### Chapter #15 Solutions - Quantum Mechanics - Robert Scherrer - 1st Edition

1. Show that the relativistic relation between energy and momentum (Equation 15.2) reduces to...for the case when v ≪ c. Get solution

2. If ϕ is an eigenfunction of both energy and momentum, then another differential equation corresponding to Equation (15.2) is...Why is this a less desirable equation than the Klein–Gordon equation? Get solution

3. (a) If J is the Schrödinger probability current, show that...(b) What are the units of J? Get solution

4. Using the Klein–Gordon equation, the continuity equation, and the expression for J from Equation (15.10), derive the Klein–Gordon probability density:... Get solution

5. Write out explicitly the full 4 × 4 matrices corresponding to α1 and α2. Get solution

6. Multiply out the matrices in the Dirac equation to express the Dirac equation as four coupled differential equations for the four components of ψ: ψ1, ψ2, ψ3 and ψ4. Get solution

7. Write down the Dirac spinor corresponding to a spin-1/2 particle at rest with spin in the +x direction and positive energy. Get solution

8. (a) The general solution for the Dirac equation can be written in the form...where ϕ1, ϕ2, χ1, and χ2 are numbers independent of r and t. To take advantage of this form for the Dirac equation, use the shorthand...and...Using this form for the solution, show that ϕ and χ satisfy the coupled equations...and...(b) Use the results from part (a) to show that the general four-component solution of the Dirac equation may be written as... Get solution

### Chapter #14 Solutions - Quantum Mechanics - Robert Scherrer - 1st Edition

1. Oxygen is the second most abundant element in the human body (by number) and the most abundant by mass. However, MRI detection of oxygen atoms is not practical. Why? Get solution

2. Electrons in hydrogen have a much larger magnetic moment (both orbital and spin) than the magnetic moment of the proton. Why then are MRI machines not tuned, for example, to the resonant frequency of the spin magnetic moment of the electron rather than the proton? Get solution

3. Consider an MRI machine with a 1.5 T static field. How far from the resonant frequency would one have to be in order for the spin-flip probability to decrease from its maximum value to each of the following?(a) one-half of the maximum value(b) zero Get solution

4. The random thermal energy of each molecule in the human body is roughly E ≈ kT, where k is Boltzmann’s constant (k = 1.38 × 10−23 J K−1) and T is the temperature. Compare this thermal energy to the potential energy experienced by a proton in a 1.5 T MRI machine. What does the answer say about the efficiency with which protons will align into lower energy states in such a magnetic field? Get solution

5. (a) Verify that the matrix corresponding to UNOT produces the correct output when applied to the | ↓ 〉 and | ↑〉 states.(b) Compute the matrix corresponding to ..., and calculate the result when it is applied to | ↓ 〉 and | ↑〉. What logical operation does ... correspond to? Get solution

6. (a) Show explicitly that the classical XOR gate cannot be inverted.(b) Calculate the matrix corresponding to the inverse of the quantum XOR gate Get solution

7. Show that .... Get solution

8. Apply UXOR to the mixed state (...) (| ↓ ↓〉 + | ↑ ↑〉) Explain what the result means. Get solution

### Chapter #13 Solutions - Quantum Mechanics - Robert Scherrer - 1st Edition

1. Consider the two-particle Hamiltonian given by...Show that the exchange operator ℰ12 commutes with H as long as the two-particle potential has the property that V(r1, r2) = V(r2, r1). Get solution

2. Two identical spin-1/2 particles with mass m are in a one-dimensional infinite square-well potential with width a, so V(x) = 0 for 0 ≤ x ≤ a, and there are infinite potential barriers at x = 0 and x = a. The particles do not interact with each other; they see only the infinite square-well potential.(a) Calculate the energies of the three lowest-energy singlet states.(b) Calculate the energies of the three lowest-energy triplet states.(c) Suppose that the particles are in a state with wave function...where x1 is the position of particle 1 and x2 is the position of particle 2. Are the particles in a triplet spin state or a singlet spin state? Explain. Get solution

3. Two identical spin-0 particles with mass m are confined inside a three-dimensional rectangular box given by 0 ≤ x ≤ a, 0 ≤ y ≤ b, and 0 ≤ z ≤ c, where a ≤ b ≤ c, and the potential barriers at the walls of the box are infinitely high. The particles do not interact with each other; they see only the potential of the box. Write down the normalized wave function ψ(x1, y1, z1, x2, y2, z2) for the ground state, and indicate the energy of the ground state. Get solution

4. (a) Two identical spin-1/2 particles are confined inside of the rectangular box from Exercise 13.3. The particles do not interact with each other; they see only the potential of the box. Write down the normalized spatial part of the lowest-energy singlet wave function. What is the energy of this state?(b) Write down the normalized spatial part of the lowest-energy triplet wave function. What is the energy of this state?(c) What is the total spin s of the ground-state wave function for the system? Get solution

5. Two identical spin-1/2 particles are confined to an infinite one-dimensional square well of width a with infinite potential barriers at x = 0 and x = a. The potential is V(x) = 0 for 0 ≤ x ≤ a. Suppose that the particles interact weakly by the potential V1(x) = Kδ(x1 − x2), where x1 and x2 are the positions of the two particles, K is a constant, and δ is the Dirac delta function. This represents a very short-range weak force between the two particles.(a) Using first-order perturbation theory, find the perturbation to the energy of the lowest-energy singlet state.(b) Show that the first-order perturbation to the energy of the lowest-energy triplet state is zero.(c) What is the physical reason for the answer in part (b)? Get solution

6. Two identical spin-1/2 particles are in the one-dimensional simple harmonic oscillator potential V(x) = (1/2) Kx2. The particles do not interact with each other; they see only the harmonic oscillator potential. The particles are in the lowest-energy triplet state (s = 1).(a) Write down the normalized spatial part of the wave function.(b) Calculate the energy of this state.(c) If the positions of both particles are measured, what is the probability that both particles will be located on the right-hand side of the minimum in the potential (i.e., the probability that both particles have x > 0)? Get solution

7. Show that the n shell in an atom can hold 2n2 electrons. Get solution

8. Sodium has Z = 11. Determine the ground-state electron configuration. Get solution

9. In Exercises 13.9–13.12, express all angular momentum states in the notation 2S+1LJ.Determine the ground-state L, S, and J values for(a) calcium which has the electron configuration...(b) yttrium which has the electron configuration... Get solution

10. In Exercises 13.9–13.12, express all angular momentum states in the notation 2S+1LJ.Consider the excited state of beryllium with the electron configuration...Determine all possible L, S, and J values. Note that the Pauli exclusion principle for the two n = 3 electrons can be ignored; why? Get solution

11. In Exercises 13.9–13.12, express all angular momentum states in the notation 2S+1LJ.Zirconium (Z = 40) consists of closed subshells plus 2 electrons in an unfilled d subshell. Derive the set of allowed L, S, and J values, and determine which state has the lowest energy. Get solution

12. In Exercises 13.9–13.12, express all angular momentum states in the notation 2S+1LJ.The electron configuration for nitrogen is...Calculate L, S, and J for the ground state. Get solution

### Chapter #12 Solutions - Quantum Mechanics - Robert Scherrer - 1st Edition

1. Show that in the Born approximation, the differentia] cross section obtained from the negative of a given potential −V(r) is exactly the same as that obtained from the potential V(r). Get solution

2. An incident particle with mass m, velocity υ, and charge ze scatters off of a charge Ze at the origin. Use the Born approximation to calculate the differential scattering cross section for the screened Coulomb potential...Then let d → ∞, so that V(r) approaches the normal Coulomb potential, and show that dσ/dΩapproaches the Rutherford scattering differential cross section... Get solution

3. (a) A particle with charge +e is incident on an electric dipole consisting of a charge of +e and a charge of −e separated by the vector d (which runs from − e to +e). The energy of the incident particle is sufficiently large to treat the dipole as a small perturbation. Calculate the differential scattering cross section dσ/dΩ as a function of the initial wave vector ki, the scattered wave vector kf, and the standard Rutherford scattering cross section (dσ/dΩ)R, given by...where K = kf −ki and Vc(r) is the Coulomb potential.(b) In the limits kid ≪ 1 and kid ≫ 1, determine whether the dipole differential cross section is larger or smaller than the Rutherford differential cross section. Explain the physical reason for these results. Get solution

4. A particle which is travelling in the +z direction scatters off of a potential consisting of four delta functions at the vertices of a square in the x-y plane at the points (−a, 0, 0), (+a, 0, 0), (0, −a, 0), and (0, +a, 0)....The potential is...where A is a constant. Use the Born approximation to calculate the differential scattering cross section. Express the answer in terms of the magnitude of the incident wave vector k and the scattering angles θ and ϕ. (This is an example where the cross section does depend on ϕ.) Get solution

5. (a) A particle of mass m and energy E scatters off of the central potential V(r) = Ar−2, where A is a constant. Use the Born approximation to calculate the differential cross section dσ/dΩas a function of E and the scattering angle θ.(b) Show that the total cross section σ is infinite. Get solution

6. A particle of mass m is incident on the potential ..., where V0 and r0 are constants with units of energy and length, respectively. The potential is independent of θ and ϕ. The energy of the particle is large, so that the potential can be treated as a small perturbation. Calculate the differential scattering cross section dσ/dΩ. Express the final answer as a function of the scattering angle θ and the energy of the particle E. Get solution

7. (a) A particle with mass m scatters off of the potential...where A is a constant. In other words, this potential forms a square in the x-y plane and is infinitesimally thin in the z direction. Use the Born approximation to calculate the differential scattering cross section. (Assume an arbitrary direction for the incident particle, and express the answer in terms of the momentum transfer vector K = kf − ki.)(b) Show that in the limit where a → ∞ (so that the scattering potential occupies the entire x-y plane), the resulting differential cross section corresponds to only two possible results: either the particle will pass through without any scattering or else it will scatter with the angle of incidence equal to the angle of reflection. Get solution

8. Suppose a particle with mass m scatters off of a finite spherical potential of radius R given by...In the limit where the energy of the incident particle is small, show that the total cross section is...where .... Get solution

### Chapter #11 Solutions - Quantum Mechanics - Robert Scherrer - 1st Edition

1. The electron in a hydrogen atom is initially in the ground state. At t = 0, a homogeneous electric field aligned in the z direction is turned on. The magnitude of the electric field decreases exponentially:...where ℰ0 and τ are constants. A measurement is made at tf = + ∞; what is the probability that the electron will be in the first excited state? Get solution

2. A system is in an eigenstate |ψi〉 with energy Ei. The perturbation...is turned on at ti = −∞ and left on until tf = + ∞. Here ... is independent of time, and α is a constant. Show that at tf = + ∞, the probability that the system has evolved into the eigenstate |ψf〉 with energy Ef is... Get solution

3. An electron is in a strong, uniform, constant magnetic field with magnitude B0 aligned in the +x direction. The electron is initially in the state | →〉 with x component of spin equal to +ħ/2. A weak, uniform, constant magnetic field of magnitude B1(where B1≪ B0) in the +z direction is turned on at t = 0 and turned off at t = t0. Let P(i → f) be the probability that the electron is in the state | ←〉 with x component of spin equal to −ħ/2 at a later time tf > t0. Show that... Get solution

4. Consider a time-dependent perturbation H1(t) which is adiabatic, i.e., slowly varying. The system is initially in the state |ψi〉 at ti = −∞. The potential is turned on, and we wish to derive the probability that the system will be in the state |ψf〉 t some later time tf. Write down the standard expression for cf in this case and use integration by parts to break the expression into two terms, one of which contains dH1/dt. Since H1 is slowly varying, this term may be taken to be 0. Then use the fact that H1(−∞) = 0 to derive the final expression for the transition probability:... Get solution

5. An electron is in a strong, static, homogeneous magnetic field with magnitude B0 in the z direction. At time t = 0, the spin of the electron is in the +z direction. At t = 0 a weak, homogenous magnetic field with magnitude B1(where B1 ≪ B0) is turned on. At t = 0 this field is pointing in the x direction, but it rotates counterclockwise in the x-z plane with angular frequency ω, so that at any later time t this field is at an angle ωt relative to the x-axis:...Calculate the probability that at a later time tf the electron spin has flipped to the −z direction. Get solution

6. A particle with mass m is in a one-dimensional infinite square-well potential of width a, so V(x) = 0 for 0 ≤ x ≤ a, and there are infinite potential barriers at x = 0 and x = a. Recall that the normalized solutions to the Schrödinger equation are...with energies...where n = 1, 2, 3,…The particle is initially in the ground state. A delta-function potential...(where K is a constant) is turned on at time t = −t1 and turned off at t = t1. A measurement is made at some later time t2, where t2 > t1.(a) What is the probability that the particle will be found to be in the excited state n = 3?(b) There are some excited states n in which the particle will never be found, no matter what values are chosen for t1 and t2. Which excited states are these? Get solution

7. A hydrogen atom is in the ground state. At t = 0 an electric field with magnitude ℰ is turned on. At t = 0 the electric field points in the x direction, and it rotates counterclockwise in the x-y plane with angular frequency ω (i.e., at any later time t the field is oriented at an angle ωt relative to the x-axis). This rotating field causes the atom to undergo a transition to an n = 2 state. Determine which of the l, ml states are possible final states and which are impossible. Get solution

8. (a) Consider an electromagnetic wave polarized in the x direction, incident on a hydrogen atom. Show that in this case, the selection rules for ml are...(b) Repeat this calculation for an electromagnetic wave polarized in the y direction. Get solution

9. A hydrogen atom in the n = 4 state emits electric dipole radiation and drops into the n = 3 state. Determine all possible transitions in terms of their initial and final values for l and j. Express the answer in spectroscopic notation. Get solution

10. (a) The electron in a hydrogen atom is initially in the state n = 5, l = 0, j = 1/2. The atom emits electric dipole radiation and drops into an n = 3 state. Determine all l, j states which are possible final states.(b) An electron in a hydrogen atom is initially in the state n = 5, l = 2, j = 5/2. It emits electric dipole radiation and drops into the state n = 4, l = l1, j = j1. From this state, it emits electric dipole radiation again and drops into the hydrogen ground state. Determine l1 and j1. Get solution

11. An electron is contained in a three-dimensional rectangular box given by 0 ≤ x ≤ a, 0 ≤ y ≤ b, and 0 ≤ z ≤ c. The solutions of the Schrödinger equation are specified by the quantum numbers nx, ny, and nz. Recall that the normalized wave function is...with energy...where nx, ny, and nz are positive integers. The electron is initially in the state nx, ny, nz. An electromagnetic wave is incident polarized in the y direction, so that the electric field vector is given by:...where E0 is a constant vector in the y direction. Use the dipole approximation and time-dependent perturbation theory to derive the selection rules for the electron to absorb the radiation and end up in the final state .... Get solution

### Chapter #10 Solutions - Quantum Mechanics - Robert Scherrer - 1st Edition

1. Suppose that the trial wave function ∣ψ(α)〉 happens to be exactly equal to the true ground-state wave function ∣ψ0〉 for some value of α. Show that in this case, the estimate of the ground-state energy given by the variational principle will be equal to the true ground-state energy. Get solution

2. Suppose that the trial wave function ∣ψ〉 used in the variational principle is orthogonal to the ground-state wave function of the Hamiltonian: 〈ψ0∣ψ(α)〉 = 0 for all values of α. Show that in this case...where E1 is the energy of the first excited state of H. Get solution

3. (a) In order to use the variational principle to estimate the ground-state energy of the one-dimensional potential V(x) = Kx4, where K is a constant, which of the following wave functions would be a better trial wave function?i. ...ii. ...Explain.(b) In order to use the variational principle to estimate the ground-state energy of the one-dimensional potential V(x) = Kx3 for x > 0 with an infinite potential barrier at x = 0, which of the following wave functions would be a better trial wave function?i. ...ii. ...Explain. Get solution

4. A particle of mass m is in the one-dimensional potential given by V(x) = Kx3 for x ≥ 0, where K is a positive constant. There is an infinite potential barrier at x = 0, so V(0) = ∞. Use the variational principle with the trial wave function ∣ψ〉 = xe−ax to estimate the ground-state energy. Get solution

5. Repeat the calculation in Example 10.1 using the trial wave function...where α is the parameter to be varied. Is the final result a better or a worse approximation to the true ground-state energy than the result of Example 10.1? Get solution

6. (a) A particle of mass m is in the one-dimensional potential given by V(x) = Kx4, where K is a positive constant. Use the variational principle with the trial wave function ... to estimate the ground-state energy.(b) The true ground-state wave function for this potential is a symmetric function of x, i.e., ψ0(−x) = ψ0(x). Use the result of Exercise 10.2, along with an appropriately chosen trial wave function, to estimate the energy of the first excited state. Get solution

7. A three-dimensional spherically-symmetric harmonic oscillator has the potential V(r) = (1/2)Kr2. The full Hamiltonian is then...[Note that the L2 operator has been written out in terms of derivatives.](a) Use the trial wave function ψ(r) = e−αr to calculate an approximation to the ground-state energy of the harmonic oscillator.(b) The exact ground-state energy for the three-dimensional harmonic oscillator is E = (3/2)ħω. What is the relative error in the estimate from part (a)? Get solution

8. Here is another approach to solve for the ground-state energy of helium.(a) Begin with the Hamiltonian of Equation (10.5), but neglect the interaction between the two electrons. Solve the Schrödinger equation in this case to derive the wave function of the two electrons and the energy.(b) Now add the interaction of the electrons as a perturbation:...Use first-order perturbation theory to calculate the change in energy, and add this change to the energy derived in part (a) to give an estimate for the total ground-state energy.(c) Is the estimate in part (b) more accurate or less accurate than the estimate from the variational principle? Get solution

9. (a) Singly-ionized lithium has a nucleus of charge +3e and two electrons. Use the variational principle to estimate the ground-state energy.(b) Now consider a nucleus of charge Ze with two electrons. Use the variational principle to estimate the ground-state energy. Get solution

### Chapter #9 Solutions - Quantum Mechanics - Robert Scherrer - 1st Edition

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