University Physics Notes: Quantum Mechanics – The Infinite Square Well
The infinite square well potential is given by:
This is illustrated below.
A particle under the influence of such a potential is free - no
forces act - between
and
and is confined to that region by the need to have an infinite energy
in order to travel outside that region. We can solve the equations
describing the motion of the particle in each region and make them
fit together at the boundaries using the conditions, for
wavefunctions
and
on
either side of the barriers that
and
In the region of the well,
and
the Schrodinger equation becomes
(1)
This has general solution
atand
since
the potential is infinite there, so
so
and
We can find the constant
by
requiring that the wavefunction
be
normalized so
We
can differentiatetwice
to get
so
from (1)