Phys 622 Problems Chapter 5

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1 1 Phys 622 Problems Chapter 5 Problem 1 The correct basis set of perturbation theory Consider the relativistic correction to the electron-nucleus interaction H LS = α L S, also known as the spin-orbit interaction. (a) Calculate the commutator of H LS with L 2 and L z. What do the results tell us about the basis set nlm l? (b) Justify replacements where needed and write down a suitable basis set. (c) Find replacements for both L 2 and L z and write down an alternative suitable basis set, indexed with eigenvalues of these two new operators. multiplets is no longer ade- Your answer will make it clear why grouping transitions in 2S+1 L J quate in heavy atoms, where this correction is large. Problem 2 Time-independent perturbation theory of degenerate states We discussed many cases of time-independent perturbations that lift degeneracy in 1st order. Consider again the perturbed Hamiltonian matrix H = E 1 0 a 0 E 1 b of problem 5.12 in a basis a b E 2 set 1, 2, 3. You already know that, because the degeneracy is not removed in 1st order (the loophole we discussed in class), the equations of time-independent perturbation theory (TIPT) will not give the correct answer and that this has something to do with choosing a correct basis set. Here you will solve this problem in a different way. states do not affect each other in 2nd order. The correct basis set is that in which the (a) Calculate the matrix V 2 = V V. You should see that 1 V 2 2 0, which means that the initial basis set is not good. Take real a, b for simplicity. cos(θ) sin(θ) 0 (b) Consider a general transformation matrix U + = sin(θ) cos(θ) 0 between two basis sets Apply the similarity transformation to V 2 to find the representation of this operator in the new basis set. Setting 1 V 2 2 = 0 will give us the condition for θ that will insure that the states 1 and 2 of the new basis set do not affect each other in 2nd order. (c) Calculate the 2nd order shifts for states 1 and 2 in this new (correct) basis set, with the usual relations of TIPT, for instance 1 = 1 V 3 2 E 1 E 3. You should now obtain the correct energy shifts as given by direct matrix H diagonalization. Problem 3 Stimulated and spontaneous emission The time-dependent two-state problem of the Rabi oscillations between the upper and a lower

2 energy level did not include contributions from the spontaneous emission, even though this process was long-known. (a) The population in the upper state is, in general, diminished by both stimulated emission dn 2 dt stimulated = B 21 n 2 ρ and spontaneous emission dn 2 dt spontaneous = A 21 n 2, where A 21 = B 21 2hν 3 /c 2 and ρ(ν, T ) = 2hν3 /c 2 is the black-body radiation energy density. Assume that the experimental e hν/kt 1 system is at T = 300 K. What is the ratio of the two rates for the energy difference of the first maser (ν = Hz between the S and A states of the ammonia NH 3 molecules) and for the energy difference (1.89 ev) of the 3p 2s transition in the hydrogen atom? Your answer will make it clear why stimulated emission was first demonstrated in the microwave energy range and only later at optical frequencies. k = J/K = evs 1 ev = J (b) A quantum-mechanical analysis of spontaneous rate processes A 21 requires the quantization of the electromagnetic field. However, the relation A 21 = B 21 2hν 3 /c 2 holds at all frequencies. This allows us an indirect way to calculate the lifetime of atomic states at optical frequencies without quantizing the field. For simplicity, consider a transition at an energy sufficiently large that the contribution to the lifetime of the stimulated emission rate can be entirely neglected. We obtained in class an expression for the (stimulated) transition rate in the dipole approximation as w 2 1 = 2πe2 2πc 2 U m 2 ω 2 2 ex 1 2 δ(e m 2 c 2 ω 2 e 2 2 E 1 ω). Replace Uδ(E 2 E 1 ω) with ρ, the density of final (photon) states for this transition. Replace 2 ex 1 2 with 2 e r 1 2 /3 from symmetry. Find an expression for B 21 in the dipole approximation. This allows calculating A 21 and the lifetime of the atomic state from τ = 1 A Problem 4 Angular distribution of radiation from atomic transitions An atom makes a transition from a higher state i to a lower state n. Its interaction with light in the electric dipole approximation can be written as V ni = ea 0ˆɛ (imω mc ni n r i ). Since E = A, this is µ E, which is the form the dipole approximation is usually encountered. c t Spontaneous emission requires a quantized EM field, but the same dipole approximation can be applied.

3 3 The Fermi golden rule gives the intensity as V ni 2. the emitted light is contained in the factor ˆɛ n r i ) 2. Therefore, the angular distribution of (a) Consider the three 2p, m = 0, ±1 1s transitions in the hydrogen atom. Write down the matrix elements 1s r 2p, m = 0, ±1. Since we are interested in the angular distribution only, you may define the dr integral as an overall factor. Hint: This can be calculated by taking the integrals. A faster way is to express x, y, x in terms of spherical harmonics Y 0,±1 1 = T (1) 0,±1 (since any cartesian vector can be expressed in terms of spherical tensors of rank 1) and then apply the property that the spherical harmonics are orthogonal. (b) Choose a spherical coordinate system with angles θ, φ defined by the position of the detector. This is the natural coordinate system for the wave vector k of the emitted light. Write one of the two polarization vectors ˆɛ perpendicular to ˆk in the x, y, z coordinate system. Calculating the inner product of the two vectors ˆɛ 1s r 2p is now straightforward, since you have all their cartesian components. Find the angular distribution of light emitted with this polarization for the three transitions. Sketch the obtained angular distribution. (c) Do the same for the other polarization component and sketch the angular distributions of the emitted light for the three transitions. Problem 5 Landé g-factor (a) Show that the expression for the Landé g-factor we obtained in class reduces to l+1 for a single-electron atom with j = l and s = 1 2. (b) Consider the multiplet 2 P 1 of a multi-electron atom. Calculate the Landé g-factors and sketch 2 the evolution of these energy levels with the applied field in the low-field range. (c) Same for the 2 P 3 2 multiplet. (d) What magnetic field is required to split the 2 P 1 2 multiplet by 10 µev? Problem 6 Spin precession equations In class, we discussed in the Heisenberg picture the motion of an electron spin in a magnetic field B = {B 1 cos(ωt), B 1 sin(ωt), B 0 } and calculated the x component of d S with the Heisenberg dt equation of motion, showing that it is equal to the x component of e S B. mc (a) Show that ( ds dt ) z = i[h, S z ] = e mc ( S B) z

4 4 (b) Same for the y component, ( ds dt ) y = i[h, S y ] = e mc ( S B) y. Problem 7 Off-resonant index of refraction (a) Consider a 2-state system with two energies E 1,2 starting in the ground state. A harmonic perturbation due to an electromagnetic field of frequency ω polarized along the x axis is applied starting at t = 0. Write down the expressions for c (1) 1 (t), c (1) 2 (t) for the system state α in 1st order time-dependent perturbation theory (TDPT). (b) Calculate α ˆr α and apply the dipole approximation to find the polarizability χ from its definition P = α eˆr α = χe and the index of refraction from n = 1 + 4πχ. (c) Your expression should have a relatively smooth dependence on ω and a characteristic divergence at ω = ω 21 = E 2 E 1. Sketch the dependence of n on ω. This result, like TDPT results in general, is valid only off-resonance. More sophisticated models are required on resonance, when the material responds strongly to the perturbation, and new damping processes have to be considered. The TDPT results are often identical to the results obtained with classical mechanics (Rutherford scattering, to be discussed later in the course, is another example). Problem 8 Free electron in a time-dependent external electric field A free electron is initially at rest. A spatially-uniform electric field increases from 0 over the time interval T as t E = {0, 0, E 0 }, and drops back to zero at t = T. T (a) Find the electron momentum p z (t) with classical mechanics and sketch its time dependence. (b) Obtain the exact solution for the canonical momentum ˆp z (t) in the Heisenberg picture. (c) Show that the canonical ˆp z and kinematic ˆπ z = m dẑ dt (d) Find the expectation value of position ẑ(t). momenta are equal. (e) Find the expectation value of momentum ˆp z (t) with time-dependent perturbation theory in 1st order. Hint: when calculating n ˆV i it may be useful to note that ẑ is the generator of translations in the p -space. Problem 9 Non-degenerate perturbation theory of a 3D quantum rotator

5 5 We have a system with the unperturbed Hamiltonian Ĥ0 = AˆL 2 + B ˆL z. This may describe a system in a magnetic field oriented along the z-axis. lm would be a good basis set to use. (a) What are the energies of the l = 1, m = +1, 0, 1 states? (b) Consider a perturbation ˆV = V 0 = const. Use the perturbation theory to find the energy shifts of states 11, 10, and 1 1 up to 2nd order. Hint: the issue of a good basis set does not appear in non-degenerate perturbation theory because the denominators E n (0) E (0) k 0. (c) Consider a perturbation ˆV = C ˆL y with C B, which may be done by applying an additional small magnetic field along the y-axis. Use the perturbation theory to find the energy shifts of states 11, 10, and 1 1 up to 2nd order. (d) More generally, show that the energy shift is E = mc2 2B for an arbitrary state lm. (e) The two fields can be added into a total field oriented at a small angle to the z-axis. Since the AˆL 2 part in the unperturbed Ĥ0 is rotationally-invariant, we can define a new z -axis along this total field and use a lm basis set. What are the exact new energy eigenstates and their exact energies in this total field? Hint: expansion for small C gives the perturbed states and their approximate energies. Problem 10 A perturbation of the 1D simple harmonic oscillator Consider a time-independent perturbation ˆV = B ˆp x B ˆp of the 1D simple harmonic oscillator with Ĥ0 = ˆp2 + mω2 ˆx 2. Its time-dependent version will later describe a perturbation due to light 2m 2 polarized along the x-axis, the most important QM perturbation of all. (a) Calculate [ ˆV, ˆN] where ˆN = â + â and represent the perturbation in the basis set n. What feature of the matrix corresponds to the commutator result obtained earlier? (b) Write down the 1st and 2nd order energy shifts of the state n. (c) Solve the problem exactly, by absorbing the perturbation into the kinetic energy with a redefined momentum operator. Compare to the results from perturbation theory. (d) Do you expect any shifts when the perturbation is ˆV = B ˆp 2 x? Calculate the 1st and 2nd order shifts with perturbation theory to confirm your answer. Hint: this problem can also be solved exactly. Problem 11 Molecule in a constant electric field (part 1) This part is related to the perturbation of a planar molecule when a constant electric filed is

6 6 applied in its plane. (a) Consider a particle in a two-dimensional potential with axial symmetry V (x, y) = V (r). The Hamiltonian is Ĥ = ˆp2 2m 0 + V (ˆr). Show that [Ĥ, ˆL z ] = 0, or ˆL z is a compatible operator. Therefore, we can label the energy eigenstates by their z axis orbital angular momentum eigenvalue m = 0, ±1,... as m. Hint: to show that [V (ˆr), ˆL z ] = 0 show that [ˆr 2, ˆL z ] = 0, where ˆ r are the unit vectors along the x, y axes. = e xˆx + e y ŷ and e x,y (b) Find the representation of the m states in the r, φ basis set, or r, φ m, where φ is the in-plane azimuthal angle, and plot φ m 2 for m = 0, 1, 2 as a function of φ for a fixed r = R. Hint: apply x ˆL z α = i φ x α. (c) Find the energies of these eigenstates m. Assume that the radius is fixed r = R, so that there is no radial coordinate r contribution to energy. Hint: apply the expression for x ˆp 2 α from the textbook section 3.6. (d) Apply a perturbation ˆV = eeˆx of dipolar form. This can be done with a constant inplane electric field E along the x axis. Find the new energies with 1st order time-independent perturbation theory. Hint: separate the dr from the dφ integral, and give a name I to the overall dr integral, which cannot be evaluated without further assumptions on the potential V (r). Also, the degeneracy does not matter because the perturbation is diagonal in the sub-space of degenerate states, separated by 2m 2 (selection rules forbid transitions between these states), and one can apply the relations of non-degenerate perturbation theory. (e) Improve the previous result by calculating the energy shifts in 2nd order and plot the energy dependence on E for states with m = 0, 1, 2 (set m 0 = 1, I = 1). (f) Obtain the perturbed eigenstates m p in 1st order, find their representations in the φ basis set, and plots of φ m p 2 for m = 0, 1, 2 as a function of φ (set m 0 = 1, I = 1) and compare to the result in part (b). Problem 12 Molecule in a constant electric field (part 2) The second part is a similar calculation done in three dimensions. (g) Consider a particle in a spherically-symmetric three-dimensional potential V (x, y, z) = V (r). The Hamiltonian is Ĥ = ˆp2 2m 0 +V (ˆr). Show that we still have [Ĥ, ˆL z ] = 0 and [Ĥ, ˆL 2 ] = 0. Therefore, we can label the energy eigenstates by the eigenvalues l, m or as l, m. (h) The representation of the l, m states in the r, θ, φ basis set, or r, θ, φ r, l, m = R(r)Y m l (θ, φ)

7 is R(r)Yl m (θ, φ). Find the energies of the lm states. Note that again the result does not depend on the form for V (r). (j) Apply a perturbation with a constant electric field E. Find the new energies for states with l = 0, 1, 2 with 1st order time-independent perturbation theory of degenerate states. Hint: orient the z axis of your coordinate system along the applied field to make the perturbation ˆV = eeẑ and apply the selection rules for ẑ. (k) Improve the previous result for states with l = 0, 1, 2, by calculating the energy shifts in 2nd order. Plot the energy dependence on E. 7 Problem 13 Quantum light states In class, you saw how permutation symmetry is assured when working with light states in the N representation and operators â, â + satisfying commutation relations. In this problem you will explore further properties of quantum light states. (a) Consider a single-mode light field and calculate ωt kz = χ to simplify further calculations. Ê from  given in class. Call the phase (b) Find the commutator [Ê(χ 1), Ê(χ 2)] for the quantized light field with different phases. In particular, show that the result is maximized when χ 1 χ 2 = π/2 or the two fields are in quadrature. (c) Take χ 1 χ 2 = π/2 and apply the uncertainty relation to find the product of the standard deviations of Ê(χ 1) and Ê(χ 2), to show that increased fluctuations of one correlate with decreased fluctuations of the other. The light quantum state with decreased fluctuations are called squeezed states. (d) Similarly, calculate the commutator [ ˆN, Ê] where ˆN = â + â. (e) Apply the uncertainty relation to relate the standard deviations of E and N and show that a laser light field of constant field strength E does not have a constant number of photons. Problem 14 Spin precession equations In class, we discussed in the Heisenberg picture the motion of an electron spin in a magnetic field B = {B 1 cos(ωt), B 1 sin(ωt), B 0 } and calculated the x component of d S with the Heisenberg dt equation of motion, showing that it is equal to the x component of e S B. mc (a) Show that ( ds dt ) z = i[h, S z ] = e mc ( S B) z

8 8 (b) Same for the y component, ( ds dt ) y = i[h, S y ] = e mc ( S B) y. Problem 15 Time-independent perturbation theory We discussed many cases of time-independent perturbations that fully lift the degeneracy in 1st order. Consider the perturbed Hamiltonian matrix representation Ĥ =. E 1 0 a 0 E 1 b in a basis set a b E 2 1, 2, 3 with E 2 > E 1 and small perturbations a, b. (a) Diagonalize the matrix to find the exact energy levels. (b) Calculate the energy levels with time-independent perturbation theory (TIPT) in 2nd order. Show that the results are different from the exact solution expanded in small a, b. This has to do with a requirement that the degeneracy be removed in 1st order. To solve this problem we must choose a basis set in which states do not affect each other in 2nd order. (c) Calculate the matrix V 2 = V V. You should see that 1 V 2 2 = 0, which means that the initial basis set is not good. Take real a, b for simplicity. cos(θ) sin(θ) 0 (d) Consider a general transformation matrix U + = sin(θ) cos(θ) 0 between two basis sets Apply the similarity transformation to V 2 to find the representation of this operator in the new basis set. Setting 1 V 2 2 = 0 will give us the condition for θ that will insure that the states 1 and 2 of the new basis set do not affect each other in 2nd order. (e) Calculate the 2nd order shifts for states 1 and 2 in this new (correct) basis set, with the usual relations of TIPT, for instance 1 = 1 V 3 2 E 1 E 3. You should now obtain the correct energy shifts as given by direct matrix H diagonalization. Problem 16 Crystal fields A particle is confined inside a two-dimensional range by a potential V 0 = 0 for 0 x, y L and V 0 = otherwise. This could be an approximation of an electron confined to a lattice site. (a) What are the energy eigenfunctions for the ground state and the 1st excited state? note if there is any degeneracy. Hint: (b) Consider a perturbation ˆV 1 = aˆxŷ for 0 x, y L and ˆV 1 = 0 otherwise. This type of perturbation may appear for an atom inside a crystal. Obtain the 1st order energy shifts for the ground state and the new energy eigenfunction. Give a physical interpretation of the result. (c) Same for 1st excited state.