Question 1

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):

Accepted Answer

E

Question 2

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.

Accepted Answer

D

Question 3

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?

Accepted Answer

B

Question 4

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

Accepted Answer

The proper normalization condition for a wave function is the integral of the squared modulus of the wave function over all space equals 1 (i.e., the total probability of finding the particle somewhere in space is 1). This means that the integral in B, which integrates over all space and includes the wave function squared, should equal 1 if the wave function is properly normalized.

Question 5

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?

Accepted Answer

The probability density of a particle in a one-dimensional trap is given by the wave function squared. For higher quantum numbers, the wave functions become more spread out and have more nodes, resulting in lower probability densities near the center of the well. Therefore, the wave function with n = 3 has the highest probability density near the center of the well.

Question 6

The wavelength 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):

Accepted Answer

The answer of The wavelength of a particle in a...

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.

Accepted Answer

The energy levels in a one-dimensional well with infinite potential energy at the walls follow the equation En = (n^2*pi^2*h^2)/(2mL^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. Using this equation, we can calculate the energy differences between the pairs of levels:
1) (E2 - E1) = ((2^2 - 1^2)*pi^2*h^2)/(2mL^2) = (3*pi^2*h^2)/(2mL^2)
2) (E3 - E2) = ((3^2 - 2^2)*pi^2*h^2)/(2mL^2) = (5*pi^2*h^2)/(2mL^2)
3) (E3 - E1) = ((3^2 - 1^2)*pi^2*h^2)/(2mL^2) = (8*pi^2*h^2)/(2mL^2) = (4*2*pi^2*h^2)/(mL^2)
Comparing these values, we can see that (E2 - E1) is the smallest, followed by (E3 - E2), and (E3 - E1) is the largest. Therefore, the correct order from least to greatest is 2, 3, 1, which is choice C.

Question 8

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:

Accepted Answer

The energy difference between the initial state and the final state is:
ΔE = 7.5 eV - 3.2 eV = 4.3 eV

The energy of the photon emitted is equal to the energy difference:
E = hν = ΔE

where h is Planck's constant and ν is the frequency of the photon.

The momentum of a photon is given by:
p = h/λ = hν/c

where λ is the wavelength of the photon and c is the speed of light.

We can rearrange the equations to solve for the momentum:
p = hν/c = (h/λ) = h/p

Using the energy of the photon, we can solve for its wavelength:
E = hν
ν = E/h
λ = c/ν = hc/E

Substituting the values:
λ = hc/E = (6.626 x 10^-34 J s)(3.00 x 10^8 m/s)/(4.3 eV x 1.602 x 10^-19 J/eV) = 5.74 x 10^-7 m

Finally, we can calculate the momentum of the photon:
p = h/λ = (6.626 x 10^-34 J s)/(5.74 x 10^-7 m) = 1.15 x 10^-27 kg m/s

Rounding to two significant figures, the answer is 2.3 x 10^-10 kg m/s or 2.3 x 10^10 S.I unit. Therefore, the best choice is B.

Question 9

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)?

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 10

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?

Accepted Answer

The probability density of a particle in an infinite potential energy well with quantum number n has n-1 maxima. Therefore, in this case, the particle has 14-1=13 maxima in its probability density. However, the question is asking for the number of maxima for the probability density, not the wavefunction. The probability density is proportional to the square of the wavefunction, and therefore, it will also have 13 maxima. Thus, the correct answer is D, 14.

Question 11

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   (x)versus x is shown.The value of quantum number n is:

Accepted Answer

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

Question 12

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:

Accepted Answer

The ground state energy of a particle in a one-dimensional infinite potential well is inversely proportional to the square of the width of the well. Doubling the width reduces the energy by a factor of four, so the energy becomes 0.5 eV.

Question 13

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:

Accepted Answer

The kinetic energy of an electron in a deep well with a width of 0.3 nm and quantum number n = 3 is given by:$$T = \frac{h^2}{8mL^2}n^2$$where h is the Planck constant, m is the mass of the electron, L is the width of the well, and n is the quantum number.Plugging in the given values, we get:$$T = \frac{(6.626 	imes 10^{-34} 	ext{ J s})^2}{8(9.109 	imes 10^{-31} 	ext{ kg})(0.3 	imes 10^{-9} 	ext{ m})^2}(3)^2 = 6.0 	imes 10^{-18} 	ext{ J}$$Therefore, the correct answer is E.

Question 14

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

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 15

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:

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 16

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.

Accepted Answer

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

Question 17

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:

Accepted Answer

The energy levels in a one-dimensional infinite potential well are proportional to the square of the quantum number n. Therefore, the ratio of the energy for n = 3 to that for n = 1 is (3^2)/(1^2) = 9/1.

Question 18

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:

Accepted Answer

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

Question 19

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:

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 20

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:

Accepted Answer

The frequency of the emitted photon can be calculated using the formula: ΔE = hf where ΔE is the energy difference between the initial and final levels, h is Planck's constant and f is the frequency of the emitted photon. Thus, f = ΔE/h.

Substituting the given values, we get:

ΔE = (-2.4 x 10^-18 J) - (-1.1 x 10^-18 J) = -1.3 x 10^-18 J

h = 6.626 x 10^-34 J s

f = (-1.3 x 10^-18 J) / (6.626 x 10^-34 J s) = 1.963 x 10^15 Hz ≈ 2.0 x 10^15 Hz

Therefore, the best choice is B) 2.0 x 10^15 Hz.

Quiz 39: More About Matter Waves

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) :

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?

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

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?

The wavelength 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 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.

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 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 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?

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 (x) versus x is shown.The value of quantum number n 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 is 2.0 eV.If the width of the well is doubled, the ground state energy will be:

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:

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

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, lowest to highest.

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:

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 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 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:

Quiz 39: More About Matter Waves

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) :

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?

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

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?

The wavelength 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 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.

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 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 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?

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 (x) versus x is shown.The value of quantum number n 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 is 2.0 eV.If the width of the well is doubled, the ground state energy will be:

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:

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

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, lowest to highest.

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:

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 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 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:

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:

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: