Download Quantum Mechanics Applied to Chemical Problems: Lecture 1 and more Quizzes Chemistry in PDF only on Docsity! Chem 131B: Lecture 1 01/05/2007 Professor Kieron Burke The course website is located at “http://eee.uci.edu/07w/40873/chemistry131b/”. On the website is a cheat sheet you can use on any in-class assignments. If there’s something you think should be included let us know. Print out copy of this cheat sheet to bring to class. You’ll need the second volume of Engel’s. We will have a quiz each Monday at 11:00am. They are worth 25% of the total grade and are ten questions each. Homeworks also are turned in on Monday. The final will consist of two parts, one covering the last third of the course, and the other covering the whole course comprehensively. Next week on Friday there will be two quizes reviewing material from Chem. 131A. This week’s homework is also posted on the site. Our subject is quantum mechanics applied to chemical problems. First two-thirds of the class covers quantum mechanics, the last third will address statistical mechanics. I. REVIEW A. The Photoelectric Effect When you shine light on a metal surface, with an electron detector handy, no signal is seen up to some binding energy, φ. Above this threshold, the kinetic energy of the electron is given by KE = hν − φ ν = c λ (1) A quantum mechanical objects has the properties of a particle and a wave. These are related by de Broglie’s equation λ = h p (2) where p is the momentum of the particle. The transition frequencies between energy levels of the hydrogen atom are given by ν = RH ( 1 n2 − 1 m2 ) RH = 13.6eV (3) where m and n are integers. RH is the Rydberg constant. One Hartree = 2RH . B. Schrodinger’s Equation The time evolution of a system described by the wavefunction Ψ(x) is h̄ ∂ ∂t Ψ(x) = −iĤΨ(x) (4) where Ĥ is the Hamiltonian operator. It may be written Ĥ = T̂ + V̂ (5) where T̂ and V̂ are the kinetic and potential energy operators respectively. T̂ may be written T̂ = p̂2 2m where p̂ = h̄ i ∂ ∂x (6) If the time and spatial evolution of the wavefunction can be factored, Ψ(x, t) = Ψ(x, 0) exp [−iE h̄ t ] (7) 2 the spatial wavefunction can be written as a solution to the stationary Schrodinger’s equation. ĤΨ(x) = EΨ(x) (8) The potential and the boundary conditions generate a set of solutions φn(x) to the Schrodinger equation. These solutions are usually chosen to have the property of orthonormality ∫ dxφ∗m(x)φn(x) = δmn (9) and completeness. Completeness implies that any function f(x) can be “expanded” in terms of the φi(x), ∀ f(x) ∃ {ci} such that f(x) = ∑ i ciφi(x) (10) The coefficients ci are given by ci = ∫ dxφ∗i (x)f(x) (11) C. Chapter 3 1. All information about a state is contained by its wavefunction. 2. Observables can be represented by operators. 3. The eigenvalues of these operators correspond to the possible results of a measurement. 4. The average (expectation) value of an observable (Â) is 〈Â〉 = ∫ Ψ∗(x)ÂΨ(x)dx D. Chapter 4 The particle in a box describes the wavefunction of a particle in a flat potential well surrounded by “walls” of infinitely high energy. It’s energy levels are described by En = n2h2 8ma2 (12) where a is the length of the box, m is the mass of the particle and h is Planck’s constant. The wavefunction solutions to Schrodinger’s equation for this system are φn(x) = √ 2 a sin (nπx a ) (13) In three dimensions the energy is given by Eijk = h2 8m ( i2 a2 + j2 b2 + k2 c2 ) (14) where a,b and c are the lengths of the box. E. Chapter 5 The commutation of two operators is defined as [Â, B̂] = ÂB̂ − B̂Â (15) Some operators have a nonzero commutator. [x̂, p̂] = ih̄ (16)
