Chemistry Homework Assignment for CHEM 5581: Variational Theory and Perturbation Theory, Assignments of Quantum Chemistry

Download Chemistry Homework Assignment for CHEM 5581: Variational Theory and Perturbation Theory and more Assignments Quantum Chemistry in PDF only on Docsity! Homework Assignment for CHEM 5581 J. M. Weber Office: JILA A709; Phone 303-492-7841; email: weberjm@jila.colorado.edu ________________________________________________________________________ 1 Problem Set 9 To be returned before class on Wednesday, November 5, 2008 (give to me in person or slide under office door). 1. Variational Theory for the Harmonic Oscillator Use the ground state wave function of a particle in a box ⎟ ⎠ ⎞ ⎜ ⎝ ⎛= a x atrial πϕ cos2 , |x| ≤ a/2 as trial wave function to approximate the ground state of a harmonic oscillator given by 22021ˆˆ xmTH ω+= using the size of the box as an adjustable parameter. a.) Calculate <T>trial (3 credits) b.) Calculate <V>trial (3 credits) c.) Sketch <T>trial, <V>trial, and the sum <E>trial as a function of the parameter a, and interpret. (4 credits) d.) Minimize <E>trial, compare with the exact solution, and check if the result is meaningful in the context of variational theory. (4 credits) Homework Assignment for CHEM 5581 J. M. Weber Office: JILA A709; Phone 303-492-7841; email: weberjm@jila.colorado.edu ________________________________________________________________________ 2 2. Perturbation Theory Part I: The Particle in a Sloped Box Given is a particle in a box with sloped bottom, where the potential energy is given by x a xV v)( Δ= for |x| ≤ a/2 V (x) = ∞ for |x| > a/2 a.) To literally make yourself a picture of the system, plot or sketch the potential energy as a function of the position for Δv/a > 0. (1 credit) b.) Write down the Hamiltonian and partition it in a suitable way to get 0Ĥ and 'Ĥ . Calculate En(0) and En(1). (5 credits) c.) Calculate 'knH for k≠n. Hint: to make your life easier, shift the box by a/2. (6 credits) d.) With the result from (c), calculate approximate values for E(2) for the ground state and the 1st excited state, setting Δv/E1(0) = 2 and expressing all energies in units of E1(0) (truncate the sum to 3 terms). Should the energy correction depend on whether the box is sloped from left to right or from right to left? Does it in your result? (6 credits)