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

B

Question 2

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

Accepted Answer

The number of maxima in the probability density function of a particle in an infinite potential energy well is equal to the quantum number (n) of the state. Therefore, for an n=14 state, there will be 14 maxima.

Question 3

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 a particle in a one-dimensional trapped system is given by the wave function squared, |ψ(x)|^2. For a given value of n, the wave function ψ(x) has n-1 nodes. As the quantum number n increases, the number of nodes increases as well, causing the probability density to be more spread out across the trap. Therefore, the probability density for n=3 is the greatest near the center of the well compared to the other options.

Question 4

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 momentum of a photon is given by p = E/c, where E is the energy of the photon and c is the speed of light in vacuum. The energy of the emitted photon is equal to the difference between the initial and final energies of the electron: ∆E = 7.5 eV - 3.2 eV = 4.3 eV. Converting to joules: ∆E = 4.3 eV * 1.6 × 10^-19 J/eV = 6.88 × 10^-19 J. The momentum of the photon is then: p = ∆E / c = (6.88 × 10^-19 J) / (3 × 10^8 m/s) = 2.30 × 10^-27 kg m/s = 2.30 × 10^-20 Ns. Converting to scientific notation: 2.3 × 10^-20 Ns = 2.3 × 10^-10 kg m/s. Therefore, the best choice is B.

Question 5

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

Accepted Answer

The answer of A particle is confined by finite potential...

Question 6

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 7

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 answer of A particle is trapped in a one-dimensional...

Question 8

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

Accepted Answer

The answer of A particle is trapped in an infinite...

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:
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 infinite potential walls is determined by the quantum mechanical properties of the system, which are independent of temperature.

Question 10

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

Accepted Answer

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

Question 11

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

Accepted Answer

The answer of Identical particles are trapped in one-dimensional wells...

Question 12

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

Accepted Answer

In a one-dimensional trap with zero potential energy in the interior and infinite potential energy at the walls, the wave function of the electron must be zero at the walls. This condition is fulfilled by standing waves, which have nodes (points where the wave function is zero) at the walls. The number of nodes in the wave function is equal to the quantum number n.

From the graph, we can see that the wave function has 4 nodes, 2 at each wall. Therefore, the value of quantum number n is 4.

Question 13

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 ground state energy for a particle in a 1-dimensional box of width L is given by $E_1=\frac{h^2}{8mL^2}$.
For a particle in a 1-dimensional box of width 2L, the energy levels are given by $E_n=\frac{n^2h^2}{32mL^2}$.
Setting $E_1=E_n$, we get $\frac{h^2}{8mL^2}=\frac{n^2h^2}{32mL^2}$, which simplifies to $n=2$. Therefore, the particle in trap B has the same energy as a particle in the ground state of trap A when $n=2$.

Question 14

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 answer of The ground state energy of an electron...

Question 15

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 infinite potential well are given by En = (n^2*h^2)/(8m*L^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 well. For this problem, the energy for n = 1 is E₁ = h^2/(8mL^2), and the energy for n = 3 is E₃ = 9*h^2/(8mL^2). The ratio of these energies is E₃/E₁ = 9/1, which is choice D.

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

Accepted Answer

The answer of The energy of a particle in a...

Question 17

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 the integral of the absolute square of the wave function over all space, which is represented by ∫|ψ(x)|^2 dx. In this case, since the particle is moving along the x-axis, the integral becomes ∫|ψ(x)|^2 dx = ∫|F(x)ψ(x)|^2 dx, where F(x) is the normalization constant. Therefore, option B is the correct choice as it represents the integral for the normalization condition of the wave function.

Question 18

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) 1.15 *10 ^- ⁷ m E) 1.0 *10 ^- ⁷m

Accepted Answer

The answer of An electron in an atom initially has...

Question 19

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 three peaks, indicating three nodes. The number of nodes is equal to n-1, so n=4. Therefore, the correct choice is C.

Question 20

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 of an electron in a one-dimensional well with finite potential energy barriers at the walls extends into the barriers because there is a small probability of finding the electron outside the well, and this is represented by the wave function extending into the barriers.

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 an infinite potential energy well. It is in the state with quantum number n = 14. How many maxima does the probability density have?

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 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 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:

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 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 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) ?

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:

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.

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.

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?

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:

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 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) :

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 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:

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:

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