Transcript Postulate 1 of Quantum Mechanics (wave function)

Lecture 2. Postulates in Quantum Mechanics
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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
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)
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 (not infinite over
a finite range), and normalized (the probability of find it somewhere is 1).
 d  (r , t )
2
 1 = <|>
probability density
(1-dim)
Born Interpretation of the Wave Function:
Probability Density
over
finite
range
Engel, 2nd Ed. p.40, last bullet
“The wave function cannot have an infinite amplitude over a finite interval.”
infinite
over
zero
range
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, and 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.