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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 Chemistry
http://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 Ψ (r, t ) that depends on the coordinates of
the particle(s) and on time.
• The probability to find the particle in the volume element d  drdt located
at r at time t is given by   (r , t )  (r , t )d. (Born interpretation)
• The wavefunction must be single-valued, continuous, finite, and normalized
(the probability of find it somewhere is 1).
 d  (r , t )
2
 1 = <|>
probability density
(1-dim)
Born Interpretation of the Wavefunction: Probability Density
Postulates 2-3 of Quantum Mechanics (operator)
• Once Ψ (r, t ) 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

A   a
eigenvalue
eigenfunction
(Operator)(function) = (constant number)(the same function)
(Operator corresponding to observable) = (value of observable)
Physical Observables & Their Corresponding Operators
Observables, Operators & Solving Eigenvalue Equations:
an example
  Aeikx
pˆ x 
 d
i dx
 d
  p x
i dx
the same function
 d
Aeikx  khAeikx  kh
i dx
constant
p x   kh
number
The Uncertainty Principle
When momentum is known precisely, the position cannot be predicted
precisely, and vice versa.
  Ae
ikx
p x   kh
  A
2
2
When the position is known precisely,
Location becomes
precise at the expense
of uncertainty in
the momentum
The Schrödinger Equation
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.
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    A d

,
= <|A|>

• For a special case when the wavefunction corresponds to an eigenstate,
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 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.