QM in 1D position-space

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 .

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