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1. Physical Chemistry Exam Help
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2. I. Effect of a δ-Function at Q = 0
on the Energy Levels of a Harmonic Oscillator
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3. A. Without doing any calculation, which energy levels are unaffected by the –αδ(Q) term?
B. Without doing any calculation, are the energy levels that are affected by the –αδ(Q) term shifted up or down?
C. 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?
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4. 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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5. 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 sizeof
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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6. II. Derivation of One Part of the Angular
Momentum Commutation Rule
A. Show that AˆBˆ,Cˆ Aˆ
Bˆ,CˆAˆ,CˆB.̂
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.
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7. B.
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8. 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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9. Answer to Problem III (continued)
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10. 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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11. 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:
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12. 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.
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13. 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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14. 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
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15. 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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16. 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.
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17. 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:
H!
Stark
  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.
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18. 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
J-dependent factors).
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19. H. Interpret the observed Stark effect and use it to estimate μel.
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20. V. 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)
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21. 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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22. (ii)
E. At t = 0, Ψ(x,0) = |1〉.
(i) Write an expression for Ψ(x,t).
(ii) Write an expression for ρ(t).
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23. (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)
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24. 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
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25. 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?
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26. Rsp (v1 = 7, v2 = 2) = Rsp (v1 = 12, v2 = 7) =
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.).
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27. 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?
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28. E. Estimate the ratio
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29. USEFUL CONSTANTS and FORMULAS
Particle in a box
Harmonic Oscillator
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30. Raising and lowering operators
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31. 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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32. Perturbation Theory
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33. Spin operators
Turning points of V(x):
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