Deck 39: More About Matter Waves

Question 1

If a wave function $\Psi$ for a particle moving along the x axis is "normalized" then:

A) $\int$$\mid$$\Psi$$\mid$ ²dt = 1
B) $\int$$\mid$$\Psi$$\mid$ ²dx = 1
C) $\partial$ $\Psi$ / $\partial$ x = 1
D) $\partial$ $\Psi$ / $\partial$ t = 1
E) $\mid$$\Psi$$\mid$ ² = 1

Answer 1

$\int$$\mid$$\Psi$$\mid$ ²dx = 1

Question 2

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

Answer 2

n = 3

Question 3

An electron is in a one-dimensional trap with zero potential energy in the interior and infinite potential energy at the walls. A graph of its probability density P(x) versus x is shown. The value of the quantum number n is:

A) 0
B) 1
C) 2
D) 3
E) 4

Answer 3

2

Question 4

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 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

Question 5

A particle is trapped in a one-dimensional well with infinite potential energy at the walls. Three possible pairs of energy levels are 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

Question 6

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

Question 7

An electron in an atom initially has an energy 7.5 eV above the ground state energy. It drops to a state with an energy of 3.2 eV 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

Question 8

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

Question 9

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 2.0 eV. If the width of the well is doubled, the ground state energy will be:

A) 0.5 eV
B) 1.0 eV
C) 2.0 eV
D) 4.0 eV
E) 8.0 eV

Question 10

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* 10¹⁷ Hz
B) 2.0 * 10¹⁵ Hz
C) 2.0 *10¹³ Hz
D) 2.0 * 10¹¹ Hz
E) 2.0 * 10⁹ Hz

Question 11

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 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

Question 12

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

Question 13

An electron in an atom initially has an energy 5.5 eV above the ground state energy. It drops to a state with energy 3.2 eV 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 *^{- 7} m
D) 1.15 * 10 ^{- 7} m
E) 1.0 *10 ^{- 7} m

Question 14

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

Question 15

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

Question 16

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/n²
D)
E) n²

Question 17

A particle is confined by finite potential energy walls to a one-dimensional trap from x = 0 to x = L. Its wave funciton in the region x > L has the form:

A) $\Psi$ (x) = A sin(kx)
B) $\Psi$ (x) = Ae^kx
C) $\Psi$ (x) = Ae^-kx
D) $\Psi$ (x) = Ae^ikx
E) $\Psi$ (x) = 0

Question 18

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

Question 19

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

Question 20

An electron is in a one-dimensional trap with zero potential energy in the interior and infinite potential energy at the walls. A graph of its wave function $\Psi$ (x) versus x is shown. The value of quantum number n is:

A) 0
B) 1
C) 4
D) 6
E) 8

Question 21

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

Question 22

Take the potential energy of a hydrogen atom to be zero for infinite seperation of the electron and proton. Then, according to the quantum theory the energy E_n of a state with principal quantum number n is proportional to:

A) n
B) n²
C) 1/n
D) 1/n²
E) none of the above

Question 23

The diagram shows the energy levels for an electron in a certain atom. Of the transitions shown, which represents the emission of a photon with the most energy?

A) I
B) II
C) III
D) IV
E) V

Question 24

Take the potential energy of a hydrogen atom to be zero for infinite separation of the electron and porton. Then the ground state energy of a hydrogen atom is -13.6 eV. 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

Question 25

The series limit for the Balmer series represents a transition m $\rightarrow$ n, where (m, n) is

A) (2,1)
B) (3,2)
C) ( $\infty$ ,0)
D) ( $\infty$ ,1)
E) ( $\infty$ ,2)

Question 26

A particle in a certian 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

Question 27

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

Question 28

If the wave function $\Psi$ is spherically symmetric then the radial probability density is given by:

A) 4 $\pi$ r² $\Psi$
B) $\mid$$\Psi$$\mid$ ²
C) 4 $\pi$ r² $\mid$$\Psi$$\mid$ ²
D) 4 $\pi$$\mid$$\Psi$$\mid$ ²
E) 4 $\pi$ r $\mid$$\Psi$$\mid$ ²

Question 29

When a hydrogen atom makes the transition from the second excited state to the ground state (at -13.6 eV) 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

Question 30

The Balmer series of hydrogen is important because it:

A) is the only one for which the Bohr theory can be used
B) is the only series which 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

Question 31

Take the potential energy of a hydrogen atom to be zero for infinite separation of the electron and porton. Then the ground state energy of a hydrogen atom is -13.6 eV. The minus 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

Question 32

The principle of complementarity is due to:

A) Einstein
B) Maxwell
C) Newton
D) Bohr
E) Schrodinger

Question 33

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

Question 34

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

Question 35

Consider the following: Of these which are spherically symmetric?

A) only I
B) only II
C) only I and II
D) only I and III
E) I, II, and III

Question 36

Take the potential energy of a hydrogen atom to be zero for infinite separation of the electron and porton. Then the ground state energy of a hydrogen atom is -13.6 eV. 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

Question 37

Take the potential energy of a hydrogen atom to be zero for infinite separation of the electron and porton. Then the ground state energy of a hydrogen atom is -13.6 eV. 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

Question 38

Which of the following sets of quantum numbers is possible for an electron in a hydogen atom?

A) n = 4,
B) n = 4,
C) n = 5,
D) n = 3,
E) n = 2,

Question 39

If P(r) is the radial probability density for a hydrogen atom then the probability that the separation of the electron and proton is between r and r + dr is:

A) P dr
B) $\mid$ P $\mid$ ² dr
C) 4 $\pi$ r²P dr
D) 4 $\pi$ r² $\mid$ P $\mid$ dr
E) 4 $\pi$$\mid$ P $\mid$ dr

Question 40

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

Question 41

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