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1. An important continuous probability distribution is the exponential distribution p(x)dix-ce-dridx, 0 x x < 00 Evaluate c, (x), Ox and the probability that x a. 2. Assume that the wavefunction V(x,1) describes the state of a particle under the influence of a scalar potential, V(x). In this case, V(x,2) is a wave packet, or a superposition of eigenfunctions. (x)(c) and (p)(1) represent the position and momentum of the center of the wavepacket at an instant in time, t. Please use the commutation relations between A. &. and p to develop the equations that describe the temporal evolution of (x)(t) and (p)(t), ic. ove and a(p) at at 3. In class, we looked at the free-electron model for a linear conjugated molecule. The 7t-orbitals of benzene may be approximated using the wavefunctions and energies of a particle on a ring. We're going to address the particle on a ring problem and then extend it to benzene. a. Suppose that a particle of mass m is constrained to move on a circle of radius r in the x-y plane. Assume that the particle's potential energy is zero. Write down the Schrödinger equation in in Cartesian coordinates b. Transform this Schrödinger equation into polar coordinates where x - rcoso and y - c. Taking r to be constant, write down the general solution '(()) to this Schrödinger equation (Note: by transforming to polar coordinates and assuming , is constant, we have reduced this 2D problem to ID). d. Write down the final expression for the normalized wavefunction and the quantized energies for the particle on a ring. e. Treat the six z-electrons of benzene as particles free to move on a ring of radius 1.42 A and calculate the wavelength of the lowest electronic transition. 4. In class we defined a Hermitian operator o as one which satisfies the following equality: - s(tu)) Vdx. a. Prove that this is equivalent to the statement that Hermitian operators have real eigenvalues. b. Prove that an equivalent definition for a Hermitian operator is given by: I r*DDdx - dx, where both f and g are arbitrary functions. In other words, if the first equality is satisfied, the second one is also satisfied. [Hint: Set y - f + cg for an arbitrary constant c, and plug into the first expression. Evaluate for c - 1 and c - i.] 5. Let - + 0.992(a + C3P2(x). a. is normalized and the Qi are orthonormal, calculate C3 (assuming C3 is real). b. Assuming that Rep, - jpj, plot the probability distribution function of the outcomes of a measurement associated with 2. c. What is ()?

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