Schrodinger
Equation
The Schrodinger Equation is a postulate, and is regarded as an axiom of
Quantum Chemistry. While no proof is possible, many of the postulates
are “reasonable” based on their congruency to experimental results.
The time-independent Schrondinger Equation (for the one dimension case)
utilizes the classical one-dimensional wave equation, deBroglie’s
Equation, and energy relationships. The final equation is:
(h/4π)
● (d2Ψ/dx2) + U(x) ● Ψ(x) = E ● Ψ(x)
where:
h
is Planck’s Constant,
U(x) is the potential energy,
Ψ(x) is the wavefunction that
describes the system,
and E is the overall energy of
the system.
Particle in One-Dimension Box or
Particle on a Line
For pedagogic purposes, the concept of the one-dimension box or
singular line is often used to generate some of complex concepts of wave
equations.
Assume there is a particle that exists between the region 0 and A. At
both 0 and A, there is an infinite potential, which forces the particle
to stay inside these bounds. The probability of find the particle
outside of these bounds is zero.
The particle experiences zero potential within these bounds, so it may
freely move around. When we apply this to the Schrodinger Equation, the
second term drops out, so the net result is:
(h/4π)
● (d2Ψ/dx2) = E ● Ψ(x)
or
(d2Ψ/dx2) + (8π2mE/h2)
● Ψ(x) = 0
where:
h
is Planck’s Constant,
Ψ(x) is the wavefunction that
describes the system,
E is the overall energy of the
system,
and m is the mass of the
particle
It can also be written in a more familiar way as:
(d2f/dx2)
+ (k) ● f(x) = 0
where
k
is a constant
Many mathematicians will recognize that the solution probably uses
cosines and sine. Hence, Ψ(x) behaves in a harmonic or sinusoidal wave.
The energy of the system is also quantized, since it can only take on
certain, discrete values, which happen to be solutions of the system.
The presence of the 8π2 forces the system to have a phase that has π in
its denominator. The net result is a system which takes on integral
values of cosines and sines. A general solution to this equation is:
Ψ(x)
= A cos kx + B sin kx
where:
A,B, are constants,
k = [(2mE)0.5]/h
h is Planck’s Constant,
and E is the overall energy of
the system.
While this result may seem fairly abstract it provides two useful
results. The first is that the energy of the system is quantized given
the boundary conditions (the figure on the left). Since the particle can
only travel in sinusoidal waves, there are only discrete energy levels
it can take on. This should make some intuitive sense. If the standing
wave was allowed to take any shape or form, it would eventually cancel
itself out as it traveled from boundary point to boundary point.
However, if it travels in a “controlled” sinusoidal path, then it will
be able to neatly fit into the box without destroying itself. The energy
of the system is the square of its position (the figure on the right).
It shows that the system can only take on certain quantized values for
total energy.
The second key result is that the function can be square normalized.
The term square normalized simply means that the function, times its
complement (including imaginary terms) must equal one when integrated
over the entire region of interest. The region of interest is variable.
In this particle in the box example, we integrate over the entire line
from 0 to A. in the real world, we would integrate over all of x, y, and
z.