- #1
LagrangeEuler
- 717
- 20
Homework Statement
##\frac{d^2\psi}{dx^2}+\frac{2m}{\hbar^2}(E-\alpha\delta(x))\psi(x)=0##
Show that ##\psi(x)## is continuous and that first derivative has discontinuity ##\frac{2m\alpha}{\hbar^2}\psi(0)##.
Homework Equations
The Attempt at a Solution
I'm not sure how to show that function ##\psi(x)## is continuous. Or that ##\psi(0^+)=\psi(0^-)##. ##\alpha\delta(x)## is even potential so ##\psi(x)## could be even function or odd function. Right. And because potential in zero is ##\infty##, ##\psi(0)=0##. Question. Could I say that
## \int^{\epsilon}_{-\epsilon}\psi(x)dx \leq 2\epsilon \psi(y)##
where ##\psi(y)## is maximum of the function ##\psi(x)## at ##(-\epsilon,\epsilon)##.
and that
## \int^{\epsilon}_{-\epsilon}\psi(x)dx \geq 0##?
and why ## \int^{\epsilon}_{-\epsilon}\psi(x)dx \geq 0##?