We are going to look at a problem

V(x)=−V0 δ(x),V0>0V(x) = -V_0\,\delta(x), \qquad V_0 > 0

Where δ(x)\delta(x) is the Dirac Delta Describe the J(x)J(x) of the particle in all states

Write the TISE

−ℏ22md2ϕdx2−V0 δ(x) ϕ=Eϕ-\frac{\hbar^2}{2m}\frac{d^2\phi}{dx^2} - V_0\,\delta(x)\,\phi = E\phi

away from origin the delta term vanished

d2ϕdx2=κ2ϕ,κ≡−2mEℏ\frac{d^2\phi}{dx^2} = \kappa^2\phi, \qquad \kappa \equiv \frac{\sqrt{-2mE}}{\hbar}

Since E<0E<0, κ\kappa is real and positive.

d2ϕdx2=κ2ϕ,κ≡−2mEℏ\frac{d^2\phi}{dx^2} = \kappa^2\phi, \qquad \kappa \equiv \frac{\sqrt{-2mE}}{\hbar}

so

ϕ(x)={Aeκxx<0Be−κxx>0\phi(x) = \begin{cases} A e^{\kappa x} & x < 0 \\ B e^{-\kappa x} & x > 0 \end{cases}