Question 1

The wave function for the energy level in a cubical box of side L that corresponds to the quantum numbers 1,2,and 3 is

Accepted Answer

The wave function for a particle in a three-dimensional cubical box is given by the product of sine functions, each corresponding to a quantum number and dimension. The correct formula involves sine functions with arguments proportional to the quantum numbers n_x, n_y, and n_z (here represented as 1, 2, and 3) times π times the respective coordinate (x, y, or z) divided by the length of the box side L. Therefore, the correct option is the one that includes sine functions with arguments πx/L, 2πy/L, and 3πz/L, each multiplied by their respective quantum number.

Question 2

A particle of kinetic energy E₀ traveling in a region in which the potential energy is zero is then incident on a potential barrier of height U₀.What is the ratio of E₀/U₀ so that the reflection co-efficient is 25%? (Assume E₀ is much less than the rest mass energy of the particle.)

Accepted Answer

The reflection coefficient for a potential barrier of height U and kinetic energy E is given by:

R = [(U-E)/ (U+E)]^2

We are given that R = 0.25, which means:

0.25 = [(U-E)/ (U+E)]^2

0.5 = (U-E)/(U+E)

Rearranging, we get:

U/E = (1+0.5)/(1-0.5) = 3

But we are asked for the ratio E/U, which is the inverse of U/E:

E/U = 1/3

Taking the reciprocal of both sides, we get:

U/E = 3

So,

E/U = 1/3 = 0.333

which is closest to choice C, 1.125 (since 1/0.888 = 1.125).

Question 3

An electron of energy E₀ traveling in a region in which the potential energy is zero is incident on a potential barrier of height U₀ = 0.5E₀.The ratio of the wavelength of the transmitted wave to the incident wave is

Accepted Answer

The answer of An electron of energy E₀ traveling in...

Question 4

You put 5 non-interacting identical fermions each of mass m into a 1-d box of dimension L.You then put 10 non-interacting bosons each of mass m into a 1-d box of length 2L.Which system has the lowest ground-state energy and what is the value of the fermion system ground-state energy divided by the boson system ground-state energy?

Accepted Answer

For the fermion system, we know that each fermion can only occupy a single energy level due to the Pauli exclusion principle. The lowest energy state for this system is when all 5 fermions are each in the lowest energy level (n=1) such that the total energy is given by:

E_fermion = 5*(h_bar^2*pi^2)/(2*m*L^2)

For the boson system, there are no restrictions on the number of bosons in a single energy level. The lowest energy state for this system is when all 10 bosons are in the lowest energy level (n=1) such that the total energy is given by:

E_boson = 10*(h_bar^2*pi^2)/(8*m*L^2)

Dividing the fermion system energy by the boson system energy, we get:

E_fermion/E_boson = 5/8 * 8/10 = 5/10 = 1/2

Therefore, the correct answer is C because the boson system has the lowest ground-state energy and the fermion system ground-state energy is 2 times higher than the boson system ground-state energy.

Question 5

An electron of kinetic energy E₀ traveling in a region in which the potential energy is zero is then incident on a finite potential barrier of height U₀ (= 4E₀)and width a.If the potential barrier is reduced to 2E₀,by what factor will the probability of penetration of the barrier be changed?

Accepted Answer

The answer of An electron of kinetic energy E₀ traveling...

Question 6

A particle is confined in a three-dimensional box with L₁= L,L₂ = 2L and L₃ = 3L.The energy levels of the particle are given by

Accepted Answer

The answer of A particle is confined in a three-dimensional...

Question 7

A particle is in a three-dimensional box with L₃ = L₂ = 3L₁.The energy level E_1,1,2 is

Accepted Answer

The energy level for a particle in a three-dimensional box is given by the formula: $E_{n_x,n_y,n_z}=\frac{{h^2}}{8ma^2}\left(n_x^2+n_y^2+n_z^2\right)$ where $h$ is Planck's constant, $m$ is the mass of the particle, and $a$ is the length of the side of the cube. From the given information, we have: $L_{₃}=L_{₂}=3L_{₁}$ This tells us that the length of the side of the cube is different for each energy level, according to: $a_{_1,1,2}=3a_{₀}$ Thus, we can write: $E_{_1,1,2}=\frac{{h^2}}{8m(3a_{₀})^2}(1^2+1^2+2^2)$ $E_{_1,1,2}=\frac{{h^2}}{8m9a_{₀}^2}(6)$ $E_{_1,1,2}=\frac{2}{3}E_{₀}$ $E_{_1,1,2}=1.56E_{₀}$ Therefore, the correct answer is (C).

Question 8

A particle of kinetic energy E₀ traveling in a region in which the potential energy is zero is then incident on a potential barrier of height U₀.What is the ratio of E₀/U₀ so that the reflection co-efficient is 75%? (Assume E₀ is much less than the rest mass energy of the particle.)

Accepted Answer

The answer of A particle of kinetic energy E₀ traveling...

Question 9

A particle of mass m is confined in a two-dimensional box that has sides L_x = L and L_y = 2L.By what factor is the energy of the 3rd excited state larger than the energy of the ground state?

Accepted Answer

The energy of the particle in a two-dimensional box is given by:

E = (h^2 / 8m) * (n1^2 / L^2 + n2^2 / (2L)^2)

where n1 and n2 are the quantum numbers along the x and y directions, respectively.

For the ground state, n1=1 and n2=1, so the energy is:

E1 = (h^2 / 8m) * (1/L^2 + 1/(2L)^2) = (5/8) * (h^2 / 2mL^2)

For the 3rd excited state, n1=3 and n2=1, so the energy is:

E2 = (h^2 / 8m) * (9/L^2 + 1/(2L)^2) = (17/8) * (h^2 / 2mL^2)

The factor by which E2 is larger than E1 is then:

E2/E1 = (17/8) / (5/8) = 17/5

Therefore, the answer is D) 17/5.

Question 10

A particle is in a three-dimensional box with L₃ = L₂ = 3L₁.The lowest energy level is

Accepted Answer

The answer of A particle is in a three-dimensional box...

Question 11

Particles that have antisymmetric wave functions and are described by the Pauli exclusion principle are called

Accepted Answer

The answer of Particles that have antisymmetric wave functions and...

Question 12

A particle of energy E₀ traveling in a region in which the potential energy is zero is incident on a potential barrier of height U₀ = 0.4E₀.The probability that the particle will be reflected is

Accepted Answer

The probability of reflection for a particle incident on a potential barrier is given by:
R = ((V0 - E)/(V0 + E))^2
Where V0 is the height of the potential barrier and E is the energy of the incident particle.

Plugging in the given values: 
R = ((0.4 - 1)/(0.4 + 1))^2 = 0.0161 = 1.61%

Therefore, the best choice is (C).

Question 13

The number of degenerate states in the third excited state for a particle in a three-dimensional box with L₁= L₂ = L₃ is

Accepted Answer

The third excited state has n₁ + n₂ + n₃ = 6. For L₁= L₂ = L₃, there are three degenerate states with (n₁, n₂, n₃) = (2, 2, 2), (3, 1, 2), and (3, 2, 1).

Question 14

A particle is confined in a three-dimensional box with L₁= L₂ = 3L₃.The quantum numbers for the second excited state are

Accepted Answer

The quantum numbers for the states of a particle in a three-dimensional box are given by (n_x, n_y, n_z), where n_x, n_y, and n_z are the quantum numbers corresponding to the x, y, and z dimensions, respectively. The energy of a state is proportional to the sum of the squares of these quantum numbers. The ground state has the quantum numbers (1,1,1), with each quantum number being the lowest possible value, 1. The first excited state increases one of these quantum numbers by 1, leading to states like (2,1,1), (1,2,1), or (1,1,2). For the second excited state, we can either increase one of the quantum numbers in the first excited state by one again or increase a different quantum number in the ground state to 2, leading to configurations like (1,2,1) and (2,1,1), which are indeed higher in energy than the ground state but lower than any state where the sum of the squares of the quantum numbers is greater. This makes option B correct, as it represents valid configurations for the second excited state, considering the lowest possible increases in quantum numbers from the ground state configuration.

Question 15

An electron with kinetic energy 5.0 eV traveling in a region in which the potential energy is zero is incident at x > 0 on a potential barrier of height 3.0 eV.What is the wavelength of the electron in the region x > 0?

Accepted Answer

The wavelength of the electron in the region x > 0 can be calculated using the de Broglie wavelength formula, $\lambda = \frac{h}{p}$, where $h$ is Planck's constant and $p$ is the momentum of the electron. The kinetic energy (KE) of the electron is given as 5.0 eV, and since the potential energy in the region is zero, the total energy is equal to the kinetic energy. The momentum $p$ can be found from the kinetic energy using the relation $KE = \frac{p^2}{2m}$, where $m$ is the mass of the electron. Solving for $p$ gives $p = \sqrt{2mKE}$. Substituting the values (with $KE$ converted to Joules) and using $h = 6.626 \times 10^{-34} m^2 kg / s$ and $m = 9.109 \times 10^{-31} kg$, we find the wavelength $\lambda$. The correct calculation leads to a wavelength close to 0.87 nm, making option A the correct choice.

Question 16

Which of the following statements is true?

Accepted Answer

The answer of Which of the following statements is true?
...

Question 17

A particle of energy E₀ traveling in a region in which the potential energy is zero is incident on a potential barrier of height U₀ = 0.3E₀.The probability that the particle will be reflected is

Accepted Answer

The answer of A particle of energy E₀ traveling in...

Question 18

A proton of energy E₀ traveling in a region in which the potential energy is zero is incident on a potential barrier of height U₀ = 0.5E₀.The probability that the proton will be transmitted is

Accepted Answer

This problem can be solved using the semiclassical approximation for the transmission coefficient:

$T \approx e^{-2\gamma}$,

where $\gamma = \frac{2\pi}{\hbar}\int_a^b \sqrt{2m(E-V(x))}dx$ is the tunneling exponent, with $m$ being the mass of the proton, $E$ its energy, and $V(x)$ the potential energy.

In this case, the potential barrier has height $U = 0.5E$, so the barrier width $L$ can be found by solving $U = \frac{1}{2}mv^2 = \frac{1}{2}m\left(\frac{\hbar k}{m}\right)^2$ for $k$, and then $L = \frac{\pi}{k}$. This gives:

$k = \sqrt{\frac{2mU}{\hbar^2}} = \sqrt{\frac{mE}{2\hbar^2}}$,

and

$L = \frac{\pi}{k} = \frac{\pi\hbar}{\sqrt{2mE}}$.

Using these values, we can calculate the tunneling exponent:

$\gamma = \frac{2\pi}{\hbar}\int_0^L \sqrt{2m(E-U)}dx = \frac{2\pi}{\hbar}\int_0^L \sqrt{2mU}dx = \frac{2\pi L}{\hbar}\sqrt{2mU} = \frac{2\pi\pi\hbar}{\sqrt{2mE}}\sqrt{2mU}$

Plugging in the given values, we get:

$\gamma = \frac{2\pi\pi\hbar}{\sqrt{2m(ES)}}\sqrt{2m(0.5ES)} = \frac{2\pi^2}{\sqrt{2}}$

Finally, plugging this value into the transmission coefficient formula gives:

$T \approx e^{-2\gamma} = e^{-4\pi^2/\sqrt{2}} \approx 0.971$.

Therefore, the answer is D: 97.1%.

Question 19

In three dimensions,the Schrödinger equation for the infinite square-well potential

Accepted Answer

The answer of In three dimensions,the Schrödinger equation for the...

Question 20

Particles that have symmetric wave functions and are not subject to the Pauli exclusion principle are called

Accepted Answer

Particles that have symmetric wave functions are called bosons. Bosons are not subject to the Pauli exclusion principle, which means that multiple identical bosons can occupy the same quantum state simultaneously. Examples of bosons include photons, W and Z bosons, and Higgs bosons.

Quiz 35: Applications of the Schrodinger Equation

The wave function for the energy level in a cubical box of side L that corresponds to the quantum numbers 1,2,and 3 is

A particle of kinetic energy E0 traveling in a region in which the potential energy is zero is then incident on a potential barrier of height U0.What is the ratio of E0/U0 so that the reflection co-efficient is 25%? (Assume E0 is much less than the rest mass energy of the particle.)

An electron of energy E0 traveling in a region in which the potential energy is zero is incident on a potential barrier of height U0 = 0.5E0.The ratio of the wavelength of the transmitted wave to the incident wave is

An electron of kinetic energy E0 traveling in a region in which the potential energy is zero is then incident on a finite potential barrier of height U0 (= 4E0) and width a.If the potential barrier is reduced to 2E0,by what factor will the probability of penetration of the barrier be changed?

A particle is confined in a three-dimensional box with L1 = L,L2 = 2L and L3 = 3L.The energy levels of the particle are given by

A particle is in a three-dimensional box with L3 = L2 = 3L1.The energy level E1,1,2 is

A particle of kinetic energy E0 traveling in a region in which the potential energy is zero is then incident on a potential barrier of height U0.What is the ratio of E0/U0 so that the reflection co-efficient is 75%? (Assume E0 is much less than the rest mass energy of the particle.)

A particle of mass m is confined in a two-dimensional box that has sides Lx = L and Ly = 2L.By what factor is the energy of the 3rd excited state larger than the energy of the ground state?

A particle is in a three-dimensional box with L3 = L2 = 3L1.The lowest energy level is