Quantum Chemistry: Our Agenda • Birth of quantum mechanics (Ch. 1)• Postulates in quantum mechanics (Ch. 3)• Schrödinger equation (Ch. 2)

• Simple examples of V(r) Particle in a box (Ch. 4-5) Harmonic oscillator (vibration) (Ch. 7-8) Particle on a ring or a sphere (rotation) (Ch. 7-8) Hydrogen atom (one-electron atom) (Ch. 9)

• Extension to chemical systems Many-electron atoms (Ch. 10-11) Diatomic molecules (Ch. 12) Polyatomic molecules (Ch. 13) Computational chemistry (Ch. 15)

Lecture 2. Postulates in Quantum Mechanics

• Engel, Ch. 2-3• 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 Expectation value (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 that depends on the coordinates of the particle(s) and on time.

•The probability to find the particle in the volume element located at r at time t is given by . (Born interpretation)

•The wavefunction must be single-valued, continuous, finite, and normalized (the probability of find it somewhere is 1).

= <|>

dtrtr ),(),(

),( trΨ

drdtd

1),(2 trd

probability density

(1-dim)

Born Interpretation of the Wavefunction: Probability Density

Postulates 2-3 of Quantum Mechanics (operator)

• Once 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

),( trΨ

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

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

eigenvalue eigenfunction

Physical Observables & Their Corresponding Operators

Observables, Operators & Solving Eigenvalue Equations:an example

ikxAe

dxd

ipx

ˆ

xpdxd

i

khkhAeAedxd

iikxikx

khpx constantnumber

the same function

The Uncertainty Principle

When momentum is known precisely, the position cannot be predicted precisely, and vice versa.

When the position is known precisely, Location becomes

precise at the expense of uncertainty in the momentum

ikxAe 22 Akhpx

The Schrödinger EquationHamiltonian 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)

Postulate 4 of Quantum Mechanics (expectation)• Although measurements must always yield an eigenvalue, the state does not have to be an eigenstate. • An arbitrary state can be expanded in the complete set of eigenvectors ( as where n . • For a system in a state described by a normalized wavefunction , the average value of the observable corresponding to is given by

= <|A|>

• For a special case when the wavefunction corresponds to an eigenstate,

dAA

Postulate 4 of Quantum Mechanics (expectation)

•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).

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

For a solution of time-independent Schrodinger equation, ,

time-independent operator

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.

Schrödinger Cat (Measurement and Superposition)