1. (a) In Example 9.2, the energy of the system can be calculated
exactly. Take ......, and calculate the exact energies. [Hint: Feel free
to use a different coordinate system; the energy levels cannot depend
on the choice of the coordinate system].(b) Take the answer in part (a)
and expand it out in powers of Bx, remembering that Bx ≪ Bz. Show that
the terms proportional to Bx and ... correspond to the answers derived
in Example 9.2. Get solution
2. A particle is in a potential V0 in its ground state |ψ0〉. A small perturbation H1 is applied to the particle. Suppose that the first order perturbation to the energy is zero: E(1) = 〈ψo|H1|ψ0〉 = 0. Show that the lowest-order effect of H1 is to decrease the energy of the ground state. Get solution
3. A particle of mass m is confined to move in a one-dimensional square well with infinite potential barriers at x = 0 and x = a, with V = 0 for 0 ≤ x ≤ a. The particle is in the ground state. A perturbation H1 = λδ(x − a/2) is added, where λ is a small constant.(a) What units does λ have?(b) Calculate the first-order perturbation E(1) due to H1.(c) Calculate the second-order perturbation E(2). The answer may be expressed as an infinite series. Get solution
4. A particle of mass m is confined to move in a narrow, straight tube of length a which is sealed at both ends with V = 0 inside the tube. Treat the tube as a one-dimensional infinite square well. The tube is placed at an angle θ relative to the surface of the earth. The particle experiences the usual gravitational potential V = mgh. Calculate the lowest-order change in the energy of the ground state due to the gravitational potential. Get solution
5. A particle of mass m is in the ground state in the harmonic oscillator potential...A small perturbation βx6 is added to this potential.(a) What are the units of β?(b) How small must β be in order for perturbation theory to be valid?(c) Calculate the first-order change in the energy of the particle. Get solution
6. In the hydrogen atom, the proton is not really a point charge but has a finite size. Assume that the proton behaves as a uniformly-charged sphere of radius R = 10−15 m. Calculate the shift this produces in the ground-state energy of hydrogen. Get solution
7. The photon is normally assumed to have zero rest mass. If the photon had a small mass, this would alter the potential energy which the electron experiences in the electric field of the proton. Instead of...we would have...where r0 is a constant with units of length. Assume r0 is large compared to the size of the hydrogen atom, so the potential energy given in Equation (9.42) differs only slightly from the standard one given by Equation (9.41 ) in the vicinity of the electron. Calculate the change in the ground state energy of hydrogen if the correct potential is given by Equation (9.42) instead of Equation (9.41). Get solution
8. Suppose that that the proton had spin 0 instead of spin 1/2.(a) How would this alter the fine structure of the energy levels of the hydrogen atom?(b) How would this alter the hyperfine structure of the energy levels of the hydrogen atom? Get solution
9. We have seen that the spin-orbit interaction splits the l ≠ 0 states in the hydrogen atom into j = l + 1/2 states (with slightly higher energy) and j = l − 1/2 states (with slightly lower energy). Suppose that the electron had spin 1. How many different energy levels would the spin-orbit interaction produce, and what would their relative energies be? Be sure to consider how the answer would depend on the value of l. Get solution
10. Equation (9.29) gives the fine-structure energy shift.(a) Show that the j = l + 1/2 state has a higher energy than the j = l − 1 /2 state.(b) Show that the change in energy, ..., is always negative. Get solution
11. An electron is in the ground state in a three-dimensional rectangular box given by 0 ≤ x ≤ a, 0 ≤ y ≤ b, and 0 ≤ z ≤ c, where V = 0 inside the box, and there are infinite potential barriers at all of the walls. A homogeneous, static electric field with magnitude ε is applied in the x direction. What is the lowest-order change in the energy of the electron? Get solution
12. A hydrogen atom in its ground state is placed in a homogeneous, static electric field with magnitude ε in the x direction.(a) Show that the first-order perturbation E(1) is 0.(b) Show that the second-order perturbation E(2) is the same as if the field was pointing in the z direction. [This is obvious from symmetry, but calculate E(2) using perturbation theory and show it explicitly.] Get solution
13. A hydrogen atom is in its ground state. A proton is fixed in space a distance R from the nucleus of the hydrogen atom, where R ≫ a0. Calculate the perturbation to the energy of the hydrogen atom due to the electric field of this proton. Get solution
14. The electron in a hydrogen atom is in a D state. A homogenous, static magnetic field is applied in the z direction.(a) Draw a diagram showing the splitting of the energy levels in the weak-field limit. Calculate the value of g for each energy level.(b) Draw a diagram showing the splitting of the energy levels in the strong-field limit. Get solution
15. (a) A particle is in a state |ψ〉 which is an eigenfunction of the Hamiltonian with energy E. A perturbation H1 is applied such that H1|ψ〉 = 0. Show that the energy of the system, is completely unchanged by this perturbation.(b) In the ground state of the helium atom, both electrons are in the l = 0 state, and the spin wave function for the two electrons is the singlet spin state (s = 0 and ms = 0). [This is a consequence of the Pauli exclusion principle, which will be discussed in Chapter 13.] A homogeneous, static magnetic field is applied in the z direction. Show that the energy of the ground state of helium is completely unaffected by this magnetic field. [Ignore the magnetic moment of the nucleus.] What is the physical reason for this? Get solution
2. A particle is in a potential V0 in its ground state |ψ0〉. A small perturbation H1 is applied to the particle. Suppose that the first order perturbation to the energy is zero: E(1) = 〈ψo|H1|ψ0〉 = 0. Show that the lowest-order effect of H1 is to decrease the energy of the ground state. Get solution
3. A particle of mass m is confined to move in a one-dimensional square well with infinite potential barriers at x = 0 and x = a, with V = 0 for 0 ≤ x ≤ a. The particle is in the ground state. A perturbation H1 = λδ(x − a/2) is added, where λ is a small constant.(a) What units does λ have?(b) Calculate the first-order perturbation E(1) due to H1.(c) Calculate the second-order perturbation E(2). The answer may be expressed as an infinite series. Get solution
4. A particle of mass m is confined to move in a narrow, straight tube of length a which is sealed at both ends with V = 0 inside the tube. Treat the tube as a one-dimensional infinite square well. The tube is placed at an angle θ relative to the surface of the earth. The particle experiences the usual gravitational potential V = mgh. Calculate the lowest-order change in the energy of the ground state due to the gravitational potential. Get solution
5. A particle of mass m is in the ground state in the harmonic oscillator potential...A small perturbation βx6 is added to this potential.(a) What are the units of β?(b) How small must β be in order for perturbation theory to be valid?(c) Calculate the first-order change in the energy of the particle. Get solution
6. In the hydrogen atom, the proton is not really a point charge but has a finite size. Assume that the proton behaves as a uniformly-charged sphere of radius R = 10−15 m. Calculate the shift this produces in the ground-state energy of hydrogen. Get solution
7. The photon is normally assumed to have zero rest mass. If the photon had a small mass, this would alter the potential energy which the electron experiences in the electric field of the proton. Instead of...we would have...where r0 is a constant with units of length. Assume r0 is large compared to the size of the hydrogen atom, so the potential energy given in Equation (9.42) differs only slightly from the standard one given by Equation (9.41 ) in the vicinity of the electron. Calculate the change in the ground state energy of hydrogen if the correct potential is given by Equation (9.42) instead of Equation (9.41). Get solution
8. Suppose that that the proton had spin 0 instead of spin 1/2.(a) How would this alter the fine structure of the energy levels of the hydrogen atom?(b) How would this alter the hyperfine structure of the energy levels of the hydrogen atom? Get solution
9. We have seen that the spin-orbit interaction splits the l ≠ 0 states in the hydrogen atom into j = l + 1/2 states (with slightly higher energy) and j = l − 1/2 states (with slightly lower energy). Suppose that the electron had spin 1. How many different energy levels would the spin-orbit interaction produce, and what would their relative energies be? Be sure to consider how the answer would depend on the value of l. Get solution
10. Equation (9.29) gives the fine-structure energy shift.(a) Show that the j = l + 1/2 state has a higher energy than the j = l − 1 /2 state.(b) Show that the change in energy, ..., is always negative. Get solution
11. An electron is in the ground state in a three-dimensional rectangular box given by 0 ≤ x ≤ a, 0 ≤ y ≤ b, and 0 ≤ z ≤ c, where V = 0 inside the box, and there are infinite potential barriers at all of the walls. A homogeneous, static electric field with magnitude ε is applied in the x direction. What is the lowest-order change in the energy of the electron? Get solution
12. A hydrogen atom in its ground state is placed in a homogeneous, static electric field with magnitude ε in the x direction.(a) Show that the first-order perturbation E(1) is 0.(b) Show that the second-order perturbation E(2) is the same as if the field was pointing in the z direction. [This is obvious from symmetry, but calculate E(2) using perturbation theory and show it explicitly.] Get solution
13. A hydrogen atom is in its ground state. A proton is fixed in space a distance R from the nucleus of the hydrogen atom, where R ≫ a0. Calculate the perturbation to the energy of the hydrogen atom due to the electric field of this proton. Get solution
14. The electron in a hydrogen atom is in a D state. A homogenous, static magnetic field is applied in the z direction.(a) Draw a diagram showing the splitting of the energy levels in the weak-field limit. Calculate the value of g for each energy level.(b) Draw a diagram showing the splitting of the energy levels in the strong-field limit. Get solution
15. (a) A particle is in a state |ψ〉 which is an eigenfunction of the Hamiltonian with energy E. A perturbation H1 is applied such that H1|ψ〉 = 0. Show that the energy of the system, is completely unchanged by this perturbation.(b) In the ground state of the helium atom, both electrons are in the l = 0 state, and the spin wave function for the two electrons is the singlet spin state (s = 0 and ms = 0). [This is a consequence of the Pauli exclusion principle, which will be discussed in Chapter 13.] A homogeneous, static magnetic field is applied in the z direction. Show that the energy of the ground state of helium is completely unaffected by this magnetic field. [Ignore the magnetic moment of the nucleus.] What is the physical reason for this? Get solution
