Lecture 2. Postulates in Quantum Mechanics Engel, Ch. 2-3 Ratner & Schatz, Ch. 2 Molecular Quantum Mechanics, Atkins & Friedman (4 th ed. 2005), Ch. 1 Introductory Quantum Mechanics, R. L. Liboff (4 th ed, 2004), Ch. 3 A Brief Review of Elementary Quantum Chemistry Wikipedia ( 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?”

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). = probability density (1-dim) Postulate 1 of Quantum Mechanics (wave function)

Born Interpretation of the Wave Function: Probability Density over finite range

This wave function is valid because it is infinite over zero range. “The wave function cannot have an infinite amplitude over a finite interval.”

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 {a n } which satisfy the eigenvalue equation. (Operator)(function) = (constant number)  (the same function) (Operator corresponding to observable)  = (value of observable)  eigenvalueeigenfunction Postulate 2 of Quantum Mechanics (measurement)

Postulate 2 of Quantum Mechanics (operator) (1-dimensional cases only) Physical Observables & Their Corresponding Operators (1D)

Physical Observables & Their Corresponding Operators (3D) Postulate 2 of Quantum Mechanics (operator)

Observables, Operators, and Solving Eigenvalue Equations: An example (a particle moving along x, two cases) constant number the same function  This wave function is an eigenfunction of the momentum operator p x  It will show only a constant momentum (eigenvalue) p x. Is this wave function an eigenfunction of the momentum operator?

The Schrödinger Equation (= eigenvalue equation with total energy operator) Hamiltonian operator  energy & wavefunction (solving a partial differential equation) (1-dim) (e.g. with ) The ultimate goal of most quantum chemistry approach is the solution of the time-independent Schrödinger equation. with (Hamiltonian operator)