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I. Effect of a δ-Function at Q = 0 on the Energy Levels of a Harmonic Oscillator

Live Exam Help A.

B.

C.

Without doing any calculation, which energy levels are unaffected by the –αδ(Q) term?

Without doing any calculation, are the energy levels that are affected by the –αδ(Q) term shifted up or down?

Without doing any calculation, among the levels that are affected by the –αδ(Q) term, is the magnitude of the energy shift larger or smaller for a low-v vs. a high-v level?

Live Exam Help D.

Based on the v-dependent magnitude of ψv(Q) at Q = 0 for the evenv states, justify your answer to part C. There are two ways to justify your answer to part C: (1) using perturbation theory, or (2) by adjusting the phase of the energy-shifted ψv(Q) at the turning points [Q±, where Ev = V(Q±)] so that ψ(± ∞) = 0.

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Your justification of your answer to part C should be based on how the magnitude of the discontinuity in d at Q = 0 affects the size of dQ the energy level shift relative to the energy of the vth harmonic oscillator level, e(v + 1/2).

E. Derive the equation in part D for the discontinuity of integrating the Schrödinger equation

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II. Derivation of One Part of the Angular Momentum Commutation Rule

Show that AˆBˆ,Cˆ  AˆBˆ,Cˆ  Aˆ,Cˆ Bˆ .

A.

Show [AB,C] = A[B,C] + [A,C]B A[B,C] = ABC – ACB [A,C]B = ACB – CAB A[B,C] + [A,C]B = ABC – CAB [AB,C] = ABC – CAB Q.E.D.

Live Exam Help B.

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III. Some Perturbation Theory All electronic properties of a molecule are parametrically dependent on the displacement coordinate, Q. This is part of the Born-Oppenheimer Approximation. We are interested in how the Q-dependence of the generic “A” property is encoded in the EvJ energy levels.

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Answer to Problem III (continued)

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IV. IR Spectroscopy Under a Deadline You have a contract with the Army Research Office (ARO) to determine the equilibrium bond length (re), vibrational frequency (ωe), and electric dipole moment (μe1) of the electronic ground state of TAt (tritium astatide). Your contract terminates tomorrow and you must write a final report today. Last night, on your desperate final attempt to record the vibrationrotation spectrum of TAt in an electric field of 100,000 Volts/cm, you obtained a spectrum unlike any you had observed previously. You suspect that this spectrum is that of the TAt v = 1 ← v = 0 transition, but you have no additional scheduled experimental time on the hyper-IPECAC facility, which is the only Astatine source (210At85 has a half life of 8.3 hours) in the world that is capable of generating the At flux needed for your experiment. Therefore you must write your final report to ARO without doing any further experiments to verify whether your spectrum is that of TAt or some other molecule. The likely other molecules include At2, T2, HAt, DAt, HT, and DT (you may ignore all other possibilities here). Your continued funding by ARO depends on the timely submittal of your report, but your career depends on its correctness.

One of your research assistants has provided you with the following possibly useful information:

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Some useful conversion formulas (Be is in cm–1 , μ is in atomic mass units, and Re is in Å, and 1 cm = 108 Å):

In the absence of an electric field, the vibrational rotational energy is given by:

Live Exam Help Before analyzing your spectrum and writing your report to ARO, it would be a good idea to make some predictions about the spectroscopic properties of TAt. PLEASE NOTE: Many parts of this question can be answered even if you are unable to answer an earlier part.

A.

Use the properties of related atoms and molecules to estimate Re and ωe for TAt. Specify the basis for the relationships that you are exploiting.

Live Exam Help B.

Compute Be from your estimated re. Let αe ≈ 0 and ωexe ≈ 0 and calculate the frequencies (in cm–1 ) of the 3 lowest-J transitions in the P branch and in the R branch of the v = 1 ← v = 0 rotation-vibration band. The P(J) line is the J – 1 ← J transition and the R(J) line is the J + 1 ← J transition. The lowest possible J-value in a 1 ∑ state is J = 0.

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C.

Compute Be from your estimated re. Let αe ≈ 0 and ωexe ≈ 0 and calculate the frequencies (in cm–1) of the 3 lowest-J transitions in the P and R branches of the v = 1 ← v

Live Exam Help You identify the following lines from the spectrum (all in cm-1 ): 1058.66, 1063.32, 1067.92, 1072.46, 1076.93, 1081.35, 1085.71, 1090.01, 1094.25, 1098.43, 1102.55, 1106.61, 1110.60, 1114.54, 1118.42, 1122.24, 1129.70, 1133.34, 1136.92, 1140.44, 1143.89, 1147.29, 1150.63, 1153.91, 1157.13, 1160.29, 1163.39, 1166.43, 1169.40, 1172.32, 1175.18, 1177.98

D.

Assign a few lines of the rotation-vibration spectrum. Two or three lines each in the R and P branches will be sufficient. Assume αe≈ 0 and ωexe≈ 0 and use your assigned lines to determine ωe and Be. Could this be TAt?

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E.

Which of the molecules At2, T2 , HAt, DAt, HT, and DT are expected to have electric dipole allowed rotation-vibration spectra? If you are undecided about HT and DT, state your reasons for and against.

HAt, DAt will definitely have a strong electric dipole allowed transition.

At2 and T2 definitely will not have any dipole allowed transition.

HT and DT will have a very weak dipole allowed transition, but it will have very large e and Be constants.

The center of electron charge will not quite coincide with the center of mass. The molecule rotates about the center of mass. There will be a small rotating electric dipole.

F.

What is the the minimum necessary spectroscopic information that could be useful in showing that your observed spectrum is not due to any of the molecules from part E that have an allowed rotation- vibration spectrum? Could your spectrum be due to any of the other likely candidate molecules?

The rotational and vibrational constants for the observed transition are too small for the molecule to be HAt or DAt. This is a huge effect.

The rotational and vibrational constants for HT and DT are vastly too large for a plausible assignment of the observed spectrum to HT or DT. The Stark effect will also be very, very small for HT and DT. The dipole moment for the putative TAt spectrum will be small, much smaller due to that for TI or DI. The small µe will be an excellent confirmation of the TAt assignment.

Live Exam Help For Question IV.G: The v = 1 ←v = 0 spectrum consists of a series of absorption lines following the selection rule ΔJ = ±1 (R and P branches). In the absence of an external electric field, all 2J + 1 MJ components of each J-level are exactly degenerate and the spectrum consists of simple R and P “lines”. When a 105 V/cm electric field is applied, a new term is added to the Hamiltonian: Stark

H!

µ.

If this field lies along the laboratory Z-direction, the MJ-degeneracy is lifted. The only non-zero integrals involving the Stark-effect Hamiltonian are

where f is a constant, the value of which depends on the units used. If µel is in Debye (D), z is in Volts/cm, and HStark is desired in cm–1, the conversion factor is f = l.6794  10–5 JM ;J1,M [(V/cm)D]–1.

At E = 105 V/cm, the lines at 1129.70 and 1122.24 cm–1 each split into two components separated by 9.0 ×10–3 cm–1. The lines at 1133.34 and 1118.42 cm–1 broaden slightly, but no splitting is resolvable. The electric field has no perceptible effect on all of the remaining lines. See next page for Question IV.G.

Live Exam Help G.

Calculate the Stark splitting for a generic diatomic molecule in J = 1 of a 1 ∑+ electronic state. The MJ= 0 component is pushed down by J = 2, MJ= 0 and pushed up by J = 0, MJ = 0. The MJ= +1 and MJ = –1 levels are both shifted downward by the same amount by their interaction with J = 2, MJ= 1, and MJ = – 1, but there exist no J = 0, MJ= ±1 levels to push these J = 2, JM = ±1 levels up. Use second-order perturbation theory to express the energy shifts in terms of μeland Be (specifically, µel 2 B times some Jdependent factors).

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H.

Interpret the observed Stark effect and use it to estimate μel.

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From (x,t) to (t) (x,t )  c1 (t )1 (x)  c2 (t )2 (x)  c3 (t )3 (x)  c4 (t )4 (x)

where {ψn} are eigenfunctions of a time-independent H(0). En is the eigen-energy associated with the ψn eigenfunction

A. (i)

(ii)

Live Exam Help B.

C. If Ψ(x,t) is normalized to 1, what combination of cn { (t)} and cn * { (t)} must be equal to 1?

D.

(i)

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(ii)

E. At t = 0, Ψ(x,0) = |1 〉 . (i) Write an expression for Ψ(x,t).

(ii)

Write an expression for ρ(t).

Live Exam Help (iii)

Is ρ time-dependent when the system is in a single energy eigenstate?

F. (3 points) Suppose we apply a pulse that terminates at t = 0. This pulse results in a flip angle of π/2 at ω12. Then Ψ(x,0) = 2–1/2 |1 〉 + 2–1/2 |2 〉 . (i) Give an expression for Ψ(x,t)

Live Exam Help G. (8 points) When Ψ(x,t) involves a superposition of energy eigenstates: (i)

(2 points) Are the population terms (diagonal elements) of ρ time-dependent? The diagonal terms are independent of time.

(ii) (3 points) Are the coherence terms (off-diagonal elements) of ρ time-dependent? The off-diagonal (coherence) terms are time-dependent. (iii) (3 points) If a ρij term is time-dependent, at what frequency does it oscillate? If ρij is time-dependent, it oscillates at ωij = (Ei − Ej) ! .

VI. Semi-classical Calculation of (35 points) Vibrational Overlap Integrals in the Diabatic Representation

Live Exam Help Diabatic potential energy curves can cross. Near-degenerate vibrational states of two crossing diabatic curves, V1(R) and V2(R), interact with each other with an interaction matrix element e1,v1 Hel (R) e2 ,v2 = e1 Hel (Rc ) e2 v1 v2 Where e1 Hel (Rc ) e2 ≡ H12 el (Rc ) and Rc is the internuclear distance at which V1(R) intersects V2(R). For this problem, V1 and V2 are both harmonic and both have the same value of ωe = 200 cm–1

Rc is chosen so that v1 = 7 is near degenerate with v2 = 2 and v1 = 12 is near degenerate with v2 = 7.

The stationary phase point, Rsp, is the value of R at which the classical mechanical momentum on V1 is the same as that on V2.

A. (5 points) What is the relationship between Rsp nd Rc for the v1 = 7, v2 = 2 pair of levels and for the v1 = 12, v2 = 7 pair of levels?

Rsp (v1 = 7, v2 = 2) = Rsp (v1 = 12, v2 = 7) =

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The stationary phase point for two near-degenerate levels is the curve-crossing point, Rc. Rsp = Rc for both (v1 = 7 and v2 = 2) and (v1 = 12 and v2 = 7).

B. (8 points) (i) (5 points) What is the distance between nodes on either side of Rc at v1 = 7 and at v2 = 2? Use the deBroglie relationship between λ(R) and the classical mechanical momentum, pv(R). Express pv(R) in terms of (Ev −V (Rsp )) and µ, the reduced mass.

(ii) (3 points) Is this node-spacing the same for v1 = 12 and v2 = 7? For near-degenerate vibrational levels, e.g. v1 = 7 and v2 = 2, [Ev – V(Rc)] is the same, so the node to node distance is the same.

C. (5 points) For a harmonic oscillator with ωe = 200 cm–1 , what is the vibrational level-independent oscillation period?

Express your answer in symbols (ωe, h, c, etc.).

Live Exam Help D.

(i) What is the probability (expressed in terms of ωe, h, and pv(R)) of finding the classical oscillator with momentum p > 0 between the Rc-centered pair of nodes for v1 = 7 on V1? [HINT: use semi-classical expressions for wavelength and velocity.]

(ii) Is this probability different from that for v2 = 2 on V2?

Live Exam Help E. Estimate the ratio

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USEFUL CONSTANTS and FORMULAS

Particle in a box

Harmonic Oscillator

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Raising and lowering operators

Live Exam Help Hydrogen atom Three-dimensional operators in spherical coordinates

Radial integrals

H atom spatial wavefunctions (where σ = Zr/a0. In atomic units a0 = 1 and σ = Zr.)

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Perturbation Theory

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Spin operators

Turning points of V(x):