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Lecture 2. Postulates in Quantum Mechanics • • • • Engel, Ch. 2-3 Ratner & Schatz, Ch. 2 Molecular Quantum Mechanics, Atkins & Friedman (4th ed. 2005), Ch. 1 Introductory Quantum Mechanics, R. L. Liboff (4th ed, 2004), Ch. 3 • A Brief Review of Elementary Quantum Chemistry http://vergil.chemistry.gatech.edu/notes/quantrev/quantrev.html • Wikipedia (http://en.wikipedia.org): Search for Wave function Measurement in quantum mechanics Schrodinger equation Postulate 1 of Quantum Mechanics (wave function) • The state of a quantum mechanical system is completely specified by the wavefunction or state function Ψ (r, t ) that depends on the coordinates of the particle(s) and on time. • The probability to find the particle in the volume element d drdt located at r at time t is given by (r , t ) (r , t )d. (Born interpretation) • The wavefunction must be single-valued, continuous, finite, and normalized (the probability of find it somewhere is 1). d (r , t ) 2 1 = <|> probability density (1-dim) Postulate 1 of Quantum Mechanics (wave function) • The state of a quantum mechanical system is completely specified by the wavefunction or state function Ψ (r, t ) that depends on the coordinates of the particle(s) and on time. • The probability to find the particle in the volume element d drdt located at r at time t is given by (r , t ) (r , t )d. (Born interpretation) • The wavefunction must be single-valued, continuous, finite (not infinite over a finite range), and normalized (the probability of find it somewhere is 1). d (r , t ) 2 1 = <|> probability density (1-dim) Born Interpretation of the Wave Function: Probability Density over finite range Engel, 2nd Ed. p.40, last bullet “The wave function cannot have an infinite amplitude over a finite interval.” infinite over zero range Postulates 2-3 of Quantum Mechanics (operator) • Once Ψ (r, t ) is known, all properties of the system can be obtained by applying the corresponding operators to the wavefunction. • Observed in measurements are only the eigenvalues a which satisfy the eigenvalue equation A a eigenvalue eigenfunction (Operator)(function) = (constant number)(the same function) (Operator corresponding to observable) = (value of observable) Physical Observables & Their Corresponding Operators Observables, Operators, and Solving Eigenvalue Equations: An example Aeikx pˆ x d i dx d p x i dx the same function d Aeikx khAeikx kh i dx constant p x kh number 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 The Schrödinger Equation Hamiltonian operator energy & wavefunction (solving a partial differential equation) with (Hamiltonian operator) (e.g. with ) (1-dim) The ultimate goal of most quantum chemistry approach is the solution of the time-independent Schrödinger equation.