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Unformatted text preview: Qhem 120A Final Exam Part I Consider a planar ring of N equally spaced tight binding sites. The location of the
nth site is speciﬁed by 15,, = nb, n = 1,2,... ,N. From periodicity of the ring, xN+1 = 301.
Let denote the nth tight binding state, i.e., the localized orbital located at :r x mm. Here, x measures the distance around the ring. X» X, X2 The orbitals, {45"}, form a complete set of single particle states in the tight binding model.
They are also an orthonormal set; i.e., 1, n=m <¢n¢m>=6nma {0, mm. The single electron Hamiltonian in the model is speciﬁed by = aénm + ﬁ<6n,m+1 + 6n,m—1) a or equivalently, = a¢m(x) + B¢m—l($) + ﬁ¢m+1($)
The parameter ﬂ is negative, i.e., ﬂ < O. The momentum operator, p, is speciﬁed by h
<¢nipl¢m) = ﬂ{6n,m+1 —‘ 5n,m——1} , or equivalently, P¢m($) = (h/2ib)l¢m+i($)  ¢m—1($)l  } is complete, any other wave function, 11), can be expressed as a linear w = ch¢n 1 Finally, because {dm combination of the an’s, 20 pts 1. Show that are the stationary single electron states. Here, I: is quantized according to To carry out the demo ' , stationary Schrodinger equation leads to
H W = EM , and that this equality is indeed satisﬁed for some certain choice of E. 10 1}. ts 2. What are the single electron eigenenergies in terms of a, ﬂ, N and b? '20 pts 3. (a) Show that p is Hermitian; and (b) Show that ibk is an eigen—function of the momentum p. (Use the same type of procedure as proposed for solving problem 1). 10 pts 4. Show that with state in, the expectation value of p is —(h/b) sin(kb). 5. (a) Is 1/1 k an eigenfunction of p? (b) Is 7,1)k an eigen—function of H ?
(c) Is 24 2 ts IbOH/Jk‘HD—k an eigen—function of p?
(d) Is 1/) oc wk + tb_k an eigen—function of H? and its uncertainty, Ap = ((p — (13))2 , for . . 2
6. Determine the expectation value of p, >1/ the state 1/) oc 1/)k + 7124;. ’24 9 ts ‘20 pts 7. For a ring with only three sites, i.e., N = 3, show that the three single electron energies are 31:021517E2=E3=0¢+lﬂl [Recall, cos(27r/3) = —1/2.] 20 pts 8. For two noninteracting electrons in this N = 3 ring, specify the normalized wave function of the ground state. Use the following notation: ¢n(i) ~— indicates electron i is at the nth tight binding site indicates electron i is in the spin up state 3+0) “
s electron i is in the spin down state. 3 _ —— indicate [0 ‘20 pts 9. Again for two non—interacting electrons in the N = 3 ring, using the notation of Problem 8, specify the spacial part of a wave function for a triplet state with energy 2Otlﬁl 24 pts 10. Suppose the N = 3 ring is oriented such that the single electron zcomponent of the dipole moment operator, pl , satisﬁes 1 1
<¢nluzl¢m> = H6nm(5n1 " 351,2 — 56,13) 2 Further, suppose there is only one electron in the ring. Determine the time dependent expectation value of )1, given that at time t = 0 its value is )1. [Recall that the time
dependent Schrodinger equation is H11) 2 ih6212/8t.] Part II Now let’s consider the mixed valence system with the Hamiltonian for a twostate system coupled to “ligand” ﬁelds, 5 :
1 2
H : HO —/J«0pg + where E K
H° = [—K E, l is the Hamiltonian for the two state system in the “le‘ftright’7 (i.e., the redox state) rep— 1 0
,uopzlu 0 _1 is the dipole operator for the two state system in that same representation. This model
ignores the kinetic energy for 8, so that 8 is simply a number (the local electric ﬁeld along the axis of the redox system), and it is not a quantal variable. Our motivation for (temporarily) not thinking about the quantum mechanics of 5 is the Born—Oppenheimer resent ation, and approximation. ‘20 pts 11. Show that the energies of the stationary states for this model are 1
E(£) = E0 + 5kg? :1: K2 + #252 20 pts 12. For the case Where K << pz/k and 5 = ip/k, what are the two corresponding
stationary state wave functions in terms of 1/13 and 2/»? Here we and 1,24 are the
normalized nonoverlapping left and right redox states, respectively. [Careful with your answers; they will depend upon the sign of 8 20 pts 13. Evaluate the appropriate matrix element(s) of [101, to show that electron transitions are allowed between the two stationary states considered in Question 12. Now let’s consider the quantum mechanics of 5, and let IVI denote its reduced or
effective mass. The low energy portions of the Born—Oppenheimer energies, parameterized
by 5, that you have determined in Problem 11 when K << [112/16 are drawn in the ﬁgure together with the lowest four energy levels associated with the dynamics of 5. 9(6) M 14. (a) Estimate AE in terms of the model parameters. (Recall the energy levels of a
harmonic oscillator are (1/2 + n)th where Lug = force constant/mass.)
(b) What type of dynamics (e.g., small amplitude vibrations, tunneling,
rotations, . . . ) gives rise to AE’?
(0) Will AE’ increase, decrease or remain approximately constant if M increases? ((1) The drawing shows AE” to be larger than AE’ . Is there a physical reason for this or could it be the other way around? Explain. 20 pts 15. Now assume AE << ,uz/k as well as K << p2/k, and consider electronic transitions
due to absorption of photons by the system when it is initally at its lowest possible
energy. Due to the dynamics of 8, there will be a band of such transitions. Employ the Franck—Condom principle to estimate the frequency (either 0.2 or 1/) at the center of that band. ...
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 Spring '10
 CHandler

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