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Here we follow the treatment of McQuarrie [1], Section
31. We start with the onedimensional classical wave equation,

(10) 
By introducing the separation of variables

(11) 
we obtain

(12) 
If we introduce one of the standard wave equation solutions for
such as (the constant can be taken care of later in the
normalization), we obtain

(13) 
Now we have an ordinary differential equation describing the spatial
amplitude of the matter wave as a function of position. The energy
of a particle is the sum of kinetic and potential parts

(14) 
which can be solved for the momentum, , to obtain

(15) 
Now we can use the de Broglie formula (4) to get
an expression for the wavelength

(16) 
The term in equation (13) can be rewritten
in terms of if we recall that
and
.

(17) 
When this result is substituted into equation (13) we
obtain the famous timeindependent Schrödinger equation

(18) 
which is almost always written in the form

(19) 
This singleparticle onedimensional equation can easily be extended to
the case of three dimensions, where it becomes

(20) 
A twobody problem can also be treated by this equation if the mass
is replaced with a reduced mass .
It is important to point out that this analogy with the classical wave
equation only goes so far. We cannot, for instance, derive the
timedependent Schrödinger equation in an analogous fashion
(for instance, that equation involves the partial first derivative
with respect to time instead of the partial second derivative). In
fact, Schrödinger presented his timeindependent equation first, and
then went back and postulated the more general timedependent
equation.
Next: The TimeDependent Schrödinger Equation
Up: The Schrödinger Equation
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David Sherrill
20060815