QM of a particle in a 1D infinite square well
A particle in the infinite square well potential
has stationary states in position basis for
with energies
General solutions to the full Schrödinger equation therefore have the form
Proof
Inside
the TISE reads or
noting that
states are forbidden (which would come up in the solutions anyway). The general solution is then we take boundary conditions
. Now giving solutions giving solutions for
with for odd and for even . The solution is not normalizable and hence is rejected as unphysical, and negative gives a rescaling of a positive solution. Thus the energies are Normalisation for odd
gives Likewise for even
we have So
.