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

Accepted Answer

The normalization condition for a wave function is given by the integral of the modulus squared of the wave function over all space, integrated with respect to the appropriate variable. In this case, the particle is moving along the x axis, so we integrate the modulus squared of the wave function over all values of x. Therefore, the correct expression for the normalization condition in this case is:
∫|ψ(x)|^2 dx = 1
This corresponds to choice B.

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

Accepted Answer

The probability density for a particle in a one-dimensional well is given by $|\psi_n(x)|^2$, where $\psi_n(x)$ is the wavefunction corresponding to the nth energy state. For higher quantum numbers, the wavefunctions become more complex and have more nodes, resulting in a decrease in probability density near the center of the well. Therefore, the probability density is greatest near the center for the lowest non-zero quantum number, n = 1. However, this option is not given in the choices. Among the given options, the wavefunction for n = 3 has only one node and has a relatively higher probability density near the center of the well compared to the higher quantum numbers. Thus, the correct answer is B.

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

Accepted Answer

The quantum number n represents the number of nodes (points where the probability density is zero) in the wave function. Looking at the graph, we can see that there are two nodes present. Therefore, the value of the quantum number n is 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

Accepted Answer

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

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

Accepted Answer

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

Comparing the energy levels given, we can see that the difference in energy between levels in the second pair is the smallest, followed by the first pair, and then the third pair. Thus, the correct order from least to greatest difference in energy is 2, 3, 1. Therefore, the answer is choice C.

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

Accepted Answer

The answer of Two one-dimensional traps have infinite potential energy...

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

Accepted Answer

In the process of dropping from an initial energy of 7.5 eV to a final energy of 3.2 eV, the electron emits a photon with energy equal to the difference in energies between the two states: 
E_photon = E_initial - E_final = (7.5 eV) - (3.2 eV) = 4.3 eV

Using the equation for the energy of a photon, E_photon = h*f, we can solve for the frequency of the photon: 
f = E_photon/h = (4.3 eV)/(4.136 x 10^-15 eV*s) = 1.04 x 10^15 Hz

The momentum of a photon is given by p = h/λ, where λ is the wavelength. We can use the equation c = f*λ to solve for the wavelength: 
λ = c/f = (3 x 10^8 m/s)/(1.04 x 10^15 Hz) = 2.88 x 10^-7 m

Finally, we can substitute the values for h and λ into the equation for momentum to find: 
p = h/λ = (6.626 x 10^-34 J*s)/(2.88 x 10^-7 m) = 2.30 x 10^-27 kg*m/s

This corresponds to answer choice B.

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

Accepted Answer

The ground state energy in this trap is determined solely by the shape of the potential energy function, and not by temperature. Therefore, it is independent of temperature.

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

Accepted Answer

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

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

Accepted Answer

The answer of An electron in an atom drops from...

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

Accepted Answer

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

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 an electron in a one-dimensional infinite potential well (or "trap") are given by:
En = (n^2*h^2)/(8mL^2)
where n is the quantum number (n = 1, 2, 3, ...), h is Planck's constant, m is the mass of the electron, and L is the width of the well. 
In this case, the potential energy is zero inside the well and infinite outside the well (i.e. at the walls), so we can assume L = 1 (arbitrary units) and the energy at the walls is infinite. 
Therefore, the energies for n = 1, 2, and 3 are:
E1 = (h^2)/(8m)
E2 = (4*h^2)/(32m) = (h^2)/(8m)*4
E3 = (9*h^2)/(72m) = (h^2)/(8m)*9
The ratio of the energy for n = 3 to that for n = 1 is then:
E3/E1 = (h^2)/(8m)*9 / (h^2)/(8m) = 9
Therefore, the answer is D) 9/1.

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 *^- ⁷m D) 1.15 * 10 ^- ⁷ m E) 1.0 *10 ^- ⁷m

Accepted Answer

The energy of the photon is equal to the difference in energy between the two states, which is 5.5 eV - 3.2 eV = 2.3 eV. The wavelength of the photon is given by the equation:λ = hc/Ewhere h is Planck's constant, c is the speed of light, and E is the energy of the photon. Substituting the values into the equation, we get:λ = (6.626 x 10^-34 J s)(3 x 10^8 m/s) / (2.3 eV * 1.6 x 10^-19 J/eV) = 5.4 x 10^-7 mTherefore, the correct answer is A.

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

Accepted Answer

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

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

Accepted Answer

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

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

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

In the region x > L, the wave function must decay exponentially because the particle is in a classically forbidden region, so the wave function takes the form Ae^-kx.

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

Accepted Answer

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

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

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.

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

Accepted Answer

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

Quiz 39: More About Matter Waves

If a wave function $\Psi$ 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?

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:

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 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.

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?

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:

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:

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 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:

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.

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 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 maxima does the probability density have?

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 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 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:

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