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ope211
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Homework Statement
I am being asked to show that the wave function ψ and dψ/dx are continuous at every point of discontinuity for a step potential. I am asked to make use of the Heaviside step function in my proof, and to prove this explicitly and in detail.
Homework Equations
d^2ψ/dx^2=2m(V(x)-E)/ħ^2Ψ
Θ(x)=1 if x>0, 0 if x<0
The Attempt at a Solution
I assumed this meant that the potential can be written as V(x)=V_0Θ(x). Therefore, if x>0,
d^2ψ/dx^2=2m(V_0-E)/ħ^2Ψ
and if x<0:
d^2ψ/dx^2=2m(-E)/ħ^2Ψ
Now, I figured that I need to show ψ_1(0)=ψ_2(0) and the two first derivatives equal each other. So:
lim ε->0 (∫ d^2ψ_1/dx^2 dx (from -ε to +ε)=∫dx ψ_1(x)2m(V_0-E)/ħ^2 (from -ε to +ε))
and
lim ε->0 (∫ d^2ψ_2/dx^2 dx (from -ε to +ε)=∫dx ψ_2(x)2m(-E)/ħ^2 (from -ε to +ε))
Taking the limit of these integrals should show that they all go to 0, but I have no idea if this is sufficient. I do not know of any other way to show this.