I have a time dependent wavefunction for inside a delta function potential well: V(x) = -a delta(x).(adsbygoogle = window.adsbygoogle || []).push({});

It reads

Psi(x,t) = (sqrt(m*a)/hbar) * exp(-m*a*abs(x)/hbar^2) * exp(-iEt/hbar)

I'm supposed to stick this back into the time dependent Schrodinger Equation and solve for E.

Taking my Psi(x,t), I found the second derivative with respect to x, and also found the time derivative. Then I plugged directly back into the time dependent Schrodinger Equation, with V(x) given as above. The problem is that I can't seem to make the delta function go away. I get

-(m*a^2)/(2*hbar^2) - a*delta(x) = E

How do I get the delta to go away? I can't just say x=0, because then the delta function is infinity.

I also tried: Plugging directly back into the time dependent Schrodinger Equation, I integrate both sides from -e to +e, where is some really small distance around x=0. Instead of using dPsi/dt, I write dPsi/dt in terms of d^2Psi/dx^2. Then I integrate both sides. The only problem with doing this, is when I integrate d^2Psi/dx^2 dx, this becomes dPsi/dx evaluated from -e to +e -- and because of the absolute value of x in Psi, this is zero. Help!!!

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# Delta function potential and Schrodinger Equation

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