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Quantum Chemistry: Our Agenda • Postulates in quantum mechanics (Ch. 3) • Schrödinger equation (Ch. 2) • Simple examples of V(r) Particle in a box (Ch. 4-5) Harmonic oscillator (vibration) (Ch. 7-8) Particle on a ring or a sphere (rotation) (Ch. 7-8) Hydrogen atom (one-electron atom) (Ch. 9) • Extension to chemical systems Many-electron atoms (Ch. 10-11) Diatomic molecules (Ch. 12-13) Polyatomic molecules (Ch. 14) Computational chemistry (Ch. 16) Lecture 3. Simple System 1. Particle in a Box References • Engel, Ch. 4-5 • Molecular Quantum Mechanics, Atkins & Friedman (4th ed. 2005), Ch. 2 • Introductory Quantum Mechanics, R. L. Liboff (4th ed, 2004), Ch. 4 • A Brief Review of Elementary Quantum Chemistry http://vergil.chemistry.gatech.edu/notes/quantrev/quantrev.html • Wikipedia (http://en.wikipedia.org): Search for Particle in a box Solving Schrödinger Equation – 1st Example. A Free (V = 0) Particle Moving in x (A Particle Not in a Box) H E 2 d 2 H 2m dx 2 2 d 2 E 2 2m dx pˆ x Solutions k 2 2 Ek 2m k Aeikx Be ikx eikx p x k A 2 2 e ikx p x k B 2 2 d i dx Aeikx d px i dx d k i dx px kh Aeikx d k i dx px kh The Uncertainty Principle When momentum is known precisely, the position cannot be predicted precisely, and vice versa. Ae ikx p x kh A 2 2 When the position is known precisely, Location becomes precise at the expense of uncertainty in the momentum Free Translation (V = 0) Confined within Boundaries: A Particle in a Box (Infinite Square Wall Potential) 2 d 2 V ( x) E 2 2m dx V 0 for 0 x L 0 and x V for x L m The same solution as the free particle but with different boundary condition. k C sin kx D cos kx A particle of mass m is confined between two walls but free inside. k 2 2 Ek 2m Applying boundary conditions k C sin kx D cos kx ( x 0) 0 ( x L) 0 D0 C sin kL 0 kL n n ( x) C sin( nx / L) n 1,2,3,... n cannot be zero. (quantum number) Normalization L 0 n 2 dx C 2 L 0 C2L 2 sin (nx / L)dx 1 2 1/ 2 2 n ( x) L sin( nx / L) n 1,2,3,... 1/ 2 2 C L n2h2 En 8mL2 Final Solution (Energy & Wave function) n2h2 En 8mL2 1/ 2 2 n ( x) L sin( nx / L) Rapidly changing Higher E node quantized zero-point energy n 1,2,3,... Energy, Wave function & Probability density node not constant over x Quantum (confinement) effect Particle in a Box: Classical vs. Quantum From Wikipedia (particle in a box) Classical Limit: Bohr’s Correspondence Principle n by increasing E (~ kT) or m or L What is the maximum value for n ? Case I: T = 300 K, m = me, L = 10 nm Case II: T = 300 K, m = 1 kg, L = 1 m Postulate 2 of Quantum Mechanics (measurement) • Once (r, t) is known, all observable properties of the system can be obtained by applying the corresponding operators (they exist!) to the wave function (r, t). • Observed in measurements are only the eigenvalues {an } which satisfy the eigenvalue equation. A a eigenvalue eigenfunction (Operator)(function) = (constant number)(the same function) (Operator corresponding to observable) = (value of observable) A Free Particle Moving along x, Two Cases A free particle not confined Aeikx pˆ x A free article confined in a box of size L d i dx d p x i dx the same function d Aeikx khAeikx kh i dx constant p x kh number eikx p x k + e ikx p x k - k Aeikx Be ikx It’s not a momentum operator’s eigenfuncition. The momentum is either or . (px ) The position x is partially known. (x L) It’s an eigenfunction of the momentum operator px Only a constant momentum px (eigenvalue) is measured. We know the momentum exactly. (px = 0) The position x is completely unknown. (x = ) x px h 0 Heisenberg’s uncertainty principle Alice, Bob, and Uncertainty Principle… Postulate 4 of Quantum Mechanics (average) • For a system in a state described by a normalized wave function , the average value of the observable corresponding to A A d is given by = <|A|> • For a special case when the wavefunction corresponds to an eigenstate, Position, Momentum and Energy of PIB momentum p Two independent quantum numbers