Download to the 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
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).
 d  (r , t )
2
 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.

A   a
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)
eikx  p x   k
  Aeikx
pˆ x 
e  ikx  p x   k
 d
i dx
 d
  p x
i dx
the same function
 d
Aeikx  khAeikx  kh
i dx
constant
p x   kh
number
 k  Aeikx  Be  ikx
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.
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.