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Dear Student,
A free particle is not subjected to any forces, its potential energy is constant. Set V(r,t)=0, since the origin of the potential energy may be chosen arbitrarily. The Schroedinger equation then becomes
.
Plane waves are possible solutions,
y(r,t)=Aexp(k×r-wt),
as long as hw=h2k2/(2m).
We have
,
using the de Broglie relations.
A plane wave represents a particle whose probability of presence is constant throughout space;
|y(r,t)|2=|A|2=constant.
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