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Download CHEM 442 Lecture 15 Problems (see reverse) 15
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CHEM 442 Lecture 15 Problems (see reverse) 15-1. Suggest a perturbation operator for a z-polarized photon with electric field amplitude E and angular frequency w . ¶Y 0 , where ¶t be the zeroth-order Hamiltonian with 15-2. Consider the time-dependent Schrödinger equation ĤY 0 = i subscript “0” means the ground state. Let Ĥ (0) { -iE known eigenfunctions Y (0) e k k (0) t/ } and eigenvalues { E } . Let Ĥ (0) k (1) be the time- dependent perturbation of 15-1 and, therefore, Ĥ = Ĥ (0) + l Ĥ (1) with l = 1. We furthermore expand the wave function in an infinite series as -iE0( 0)t/ + lY(1) +… . Substituting these in the time-dependent Schrödinger 0 ¶Y 0 equation ĤY 0 = i and collecting those terms that are proportional to l , find the ¶t Y0 = Y(0) e 0 first-order perturbation equation. What are the unknowns in this equation? ¥ () 15-3. Expand the first-order correction to the wave function as Y (1) = å Ck t Y (0) e- iEk 0 k k=0 (0) t/ and substitute it into the first-order perturbation equation derived in 15-2. Multiply the equation by Y (0)* from the left and integrate the result over the whole space. Simplify the n integrals using orthogonality, òY (0)* k òY (0)* k Y(0) l dt = 0 (k ¹ l) , and normalization, (1) Y(0) . Show the k dt = 1 . Substitute the perturbation operator of 15-1 into Ĥ energy conservation condition between the system and a photon: En(0) - E0(0) = ± w . 15-4. On the basis of 15-3, derive Fermi’s golden rule in the form of 2 (0) 2 wn¬0 µ ò Y (0)* n ẑY 0 dx E , where wn¬0 is the probability of transition from the state 0 to state n. What is the name of the integral? 15-5. Consider the particle in a box along x axis, Y (0) n = ò L 0 2 np sin x . Evaluate L L (0) with n = 2, 3, and 4. Discuss optical transitions between state 1 to state n Y (0)* n xY1 dx on this basis. Use ò p 0 x sin x sin nx dx = - { 2n 1+ ( -1) (n 2 - 1) 2 n } where n is an integer greater than 1. 15-6. Using the recursion relationship of the Hermite polynomials, show that a Nn N xY n = Y n+1 + na n Y n-1 , where Y n is the nth harmonic-oscillator wave 2 N n+1 N n-1 function (n = 0, 1, 2, …), N n is the corresponding normalization coefficient, and a =( 2 / mk ) . 1/4 15-7. Consider the harmonic oscillator wave functions along x axis, Y (0) . Evaluate n ò ¥ -¥ (0) with n = 1, 2, and 3. Discuss optical transitions between state 0 to state Y (0)* n xY 0 dx n on this basis. Use 15-6. 15-8. In the spherical coordinates, x = rsinq cosj , y = rsinq sinj , z = r cosq . Some loworder spherical harmonics are Y0,0 = p 2p 0 0 ò ò Y1,0* xY0,0 sinq dq dj , p 2p 0 0 ò ò 1 3 , Y1,0 = cosq . Evaluate 4p 4p Y1,0* yY0,0 sinq dq dj , and p 2p 0 0 ò ò Y1,0* zY0,0 sinq dq dj .