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Up: 1.3 Prototype problems
Next: 1.4 Numerical discretization
1.3.5 Schrödinger
Slide : [
Schrödinger 
VIDEO
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Choosing units where Planck's constant
and the mass
,
the timedependent Schrödinger equation clearly appears as a
special type of wave / diffusion equation:
In quantum mechanics, the Schrödinger equation is used to evolve the
complex wavefunction
to describe the probability
of finding a particle in a
given interval
.
Take the simplest example of a free particle modeled with a wavepacket
in a periodic domain and assume a constant potential
.
The JBONE applet below shows the evolution
of a low energy
(long wavelength) particle
that is initially known with a rather good accuracy in space (narrow
Gaussian envelope):
JBONE applet: press Start/Stop
to run the simulation showing how a well localized low energy
wavepacket spreads out in time, in agreement with Heisenberg's
uncertainty principle in quantum mechanics.

the wavefunction
(blue line) starts to oscillate, the
probability density
spreads out (black line) and the position
increasingly becomes uncertain... reproducing the famous uncertainty
principle.
SYLLABUS Previous: 1.3.4 Wavebreaking
Up: 1.3 Prototype problems
Next: 1.4 Numerical discretization
