A. Increase the momentum of the particle B. Decrease the momentum of the particle C. Decrease the well width D. Increase the well depth E. Decrease the well depth

D

A particle is trapped in an infinite potential energy well. It is in the state with quantum number n = 14. How many maxima does the probability density have? A. none B. 7 C. 13 D. 14 E. 15

D

An electron is in a one-dimensional trap with zero potential energy in the interior and infinite potential energy at the walls. The ratio E3/E1 of the energy for n = 3 to that for n = 1 is: A. 1/3 B. 1/9 C. 3/1 D. 9/1 E. 1/1

D

An electron is in a one-dimensional well with finite potential energy barriers at the walls. The matter wave: A. is zero at the barriers B. is zero everywhere within each barrier C. is zero in the well D. extends into the barriers E. is discontinuous at the barriers

D

Consider the following: 1. the probability density for an l= 0 state 2. the probability density for a state with l!=0 3. the average of the probability densities for all states in an l!= 0 subshell Of these which are spherically symmetric? A. only 1 B. only 2 C. only 1 and 2 D. only 1 and 3 E. 1, 2, and 3

D

Identical particles are trapped in one-dimensional wells with infinite potential energy at the walls. The widths L of the traps and the quantum numbers n of the particles are 1. L =2Lo, n =2 2. L =2Lo, n =4 3. L =3Lo, n =3 4. L =4Lo, n =2 Rank them according to the kinetic energies of the particles, least to greatest. A. 1, 2, 3, 4 B. 4, 3, 2, 1 C. 1 and 3 tied, then 2, 4 D. 4, 2, then 1 and 3 tied E. 1, 3, then 2 and 4 tied

D

Take the potential energy of a hydrogen atom to be zero for infinite separation of the electron and proton. Then the ground state energy is −13.6eV. The negative sign indicates: A. the kinetic energy is negative B. the potential energy is positive C. the electron might escape from the atom D. the electron and proton are bound together E. none of the above

D

Take the potential energy of a hydrogen atom to be zero for infinite separation of the electron and proton. Then the ground state energy of a hydrogen atom is −13.6eV. When the electron is in the first excited state its excitation energy is: A. 0 B. 3.4 eV C. 6.8 eV D. 10.2 eV E. 13.6 eV

D

Take the potential energy of a hydrogen atom to be zero for infinite separation of the electron and proton. Then, according to quantum theory the energy En of a state with principal quantum number n is proportional to: A. n B. n^2 C. 1/n D. 1/(n^2) E. none of the above

D

The ground state energy of an electron in a one-dimensional trap with zero potential energy in the interior and infinite potential energy at the walls: A. is zero B. decreases with temperature C. increases with temperature D. is independent of temperature E. oscillates with time

D

The principle of complementarity is due to: A. Einstein B. Maxwell C. Newton D. Bohr E. Schrodinger

D

When a hydrogen atom makes the transition from the second excited state to the ground state (at −13.6eV) the energy of the photon emitted is: A. 0 B. 1.5 eV C. 9.1 eV D. 12.1 eV E. 13.6 eV

D

A particle is trapped in an infinite potential energy well. It is in the state with quantum number n = 14. How many nodes does the probability density have (counting the nodes at the ends of the well)? A. none B. 7 C. 13 D. 14 E. 15

E

An electron is trapped in a deep well with a width of 0.3 nm. If it is in the state with quantum number n = 3 its kinetic energy is: A. 6.0×10^−28 J B. 1.8×10^−27 J C. 6.7×10^−19 J D. 2.0×10^−18 J E. 6.0×10^−18 J

E

The energy of a particle in a one-dimensional trap with zero potential energy in the interior and infinite potential energy at the walls is proportional to (n = quantum number): A. n B. 1/n C. 1/(n^2) D. √n E. n^2

E

The series limit for the Balmer series represents a transition m → n, where (m,n) is A. (2,1) B. (3,2) C. (∞,0) D. (∞,1) E. (∞,2)

E