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). = <|> 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. 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) constant number the same function 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.

Postulate 3 of Quantum Mechanics (eigenvalue) Observed in measurements are only the eigenvalues {ai } which satisfy the eigenvalue equation. Although measurements must always yield an eigenvalue, a state (a wave function) does not have to be an eigenstate (eigenfunction). An arbitrary state (wave function) can be expanded in the complete set of eigenfunctions ( as where n  . “superposition” After measurement, the fuzzy, superposed wave function is “collapsed” into an eigenfunction. (The measurement action itself affects the state.) eigenvalue ai

Postulate 3 of Quantum Mechanics (eigenvalue) An arbitrary state can be expanded in the complete set of eigenvectors ( as where n   (superposition). We know that the measurement will yield one of the values ai, but we don't know which one. However, we do know the probability that eigenvalue ai will occur ( , if the eigenfunctions form an orthonormal set). An example: Superposition of two different states of definite energy E1 and E2 = ij

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 is given by = <|A|> For a special case when the wavefunction corresponds to an eigenstate,

Postulate 4 of Quantum Mechanics (expectation) : normalized : orthogonal : not orthogonal

Postulate 5 of Quantum Mechanics (time dependence) The evolution in time of a quantum mechanical system is governed by the time-dependent Schrodinger equation. Hamiltonian again Plank constant again For a solution of time-independent Schrodinger equation, , time-independent operator

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.

Schrödinger Cat (Measurement and Superposition) Schrödinger wrote (1935): One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The psi-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts. It is typical of these cases that an indeterminacy originally restricted to the atomic domain becomes transformed into macroscopic indeterminacy, which can then be resolved by direct observation. That prevents us from so naively accepting as valid a "blurred model" for representing reality. In itself it would not embody anything unclear or contradictory. There is a difference between a shaky or out-of-focus photograph and a snapshot of clouds and fog banks.

How to put an elephant in a fridge? (Quantum mechanics version 1)

How to put an elephant in a fridge?