Molecular Spectra and Molecular Structure - 1977 PS1

Problem Set #1 1. Carry out the angular momentum algebra indicated � � in Section 2 (of Steinfeld, Chapter 1); that is, � � show explicitly that 11z |j x |11z = 0, and that J2x − �J x �2 � 0. 2. (a) Given the matrix elements of the coordinate x for a harmonic oscillator: � � ψ�v xψv� dx = 0 unless v� = v ± 1 �v|x|v � = and �v + 1|x|v� = (2β)−1/2 (v + 1)1/2 �v − 1|x|v� = (2β)−1/2 (v)1/2 where β = 4π2 mω/h, and ω is the vibrational frequency. Evaluate the non-zero matrix elements of x2 , x3 , and x4 ; that is, evaluate the integrals � r � ψv xr ψv� dx �v|x |v � = for r = 2, 3, and 4 (without actually doing the explicit integrals, of course!). (b) From the results of (a), evaluate the average values of x, x2 , x3 , and x4 in the vth vibrational state. Is it true that x2 = (x)2 , or that x4 = (x2 )2 ? What conclusions can you draw about the results of a measurement of x in the vth vibrational state? 3. Let �� � ��H11 0 0 ��� in the φi basis, and H = �� 0 d V �� �� � 0 V −d� �� � 0 0 �� ��1 � U = ��0 cos θ − sin θ�� �� � 0 sin θ cosθ � � �1/2 � �1/2 d d cos θ = 1/2 + sin θ = 1/2 − 2(d2 + V 2 )1/2 2(d2 + V 2 )1/2 (a) Calculate U† HU. Page 2 (b) List the three energy eigenvalues and associated eigenfunctions expressed as the linear combi­ nations of the φi basis functions. (c) Show that the eigenfunctions are normalized and mutually orthogonal. (d) Suppose that µ is the electric dipole transition moment operator, and that its matrix represen­ tation in the φi basis is �� � �� 0 µ12 µ13 ��� 0 �� . µ = ��µ12 0 �� � µ13 0 0� Express µ in the energy eigenbasis. (e) What are the transition probabilities between energy eigenstates, expressed in terms of µ12 , µ13 , V, d, and H11 ? (f) There are several interesting limiting cases. Suppose φ2 and φ3 are “unperturbed” states which have equal transition probabilities, |µ12 |2 = |µ13 |2 , from φ1 . V is a perturbation term which causes φ2 and φ3 to mix. The U23 and U32 elements are the mixing coefficients. (i) Let |d/V| � 1. The perturbation is turned off. What are the ψ1 → ψ2 and ψ1 → ψ3 transition probabilities? (ii) Let |d/V| � 1. The perturbation is on full strength. What are the ψ1 → ψ2 and ψ1 → ψ3 transition probabilities? Note that one transition has accidentally become forbidden, even though the final state is composed of equal parts of two basis functions, each of which have allowed transitions from φ1 = ψ1 . This is a quantum interference effect that is often observed experimentally. (iii) Let d > 0, V > 0, and µ12 = µ13 . Which transition is allowed, the one to the upper or lower of the mixed levels? (iv) Note that if µ12 = −µ13 , the opposite transition becomes allowed. What happens when the sign of V is switched? What happens when the sign of d changes? Answer in terms of transitions to upper vs. lower eigenstates rather than ψ2 or ψ3 . (v) Comment on when changing the sign of an off-diagonal matrix element might be expected to have observable consequences. 4. Several methods for computer diagonalization of large matrices are discussed in the handout “Notes on matrix Methods” pages 12–18. Unfortunately there are several typographical errors. (a) What is the error in Equation 3.10? (b) The discussion of Method IIB is rather confused. Describe the procedure for construction of a tridiagonal matrix. Which matrix element is annihilated first? How does the rotation differ from that performed in Method IIA? Why is it possible to arrange for annihilation of various elements without spoiling any of the previously zeroed off-diagonal elements? Why is it not possible to reduce the matrix beyond tridiagonal form? (c) Suppose you have a 5 × 5 tridiagonal matrix and apply Method IIA to annihilate the M34 and M43 elements. What happens to the elements outside of the 2 × 2 (3, 4) block?