PH 253 / LeClair Spring 2013 Problem Set 7: Molecules Instructions: 1. Answer all questions below. Show your work for full credit. 2. All problems are due 16 April 2013 by the end of the day. 3. You may collaborate, but everyone must turn in their own work. 1. Variational Principle I. The energy of a system with wave function is given by E[ ] = R R H dV j j2 dV (1) where H is the energy operator. The variational principle is a method by which we guess a trial form for the wave function , with adjustable parameters, and minimize the resulting energy with respect to the adjustable parameters. This essentially chooses a best t” wave function based on our guess. Since the energy of the system with the correct wave function will always be minimum, our guess will always lead to an energy which is slightly too high, but the variational principle allows us to get as close as possible to the correct energy with our trial wave function. Pretend we don’t know the ground state wave function for hydrogen, but decided to guess the following form for : (r) = 2 + r2 (2) (a) Use the variational principle and normalization to nd the values of and that give the minimum energy for this trial wave function. Note that since the trial function is spherically symmetric, dV =4r2 dr and r2 = 1 r2 @ @r r2 @ @r . (b) Compare this result to the correct ground state energy of hydrogen and sketch/plot your best guess for with the correct ground state wave function. 2. Variational Principle II. The energy operator for a simple harmonic oscillator in one dimension is H = – h2 2m d2 dx2 + 12m!2×2 (3) Presume we don’t know the proper wave function, but guessed a wave function of the form (r) = 2 + x2 (4) (a) Use the variational principle and normalization to nd the values of and that give the minimum energy for this trial wave function. Since this is a one dimensional problem, take dV =dx. (b) Compare this result to the correct ground state energy of the simple harmonic oscillator and…