Question 1

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

Accepted Answer

The frequency of the emitted photon is given by the difference in energy between the initial and final states of the electron, divided by Planck's constant. 
ΔE = (-2.4 x 10^-18 J) - (-1.1 x 10^-18 J) = -1.3 x 10^-18 J 
ν = ΔE/h = (-1.3 x 10^-18 J)/(6.626 x 10^-34 J s) = 1.96 x 10^15 Hz ≈ 2.0 x 10^15 Hz, which corresponds to choice B.

Question 2

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

Accepted Answer

The energy levels in a one-dimensional well are given by $E_n = \frac{n^2h^2}{8ma^2}$, where $n$ is a positive integer, $h$ is Planck's constant, $m$ is the mass of the particle, and $a$ is the width of the well. The energy difference between two levels is given by $\Delta E = E_f - E_i = \frac{h^2}{8ma^2}(f^2 - i^2)$.

For the three pairs of energy levels, we have:

Pair 1: $\Delta E = \frac{h^2}{8ma^2}(2^2 - 1^2) = \frac{3h^2}{8ma^2}$

Pair 2: $\Delta E = \frac{h^2}{8ma^2}(3^2 - 2^2) = \frac{5h^2}{8ma^2}$

Pair 3: $\Delta E = \frac{h^2}{8ma^2}(3^2 - 1^2) = \frac{8h^2}{8ma^2} = \frac{h^2}{ma^2}$

Therefore, the pairs should be ordered from least to greatest energy difference, which is:

Pair 1, Pair 2, Pair 3

So choice C is the correct answer.

Question 3

A particle is confined by finite potential energy walls to a one-dimensional trap from x = 0 to x = L. Its wave function 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

Accepted Answer

For a particle in a one-dimensional trap with finite potential energy walls, outside the region of confinement (x > L), the wave function decays exponentially, indicating that the particle has a non-zero probability of being found outside the walls, but this probability decreases as one moves further away. This behavior is correctly described by an exponentially decaying function, which is represented by choice C, "Ae^(-kx)", where A is a normalization constant, and k is related to the decay rate of the wave function outside the potential walls.

Question 4

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

Accepted Answer

The normalization condition for a wave function is given by integrating the product of the wave function and its complex conjugate with respect to the relevant variable over all space. In this case, since the particle is moving along the x axis, we integrate over x. Therefore, the correct normalization condition is:

∫ |ψ(x)|^2 dx = 1

This is equivalent to choice B in the multiple-choice options.

Question 5

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

Accepted Answer

The energy difference between the initial and final states of the electron is:
ΔE = 7.5 eV - 3.2 eV = 4.3 eV
The energy of the emitted photon is equal to the energy difference, therefore:
E = 4.3 eV
Using the relation E = pc (where p is the momentum of the photon and c is the speed of light), we can find the momentum of the photon:
p = E/c = (4.3 eV)(1.6 × 10^-19 J/eV)/(3.0 × 10^8 m/s) = 2.3 × 10^-27 kg·m/s
Therefore, the answer is B, with a momentum of 2.3 × 10^-27 kg·m/s.

Question 6

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 * 10 ^- ⁷m D) 1.2 *10 ^- ⁷ m E) 1.0 *10 ^- ⁷m

Accepted Answer

The energy difference between the initial state and the final state is:

ΔE = 5.5 eV - 3.2 eV = 2.3 eV

The energy of the photon emitted is equal to the energy difference, so:

E = ΔE = 2.3 eV

We can convert this to joules using the conversion factor 1 eV = 1.6 x 10^-19 J:

E = 2.3 eV x 1.6 x 10^-19 J/eV = 3.68 x 10^-19 J

The energy of a photon is related to its wavelength by the equation:

E = hc/λ

where h is Planck's constant and c is the speed of light. Rearranging this equation to solve for λ, we get:

λ = hc/E

Plugging in the values for h, c, and E that we have, we get:

λ = (6.63 x 10^-34 J s) x (3.00 x 10^8 m/s) / (3.68 x 10^-19 J) = 5.4 x 10^-7 m

Converting meters to nanometers, we get:

λ = 540 nm

Therefore, the wavelength of the photon emitted is 5.4 x 10^-7 m or 540 nm. Choice A is the correct answer.

Question 7

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) 0
B) 7
C) 13
D) 14
E) 15

Accepted Answer

The number of nodes in the probability density of a particle in a quantum state with quantum number n = 14 is n + 1. This includes the nodes at the ends of the well. Therefore, for n = 14, there are 14 + 1 = 15 nodes.

Question 8

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

Accepted Answer

The matter wave extends into the barriers because according to quantum mechanics, there is always a non-zero probability of finding the particle outside the well, even if it is unlikely. Therefore, the wave function must extend into the barriers in order to account for this non-zero probability.

Question 9

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

Accepted Answer

The answer of An electron is trapped in a deep...

Question 10

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

Accepted Answer

The energy levels of a particle in a one-dimensional infinite potential well are given by $E_n=\frac{n^2h^2}{8mL^2}$, where $n$ is the quantum number, $h$ is Planck's constant, $m$ is the mass of the particle, and $L$ is the width of the well.

For trap A, the ground state energy is $E_1=\frac{h^2}{8mL^2}$.

For trap B, the ground state energy is $E_1'=\frac{h^2}{8m(2L)^2}=\frac{1}{4}\frac{h^2}{8mL^2}=\frac{1}{4}E_1$.

To find the quantum number $n$ such that a particle in trap B has the same energy as a particle in the ground state of trap A, we set the energies equal to each other:

$E_n=\frac{n^2h^2}{8mL^2}=E_1'=\frac{1}{4}E_1=\frac{h^2}{32mL^2}$

Solving for $n$, we get:

$n=\sqrt{\frac{E_1'}{E_1}}=\sqrt{\frac{1}{4}}=\frac{1}{2}$

But $n$ must be a positive integer, so the only option is $n=2$. Therefore, the correct answer is (B).

Question 11

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

Accepted Answer

The probability density of the particle is proportional to the square of the wave function amplitude. For a particle confined in a well, the wave functions are given by $\psi_n(x) = \sqrt{\frac{2}{a}} \sin\left(\frac{n\pi x}{a}\right)$, where $a$ is the width of the well. The probability density is proportional to $|\psi_n(x)|^2 = \frac{2}{a} \sin^2\left(\frac{n\pi x}{a}\right)$.

For a particle in the ground state, $n=1$, the probability density is zero at the center of the well, which is at $x=a/2$. For higher energy levels with larger $n$, the probability density oscillates more rapidly and becomes zero at the walls of the well. Therefore, the probability density is greatest near the center of the well for the first excited state with $n=2$. Choice (B) is the correct answer.

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

Accepted Answer

The energy levels for a particle in a one-dimensional box are given by the formula E_n = n^2 h^2 / 8mL^2, where n is the quantum number, h is Planck's constant, m is the mass of the particle, and L is the length of the box. For this problem, we have L = 1 (since it is a one-dimensional box) and m is the mass of an electron.

For n = 1, E_1 = 1^2 h^2 / 8m = h^2 / 8m
For n = 3, E_3 = 3^2 h^2 / 8m = 9h^2 / 8m

The ratio E_3 / E_1 is:
E_3 / E_1 = (9h^2 / 8m) / (h^2 / 8m) = 9

Therefore, the answer is D) 9/1.

Question 13

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

Accepted Answer

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 solely determined by the size of the trap and the mass of the electron, but is independent of temperature. Therefore, the answer is D) is independent of temperature.

Question 14

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

Accepted Answer

The probability density graph shows a clear pattern of peaks and nodes, which can be used to determine the quantum number n. Each peak corresponds to a maximum probability density, and each node corresponds to zero probability density. The number of peaks (excluding the one at x=0) corresponds to n-1, and the number of nodes corresponds to n. Therefore, in this case, there are 2 peaks, which corresponds to n-1=1, and there are 3 nodes, which corresponds to n=2. Thus, the answer is C, n=2.

Question 15

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, lowest to highest.
A) 3 and 4 tied, then 2, then 1
B) 1, 2, then 3 and 4 tied
C) 1 and 2 tied, then 3, then 4
D) 4, 3, 2, 1
E) 3, 1, 2, 4

Accepted Answer

The answer of Four different particles are trapped in one-dimensional...

Question 16

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, then 4
D) 4, then 1 and 3 tied, then 2
E) 2, then 1 and 3 tied, then 4

Accepted Answer

The kinetic energy of a particle in a one-dimensional box is given by $E_n=\frac{n^2\pi^2\hbar^2}{2mL^2}$. As the width of the traps is the same for all particles, the kinetic energy is proportional to $n^2$. Thus, particle 4 with $n=4$ has the greatest kinetic energy, followed by particles 1 and 3 with $n=1$ and $n=3$ tied for second, and particle 2 with $n=2$ having the least kinetic energy. Therefore, option D is the correct answer.

Question 17

An electron confined in a one-dimensional infinite potential well has an energy of 180 eV. What is its wavelength?
A) 65 pm
B) 91 pm
C) 130 pm
D) 370 pm
E) 520 pm

Accepted Answer

The energy of an electron in a one-dimensional infinite potential well is given by:
$E = \frac{n^2h^2}{8ma^2}$, where n is the quantum number, h is Planck's constant, m is the mass of the electron, and a is the width of the well.

We can rearrange this equation to solve for the wavelength, λ:
$\lambda = \frac{h}{p}$, where p is the momentum of the electron, given by:
$p = \sqrt{2mE}$

Substituting the given values:
$E = 180$ eV $= 2.88 	imes 10^{-17}$ J
$m = 9.11 	imes 10^{-31}$ kg
$a$ is not given, but it is not needed to solve for λ.
First, we convert the energy to joules, and then solve for the momentum and wavelength:

$p = \sqrt{2mE} = \sqrt{2(9.11 	imes 10^{-31}	ext{ kg})(2.88 	imes 10^{-17}	ext{ J})} \approx 1.27 	imes 10^{-24}	ext{ kg}\cdot	ext{m/s}$

$\lambda = \frac{h}{p} = \frac{6.63 	imes 10^{-34}	ext{ J}\cdot	ext{s}}{1.27 	imes 10^{-24}	ext{ kg}\cdot	ext{m/s}} \approx 91	ext{ pm}$

Therefore, the answer is B.

Question 18

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) 0
B) 7
C) 13
D) 14
E) 15

Accepted Answer

The probability density for a particle in an infinite potential energy well with quantum number n can be represented by the equation: P(x) = (2/L)sin^2 (n*pi*x/L), where L is the length of the well. For n = 14, there are 14 half-wavelengths of the sine function within the length of the well. The maxima of the sine function occur at every half-wavelength, so there are a total of 14 maxima in the probability density function. Thus, the answer is D.

Question 19

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

Accepted Answer

The ground state energy of an electron in a one-dimensional trap with infinite potential at the walls and zero potential energy inside is given by:

E1 = π^2ħ^2/2mL^2

Where ħ is the reduced Planck constant, m is the mass of the electron, and L is the width of the well.

If the width of the well is doubled, the new ground state energy E'1 is given by:

E'1 = π^2ħ^2/2m(2L)^2 = E1/4

Therefore, the ground state energy is reduced to one-fourth of its original value, which is 0.5 eV. Therefore, the correct answer is (A).

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) 2
D) 4
E) 8

Accepted Answer

The answer of An electron is in a one-dimensional trap...

Quiz 39: More About Matter Waves

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

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

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:

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 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) ?

An electron is in a one-dimensional well with finite potential energy barriers at the walls. The matter wave:

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:

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 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?

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:

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:

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:

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, lowest to highest.

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.

An electron confined in a one-dimensional infinite potential well has an energy of 180 eV. What is its wavelength?

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?

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: