PHYS 402, Atomic and Molecular Physics Spring 2017, final exam, solutions 1. Hydrogenic atom energies: Consider a hydrogenic atom or ion with nuclear charge Z and the usual quantum states φ nlm. (a) (2 points) Consider only the electro-static interaction between an infinitely heavy point nucleus and a non-relativistic electron, such that the Hamiltonian of the system is Ĥ 0 = 2 2m 2 Ze2 4πɛ 0 1 r. What are the energies E nlm of states φ nlm? [max 1 line] (b) (3 points) We now add spin into the picture with states φ nlmms = φ nlm χ ms, and consider relativistic corrections to the Hamiltonian above. Write the new terms that arise in the Hamiltonian and discuss their physical meaning. [max 8 lines] (c) (2 points) After changing to a total (orbital+spin) angular momentum basis φ nljmj, what are the energies belonging to states φ nljmj, taking into account the corrections found in (b)? Define all quantities. [max 3 lines] (d) (3 points) Give an example of a hypothetical species of ion, in which the fine-structure energy level differences would have the same order of magnitude as the original (nonrelativistic) energy level differences? Explain. [max 2 lines] (e) (2 points) How can your result from part (a) be modified for it to apply to Rydberg states of Alkali atoms? Define all variables. What can you say about energy of 40s compared to 40p for such an atom? [max 3 lines]. (a) E nlm = E n = Ry n 2 (b) See Eq. (2.1) of the lecture and discussion below. ( ) (c) See Eq. (2.4) of the lecture: E nj = E n 1 + (Zα)2 n ( n 2 j+1/2 3 4 ) (d) From the formula in (c), we see that the correction can be of order E n if Z = 1/α 137 (and n = 1). So an element with Z = 138 (which does not exist in the periodic table) stripped of all but one electron would work. Ry (e) We write E n,l = (n δ l,n, where δ ) 2 l,n is the quantum defect (of order one but larger for s than for p states). Hence E 40s < E 40p. 2. Atom-light coupling: Consider a hydrogenic atom for which the electronic ground state g is coupled to an excited state e with mono-chromatic laser light at frequency ω 0 = E 0 / =
PHYS 402, Atomic and Molecular Physics - Page 2 of 6 (E e E g )/, where E g,e are the electron energies in these two states. Let the electric field amplitude of the laser be F. (a) (2 points) Write the effective Hamiltonian for these two levels in the presence of laser light in the dipole and rotating wave approximation. Focus on the situation described above. Define and discuss all variables. [max 4 lines] (b) (3 points) Suppose the atom is initially in g, when the laser light is suddenly switched on and remains on. Discuss qualitatively what happens to the populations in states g and e. What changes if we increase the laser intensity by a factor of four? [max 6 lines] (c) (4 points) Ignoring the laser for the moment: Suppose there is a weak static magnetic field B = B 0ˆk along the z-axis (weak compared to fine structure). Consider the specific case g = 1s 1/2 [m j = 1/2] and e = 5p 1/2 [m j = 1/2] in Hydrogen. How does the energy difference E B = E e E g change in the presence of a magnetic field, compared to E 0 (without magnetic field)? Write a formula and discuss its origin (define all variables). [max 6 lines] (d) (3 points) We now consider both laser and magnetic field. Assume the only effect of the magnetic field is to change the energy difference from E 0 to E B found in (c). However we keep the laser frequency at ω 0 = E 0 /. How does the Hamiltonian in (a) have to be modified in this situation? [max 4 lines] (e) (2 points) Consider again (b), now with this magnetic field. How does an increased magnetic field strength B 0 affect the maximum population that reaches e. Explain your answer. [max 2 lines] (a) [ H eff = 0 Ω/2 Ω/2 0 ]. (1) We used the Rabi-frequency Ω = e ˆd F g, note the detuning is zero here. (b) We get sinusoidal population oscillations P g (t) cos 2 (tω/2), P e (t) sin 2 (tω/2). When the laser intensity is quadrupled, the field strength is doubled, so is the Rabifrequency. Thus the population oscillations speed up by a factor of two. (c) The two levels now experience an additional Zeeman shift, so E B E 0 = g e µ B ( 1 2 )B 0 g g µ B ( 1 2 )B 0 = µ B B 0 (g g + g e )/2, where the g x are the Lande g-factors of states x { g, e } and µ B is the Bohr magneton. Inserting the g x, we get E B E 0 = 4 3 µ BB 0. (d) Now the Hamiltonian is [ H eff = 0 Ω/2 Ω/2 ]. (2) with = [ E B E 0 ]. (e) We now no longer reach P e = 1 due to the non-vanishing detuning. 3. Molecular energy scales: Let us estimate the relative magnitudes of electronic-, vibrationaland rotational energies in di-atomic, homo-nuclear molecules.
PHYS 402, Atomic and Molecular Physics - Page 3 of 6 (a) (3 points) Use the relation ( x)( p) based on the Heisenberg uncertainty principle to estimate the electronic energy E el given the bond length R 0 of the molecule and the electron mass m e. [Hint: Use also the virial theorem, E el E kin, where E kin is the kinetic energy]. Compare your findings with electronic energies of the hydrogen atom, after expressing the latter in terms of only the electron mass and the Bohr radius a 0. [max 15 lines] (b) (3 points) Estimate vibrational energies E vib in terms of E el, m e and the nuclear mass M. Here you should assume that the oscillator/vibrational spring constant K is given by E el KR0 2. [max 4 lines] (c) (3 points) Estimate rotational energies E rot in terms of E el. [max 4 lines] (d) (3 points) In terms of the ratio m e /M, discuss the hierarchy of energy scales that you obtained. How does this relate to the spectral range of electromagnetic waves that can induce pure-rotational / vibrational / electronic transitions? [max 8 lines] (a)-(c) See solution for assignment 5. (d) We find E rot m e /ME el, E vib m e /ME el. Since m e /M 1, we have E rot < E vib < E el. Electronic transitions are typically visible or UV as for atoms, vibrational transitions infra-red and rotational transitions microwave / THz. 4. Molecular states and transitions: Consider ro-vibrational states of a homonuclear, diatomic molecule (see figure 1) in the Born-Oppenheimer (BO) approximation as in the lecture. (a) (2 points) Sketch the energy of a typical bonding BO surface in terms of the relevant coordinate(s), indicate the (probability density for the) nuclear wavepacket in the vibrational groundstate. Make sure your sketch has axis labels and define all ingredients. Indicate the equilibrium bond length. [max 4 lines] (b) (2 points) Discuss the harmonic approximation and how it results in vibrational states with quantum number ν that are equivalent to quantum harmonic oscillator states. When does this approximation break down? [max 8 lines] (c) (4 points) Write the position space representation (explicit wave-function) of the part of the nuclear wave function describing rotations of the inter-nuclear axis in the lab-frame. Which quantum numbers govern these rotations? What are their allowed ranges? [max 4 lines] (d) (2 points) How can we approximately write the total energy of the molecule taking into account all three degrees of freedom discussed in (a-c). Define your variables. [max 4 lines] (e) (2 points) Can a homonuclear, di-atomic molecule undergo purely ro-vibrational transitions in the dipole approximation? If yes, write the selection rules. If not, explain why not. [max 4 lines] (f) (2 points) Describe the rules for the probability of vibrational state changes from ν to ν accompanying an electronic transition. Give a formula and illustrate it with a separate diagram similar to the one in (a). [max 8 lines]
PHYS 402, Atomic and Molecular Physics - Page 4 of 6 (a) See figure on page 73 of lecture notes. (b) We taylor expand the BO-surface around its minimum to obtain a parabolic potential. Solutions are then identical to those of the simple harmonic oscillator, except they are now centered around the equilibrium bond length R 0. This works well for low vibrational quantum numbers, but breaks down for high ones, since we reach the nonparabolic part of the BO-surface. (c) Y KmK (θ, ϕ), where the Y are spherical harmonics. K is the rotational angular momentum quantum number, and m K the one for its z-component. K is not limited mathematically, but for too large K molecules dissociate. However m K K, in analogy to m l l for atomic electronic states. (d) E = E s (R 0 ) + E vib + E rot, with E s (R 0 ) the electronic energy at the equilibirum bond length, E vib = ω(ν +1/2), where ω = d 2 E s (R 0 )/dr 2 R=R0 /µ, µ is the reduced mass, E rot = 2 /(2µR0 2 )K(K + 1). (e) No, because for symmetry reasons it neither has a permanent dipole moment, nor can it get a dipole moment under changes of inter nuclear distance R. (f) These are governed by the Frank-Condon factor, F CFνν ss = 0 Fν s (R)F s ν (R)dR, i.e. the overlapp of the vibrational state ν on the initial electronic surface s with the final one ν on the final surface s. See sketch on page 80 of lecture notes. 5. Atomic field ionisation: Consider the electron of hydrogen with the addition of a strong static electric field F = Eˆk. You can work in atomic units for simplicity, if you like. (a) (2 points) Write down and then sketch the potential energy V (z) of the electron along the z-axis (this axis passes through the nucleus and is oriented along the electric field direction). (b) (3 points) Your sketch in (a) should show a local maximum. Determine the location z of that local maximum and the energy there V (z ). (c) (2 points) Suppose an electron initially is in a Hydrogen eigenstate with principal quantum number n. From which critical field strength E crit would you expect this electron to ionize (leave the proton) if you now switch on the electric field. Base your answer only on the result of (b) and the usual unperturbed Hydrogen energies E n (0). Neglect tunneling. (d) (2 points) Assume the most strongly Stark shifted states for a given n have an energy shift E = ± 3 2 n(n 1)ea 0E. Estimate the critical field strength E crit for ionization from this expression [Hint: You can assume a state will ionize if it is shifted to positive energies (=unbound state)]. Compare with the result from (c) and discuss similarities and why there are differences. (e) (3 points) Field ionisation is also the basis for high-harmonic generation of femtosecond laser pulses. Briefly describe the three step model of high-harmonic generation in intense laser fields, using diagrams as in (a) [max 6 lines].
PHYS 402, Atomic and Molecular Physics - Page 5 of 6 (a) V (z) = 1/ z Ez. See first figure on page 86 of lecture notes. (b) dv (z )/dz = 0, which gives z = 1/ E. Thus V (z ) = 2 E. (c) We guess the electron will ionize for sure, if its energy is higher than this maximum. Thus 1 2n 2 = 2 E crit, E crit = 1/(16n 4 ). (d) We solve 1 + 3 2n 2 2 n(n 1)E = 0, which gives E crit = 1/(3n 3 (n 1)), which gives the same scaling for large n as in (c), but different numerical prefactors. The latter is expected since perturbation theory is not really valid anymore in this case. (e) See description and figures on page 86 of lecture notes. 6. (6 points) How many non-degenerate energy levels of the Hydrogen electron with n = 3, l = 2 and s = 1/2 are there in a strong magnetic field (splitting large compared to the fine-structure)? From which formula did you get this? [max 1 line]. (a) This is the normal Zeeman effect regime, with E = B 0 µ B (m l + 2m 2 ). Since 2 m l 2 and m s = 1/2, 1/2, we can assemble 7 different energies, from 3B 0 µ B to 3B 0 µ B in steps of B 0 µ B. However when checking the assignment of (m l, m s ) to these seven levels, we find that only 4 are non-degenerate (belong to a single (m l, m s ) pair). 7. (6 points) What is the shape and the width of a natural spectral-line corresponding to an atomic ground to excited state transition? How is this shape called? Write an equation and define all your variables in it. Discuss the relation to the finite life-time τ of the upper level involved in the transition? [max 6 lines]. (a) c(ω) 2 1 (ω ω 0 ) 2 +(Γ/2) 2, the Lorentzian line shape. ω 0 is the transition frequency and Γ the decay rate. The decay rate Γ = 1/τ is related to the life-time. 8. (6 points) Describe what is meant by Born-Oppenheimer approximation in molecular physics, why it is usually justified, and under which circumstances it may fail. Discuss the difference between Born-Oppenheimer approximation and Born-Oppenheimer separation [max 8 lines].
PHYS 402, Atomic and Molecular Physics - Page 6 of 6 (a) We first solve the electronic SE Ĥel(R) φ q (R) = U q (R) φ q (R), giving us the BOsurface U q (R). In the BO-approximation we then assume nucleii move on one BOsurface s only: [ 2 2µ 2 + E s (R)]F s (R) = EF s (R). This is usually justified because electrons are much lighter than nuclei, but anyway fails whenever two surfaces become degenerate E s (R) = E s(r). BO-separation, in contrast, is an exact rewriting of the nuclear SE taking into account all electronic states and non-adiabatic coupling terms. 9. (6 points) Describe qualitatively how a covalent molecular bond arises. Give one example of a covalently bonding orbital in the H 2 + molecular ion. Write the equation for this orbital and define all your variables [max 8 lines]. (a) It arises because in the bonding orbital the electron(s) are more likely to be found in between the positive nuclear charges, thus resulting in overall attraction. For example we can approximately write φ g = [φ 1s (r A ) + φ 1s (r B )]/ 2, for the ground state orbital in H 2 +. See Eq. (4.9) for the definition of variables. 10. (6 points) Write the Bose-Einstein distribution function. Define all your variables. For which state does it diverge? Explain Bose-Einstein condensation qualitatively. [max 8 lines]. [max 8 lines]. (a) See equation (4.35) of the lecture and discussion around. In Bose-Einstein condensate the ground-state becomes macroscopically occupied N 0 /N 1, since n i N, when summing over all other states than the ground state. 11. (6 points) Roughly describe the operation principle of modern atomic clocks [max 6 lines]. (a) They measure the transition frequency between hyperfine split ground states of ultracold alkali atoms, which is in the microwave range, see lecture p. 91.