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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 Six Postulates of Quantum Mechanics Postulate 1 of Quantum Mechanics (wave function) • The state of a quantum mechanical system is completely specified by the wave function or state function (r, t) that depends on the coordinates of the particle(s) and on time. – a mathematical description of a physical system • The probability to find the particle in the volume element d = dr dt located at r at time t is given by (r, t)(r, t) d . – Born interpretation * Let’s consider a wave function of one of your friend (as a particle) as an example. Draw P(x, t). “Where would he or she be at 9 am / 10 am / 11 am tomorrow?” Postulate 1 of Quantum Mechanics (wave function) • The wave function 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 “The wave function cannot have an infinite amplitude over a finite interval.” This wave function is valid because it is infinite over zero range. 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) Postulate 2 of Quantum Mechanics (operator) Physical Observables & Their Corresponding Operators (1D) (1-dimensional cases only) Postulate 2 of Quantum Mechanics (operator) Physical Observables & Their Corresponding Operators (3D) Observables, Operators, and Solving Eigenvalue Equations: An example (a particle moving along x, two cases) eikx p x k Aeikx pˆ x e ikx p x k d i dx d p x i dx the same function d Aeikx khAeikx kh i dx constant p x kh number k Aeikx Be ikx Is this wave function an eigenfunction of the momentum operator? This wave function is an eigenfunction of the momentum operator px It will show only a constant momentum (eigenvalue) px. The Schrödinger Equation (= eigenvalue equation with total energy operator) 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.