1. A particle is confined inside a rectangular box given by 0 ≤ x ≤
a, 0 ≤ y ≤ b,and 0 ≤ z ≤ c. The solution to the Schrödinger equation
is...where A is a constant. The energy levels are given by...(a)
Normalize the wave functions. You should obtain ..., where V = abc is
the volume of the box.(b) Suppose the particle is in the ground state.
Calculate the probability of finding the particle in the lower fourth of
the box, i.e., in the region z ≤ c/4. Get solution
2. A particle is confined inside a cubic box with edge of length a. Show that there are six different wave functions that have E = 14(ħ2π2/2ma2). (This is called sixfold degeneracy.) Get solution
3. (a) A particle is confined inside a rectangular box with sides of length a, a, and 2a. What is the energy of the first excited state? Is this state degenerate? If so, determine how many different wave functions have this energy.(b) Now assume the rectangular box has sides of length a, 2a, and 2a. What is the energy of the first excited state? Is this state degenerate? If so, determine how many different wave functions have this energy. Get solution
4. (a) A particle with mass m and energy E is inside a square tube with infinite potential barriers at x = 0, x = a, y = 0, and y = a. The tube is infinitely long in the z direction. Inside the tube, V = 0. The particle is moving in the +z direction. Solve the Schrödinger equation to derive the allowed wave functions for this particle. Do not try to normalize the wave functions, but make sure they correspond to motion in the +z direction.(b) Energy should not be quantized in this case because the particle is not in a bound state. Use the answer from part (a) to show that this is indeed the case. Get solution
5. Show that Lx and Ly are Hermitian. Get solution
6. Verify that [X, Y] = 0 and [Px, Py] = 0. Get solution
7. Calculate these commutators:... Get solution
8. Show that ħ has units of angular momentum. Get solution
9. The operator Q obeys the commutation relation [Q, H] = E0Q, where E0 is a constant with units of energy. Show that if ψ(x) is a solution of the time-independent Schrödinger equation with energy E, then Qψ(x) is also a solution of the time- independent Schrödinger equation, and determine the energy corresponding to Qψ(x). Get solution
10. The simple harmonic oscillator in one dimension can also be solved by the method of ladder operators. This solution is simpler and more elegant than the one in Chapter 4.(a) For a particle of mass m in a one-dimensional simple harmonic oscillator potential ..., the Hamiltonian operator is...Define the ladder operators a− and a+ to be given by...and...Show that...and...where ....(b) Suppose that ψ (x) is a solution of the time-independent Schrödinger equation for the harmonic oscillator with energy E. Show that a+ψ(x) is also a solution of the time-independent Schrödinger equation for the harmonic oscillator with energy E + ħω. Show that a−ψ(x) is a solution with energy E − ħω.(c) Show that a+a− = H − ħω/2.(d) There is no upper bound on the possible values for E, but there is a lower bound; the energy cannot be negative. This means that if ψ0(x) is the ground state wave function, then a−ψ0(x) = 0. Using the relation derived in part (c), show that the ground state wave function has energy ħω/2.(e) Write out the equation a−ψ0(x) = 0 as a differential equation and solve it to find the ground-state wave function. Get solution
11. A particle is confined in a cubic box with edge of length a, with V = 0 inside the box. The particle is in its ground state; determine whether or not the particle is in an eigenstate of Lz. Get solution
12. Consider a three-dimensional system with wave function ψ. If ψ is in the l = 0 state, we already know that Lzψ = 0. Show that Lxψ = 0 and Lyψ = 0 as well. (Note this is the only exception to the rule that a wave function cannot be simultaneously an eigenfunction of Lx, Ly, and Lz.) Get solution
13. A particle is in an eigenstate of L2 and Lz, with quantum numbers l and ml. By symmetry, we must have .... Show that .... Get solution
14. The “radius of the hydrogen atom” is often taken to be on the order of about 10−10 m. If a measurement is made to determine the location of the electron for hydrogen in its ground state, what is the probability of finding the electron within 10−10 m of the nucleus? Get solution
15. (a) The electron in a hydrogen atom is in the l = 1 state having the lowest possible energy and the highest possible value for ml. What are the n, l, and ml quantum numbers?(b) A particle is moving in an unknown central potential. The wave function of the particle is spherically symmetric. What are the values of l and ml? Get solution
16. The deuteron is a nucleus of “heavy hydrogen” consisting of one proton and one neutron. As a simple model for this nucleus, consider a single particle of mass m moving in a fixed spherically-symmetric potential V(r), defined by V(r) = −V0 for r r0 and V(r) = 0 for r > r0. This is called a spherical square-well potential. Assume that the particle is in a bound state with l = 0.(a) Find the general solutions R(r) to the radial Schrödinger equation for r r0 and r > r0. Use the fact that the wave function must be finite at 0 and ∞ to simplify the solution as much as possible. (You do not have to normalize the solutions.)(b) The deuteron is only just bound; i.e., E is nearly equal to 0. Take m to be the proton mass, m = 1.67 × 10−27 kg, and take r0 to be a typical nuclear radius, r0 = 1 × 10−15 m. Find the value of V0 (the depth of the potential well) in MeV (1 MeV = 1.6 × 10−13 J). (Hint: The continuity conditions at r0 must be used. The radial wave function R(r) and its derivative R′(r) must both be continuous at r0; this is equivalent to requiring that u(r) and u′(r) must both be continuous at r0, where u(r) = rR(r). The resulting equations cannot be solved exactly but can be used to derive the value for V0.) Get solution
17. Determine all potentials V(r, θ, ϕ) for which it is possible to find solutions of the time-independent Schrödinger equation which are also eigenfunctions of the operator Lz. Get solution
18. A particle with mass m is confined inside of a spherical cavity of radius r0. The potential is spherically symmetric and can be written in the form: V(r) = 0 for r r0, and V(r) = ∞ for r = r0; in other words, there is an infinite potential barrier at r = r0. The particle is in the l = 0 state.(a) Solve the radial Schrödinger equation and use the appropriate boundary conditions to find the ground state radial wave function R(r) and the ground state energy.(b) What is the pressure exerted by the particle (in the l = 0 ground state) on the surface of the sphere? Get solution
19. A particle of mass m is in a three-dimensional, spherically-symmetric harmonic oscillator potential given by V(r) = (1/2)Kr2. The particle is in the l = 0 state. Find the ground-state radial wave function R(r) and the ground-state energy. Get solution
20. Deuterium is an isotope of hydrogen with a nucleus consisting of one proton and one neutron. Let λ(D)2→1 be the wavelength of the photon emitted when the electron in a deuterium atom drops from the n = 2 state to the n = 1 state, and let λ(H)2→1 be the corresponding energy for ordinary hydrogen. Calculate λ(D)2→1 − λ(H)2→1. Get solution
2. A particle is confined inside a cubic box with edge of length a. Show that there are six different wave functions that have E = 14(ħ2π2/2ma2). (This is called sixfold degeneracy.) Get solution
3. (a) A particle is confined inside a rectangular box with sides of length a, a, and 2a. What is the energy of the first excited state? Is this state degenerate? If so, determine how many different wave functions have this energy.(b) Now assume the rectangular box has sides of length a, 2a, and 2a. What is the energy of the first excited state? Is this state degenerate? If so, determine how many different wave functions have this energy. Get solution
4. (a) A particle with mass m and energy E is inside a square tube with infinite potential barriers at x = 0, x = a, y = 0, and y = a. The tube is infinitely long in the z direction. Inside the tube, V = 0. The particle is moving in the +z direction. Solve the Schrödinger equation to derive the allowed wave functions for this particle. Do not try to normalize the wave functions, but make sure they correspond to motion in the +z direction.(b) Energy should not be quantized in this case because the particle is not in a bound state. Use the answer from part (a) to show that this is indeed the case. Get solution
5. Show that Lx and Ly are Hermitian. Get solution
6. Verify that [X, Y] = 0 and [Px, Py] = 0. Get solution
7. Calculate these commutators:... Get solution
8. Show that ħ has units of angular momentum. Get solution
9. The operator Q obeys the commutation relation [Q, H] = E0Q, where E0 is a constant with units of energy. Show that if ψ(x) is a solution of the time-independent Schrödinger equation with energy E, then Qψ(x) is also a solution of the time- independent Schrödinger equation, and determine the energy corresponding to Qψ(x). Get solution
10. The simple harmonic oscillator in one dimension can also be solved by the method of ladder operators. This solution is simpler and more elegant than the one in Chapter 4.(a) For a particle of mass m in a one-dimensional simple harmonic oscillator potential ..., the Hamiltonian operator is...Define the ladder operators a− and a+ to be given by...and...Show that...and...where ....(b) Suppose that ψ (x) is a solution of the time-independent Schrödinger equation for the harmonic oscillator with energy E. Show that a+ψ(x) is also a solution of the time-independent Schrödinger equation for the harmonic oscillator with energy E + ħω. Show that a−ψ(x) is a solution with energy E − ħω.(c) Show that a+a− = H − ħω/2.(d) There is no upper bound on the possible values for E, but there is a lower bound; the energy cannot be negative. This means that if ψ0(x) is the ground state wave function, then a−ψ0(x) = 0. Using the relation derived in part (c), show that the ground state wave function has energy ħω/2.(e) Write out the equation a−ψ0(x) = 0 as a differential equation and solve it to find the ground-state wave function. Get solution
11. A particle is confined in a cubic box with edge of length a, with V = 0 inside the box. The particle is in its ground state; determine whether or not the particle is in an eigenstate of Lz. Get solution
12. Consider a three-dimensional system with wave function ψ. If ψ is in the l = 0 state, we already know that Lzψ = 0. Show that Lxψ = 0 and Lyψ = 0 as well. (Note this is the only exception to the rule that a wave function cannot be simultaneously an eigenfunction of Lx, Ly, and Lz.) Get solution
13. A particle is in an eigenstate of L2 and Lz, with quantum numbers l and ml. By symmetry, we must have .... Show that .... Get solution
14. The “radius of the hydrogen atom” is often taken to be on the order of about 10−10 m. If a measurement is made to determine the location of the electron for hydrogen in its ground state, what is the probability of finding the electron within 10−10 m of the nucleus? Get solution
15. (a) The electron in a hydrogen atom is in the l = 1 state having the lowest possible energy and the highest possible value for ml. What are the n, l, and ml quantum numbers?(b) A particle is moving in an unknown central potential. The wave function of the particle is spherically symmetric. What are the values of l and ml? Get solution
16. The deuteron is a nucleus of “heavy hydrogen” consisting of one proton and one neutron. As a simple model for this nucleus, consider a single particle of mass m moving in a fixed spherically-symmetric potential V(r), defined by V(r) = −V0 for r r0 and V(r) = 0 for r > r0. This is called a spherical square-well potential. Assume that the particle is in a bound state with l = 0.(a) Find the general solutions R(r) to the radial Schrödinger equation for r r0 and r > r0. Use the fact that the wave function must be finite at 0 and ∞ to simplify the solution as much as possible. (You do not have to normalize the solutions.)(b) The deuteron is only just bound; i.e., E is nearly equal to 0. Take m to be the proton mass, m = 1.67 × 10−27 kg, and take r0 to be a typical nuclear radius, r0 = 1 × 10−15 m. Find the value of V0 (the depth of the potential well) in MeV (1 MeV = 1.6 × 10−13 J). (Hint: The continuity conditions at r0 must be used. The radial wave function R(r) and its derivative R′(r) must both be continuous at r0; this is equivalent to requiring that u(r) and u′(r) must both be continuous at r0, where u(r) = rR(r). The resulting equations cannot be solved exactly but can be used to derive the value for V0.) Get solution
17. Determine all potentials V(r, θ, ϕ) for which it is possible to find solutions of the time-independent Schrödinger equation which are also eigenfunctions of the operator Lz. Get solution
18. A particle with mass m is confined inside of a spherical cavity of radius r0. The potential is spherically symmetric and can be written in the form: V(r) = 0 for r r0, and V(r) = ∞ for r = r0; in other words, there is an infinite potential barrier at r = r0. The particle is in the l = 0 state.(a) Solve the radial Schrödinger equation and use the appropriate boundary conditions to find the ground state radial wave function R(r) and the ground state energy.(b) What is the pressure exerted by the particle (in the l = 0 ground state) on the surface of the sphere? Get solution
19. A particle of mass m is in a three-dimensional, spherically-symmetric harmonic oscillator potential given by V(r) = (1/2)Kr2. The particle is in the l = 0 state. Find the ground-state radial wave function R(r) and the ground-state energy. Get solution
20. Deuterium is an isotope of hydrogen with a nucleus consisting of one proton and one neutron. Let λ(D)2→1 be the wavelength of the photon emitted when the electron in a deuterium atom drops from the n = 2 state to the n = 1 state, and let λ(H)2→1 be the corresponding energy for ordinary hydrogen. Calculate λ(D)2→1 − λ(H)2→1. Get solution
