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 Chemistryhttp://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?”

•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).

= <|>

1),(2 trd

probability density

(1-dim)

Postulate 1 of Quantum Mechanics (wave function)

Born Interpretation of the Wave Function: Probability Density

over finite rang

e

This wave function is validbecause 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 {an } which

satisfy the eigenvalue equation.

(Operator)(function) = (constant number)(the same function)

(Operator corresponding to observable) = (value of observable)

eigenvalue eigenfunction

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)

ikxAe

dx

d

ipx

ˆ

xpdx

d

i

khkhAeAedx

d

iikxikx

khpx constantnumber

the same functionikxikx

k BeAe

kpe xikx

kpe xikx

This wave function is an eigenfunction of the momentum operator px It will show only a constant momentum (eigenvalue) px.

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)