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Let us come back to de Broglie's idea to describe a particle as a wave or better as a superposition
of waves.
We assume that a particle with energy can be described by a function that is a superposition of
plane waves,
We have used the relation between momentum and wave vector, , and
the relation between energy and angular frequency,
. As with waves, the
angular frequency in general depends on the wave length and therefore
.
For simplicity, we adopted a one-dimensional version.
Note that the time evolution of a single plane wave
goes with the minus sign.
We would like to know the time evolution of the function , i.e. to find its equation of motion.
Equations of motion often represent fundamental laws in physics, like Newton's , which is a second
order differential equation
. We therefore differentiate (1.17)
with respect to time (we write
for
etc.
So far we have only considered a particle that only has kinetic energy
.
In general, a particle can have both kinetic energy and potential energy .
Example: A harmonic oscillator with
angular frequency and mass in one dimension has the potential energy
.
We now postulate that the above equation for a free particle (zero potential energy),
has to be generalized by replacing with the total energy for a particle
in a non-zero potential. Then, the equation of motion becomes
The equation
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(20) |
is called Schrödinger equation and is one of the most important equations of physics at all.
We only have given the one-dimensional version of it so far, the generalization to two or three dimensions
is not difficult: the variables and become vectors and .
Instead of the differential operator
, one has
in two or
in three
dimensions. This is nothing else but the Laplace operator
.
The Schrödinger equation reads
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(21) |
Next: Interpretation of the Wave
Up: Waves, particles, and wave
Previous: Introduction
  Contents
Tobias Brandes
2004-02-04