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A complex scalar field that describes a quantum mechanical system. The square of the modulus of the wave function gives the probability of the system to be found in a particular state. DO NOT USE THIS TAG for classical waves.
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How to find the wavefunction that solves an infinite square well with a delta function well ...
Then for $0<x<L/2$ we have $$\psi(x)=C\sinh(\kappa x)$$
and for $L/2<x<L$ we have
$$\psi(x)=-C\sinh(\kappa(x-L))$$
Now, applying the calculation for the discontinuity in the derivative of the wavefunction … even possible to have a solution for $E=0$:
For $0<x<L/2$ we have:
$$\psi(x) = Cx$$
And for $L/2 < x<L$ we have:
$$\psi(x)= C(L-x)$$
The dirac potential allows a discontinuity of the derivative of the wavefunction …