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The questions for an exam in quantum physics of atoms and molecules (phy-30009) at keele university, 2010/11. The exam covers topics such as particle in a ring potential, x-ray emission and absorption spectra, and energy levels of atoms. Candidates are required to answer three questions. Detailed instructions and calculations for each question.
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Level III
Tuesday 10th^ May 2011, 16.00-18.
PHYSICS/ASTROPHYSICS
PHY-
QUANTUM PHYSICS OF ATOMS AND MOLECULES
Candidates should attempt to answer THREE questions.
ψ(θ) =
√√ √√ 1 2 π e
inθ
where n = 0, ± 1 , ± 2 ,... is any integer. The energies of the states are En = n
(^2) ¯h 2 2 mR^2. (a) At time t = 0 the particle has equal probability of being at all positions within one half of the ring. i. Show that the wavefunction at time t = 0 can be written as
Ψ(θ, 0) =
√^1 π 0 ≤^ θ < π 0 π ≤ θ < 2 π [20] ii. Calculate the probability of observing the particle in the ground state at time t > 0. [40] (b) State and explain the value of the degeneracy of the ground state and first excited state in the case of an electron confined to a ring. [20] (c) State and explain the boundary conditions that apply to the energy eigenfunctions in this case. [20]
/Cont’d
(a) Sketch the energy levels for the electrons in both the 6s^2 and 6s^1 5d^1 states using spectroscopic notation to label each state. Your sketch should indicate the correct ordering of the energy levels according to Hund’s rules and the relative spacing of dif- ferent levels. You do not need to provide an energy scale for your sketch. [25] (b) Explain why the 6s^1 5d^1 state has a higher energy than the 6s^2 ground state despite one electron having a lower principal quan- tum number. [25] (c) State the transition rules that govern allowed transitions from the 6s^1 5d^1 state. Hence, list the transitions that are not for- bidden from the 6s^1 5d^1 state to the 3 F 2 , 3 , 4 triplet of the 6s^1 5f^1 state [20] (d) For each of the 6s^1 5d^1 energy levels, calculate the number of components into which the level is split by a weak magnetic field and state whether the splitting is due to the normal or anomalous Zeeman effect. [15] (e) It is observed that for states with the same total spin angular momentum but different values of the total orbital angular mo- mentum, LT^ , states with larger LT^ values have lower energy. Give a simple physical explanation for this observation. [15]
/Cont’d
EJ = ¯h
2 2 I J(J^ + 1),^ J^ = 0,^1 ,^2 ,... , where I = (^) mm^1 m^2 1 +^ m 2
a^2 is the moment of inertia for masses m 1 and m 2 separated by a dis- tance a.
(a) Sketch an energy level diagram for the vibration states v = 0 (ground state) and v = 1 and the rotation states J = 0, 1 , 2. Use your sketch to indicate approximately the relative spacing of different energy levels and the allowed transitions between the v = 0 and v = 1 states for a molecule such as HCl. [20] (b) Sketch the absorption spectrum resulting from the transitions in part (a). Indicate the relevant quantum numbers for the initial and final state of the transitions where possible. [20] (c) Describe and explain how the spectrum in part (b) changes when the temperature of the gas is reduced. [20] (d) The bond length in HCl is 0.127 nm. Estimate the frequency spacing between adjacent lines in the spectrum in part (b).[20] (e) The HCl molecule is a non-rigid rotator. Describe and explain how this affects the appearance of the spectrum in part (b). [20]
/Cont’d
