Solving Delta Potential Barrier Problem with Schrodinger Equation

[Petar Mali]

Homework Statement

In delta potential barrier problem Schrodinger equation we get

\psi(x)=Ae^{kx}, x<0

\psi(x)=Ae^{-kx}, x>0

We must get solution of

lim_{\epsilon \rightarrow 0} \int^{\epsilon}_{-\epsilon}\frac{d^2\psi}{dx^2}dx

Homework Equations

The Attempt at a Solution

lim_{\epsilon \rightarrow 0} \int^{\epsilon}_{-\epsilon}\frac{d^2\psi}{dx^2}dx=lim_{\epsilon \rightarrow 0} \frac{d\psi}{dx}|^{\epsilon}_{-\epsilon} and get the solution

I can say that the whole function is

\psi(x)=Ae^{-k|x|}

I don't have first derivative in 0.

lim_{\epsilon \rightarrow 0} \int^{\epsilon}_{-\epsilon}\frac{d^2\psi}{dx^2}dx=lim_{\epsilon \rightarrow 0} \int^{0}_{-\epsilon}\frac{d^2\psi}{dx^2}dx+lim_{\epsilon \rightarrow 0} \int^{\epsilon}_{0}\frac{d^2\psi}{dx^2}dx=0Why I don't get same solution different then zero like in case

lim_{\epsilon \rightarrow 0} \int^{\epsilon}_{-\epsilon}\frac{d^2\psi}{dx^2}dx=lim_{\epsilon \rightarrow 0} \frac{d\psi}{dx}|^{\epsilon}_{-\epsilon}

?

Homework Statement

Homework Equations

The Attempt at a Solution

 

[gabbagabbahey]

First, I'd like to point out how disappointed I was when I clicked on this thread and found that it wasn't a Chuck Norris movie

I can say that the whole function is

\psi(x)=Ae^{-k|x|}

I don't have first derivative in 0.

No you cant. The wavefunction \psi(x)=\left\{\begin{array}{lr}Ae^{kx}, &amp; x&lt;0 \\ Ae^{-kx}, &amp; x&gt;0\end{array}\right.[/itex] is undefined at x=0 (as it should be for a delta function potential). The wavefunction \psi(x)=Ae^{-k|x|} <b>is</b> defined at x=0; the two wavefunctions are not equivalent.<br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> lim_{\epsilon \rightarrow 0} \int^{\epsilon}_{-\epsilon}\frac{d^2\psi}{dx^2}dx=lim_{\epsilon \rightarrow 0} \int^{0}_{-\epsilon}\frac{d^2\psi}{dx^2}dx+lim_{\epsilon \rightarrow 0} \int^{\epsilon}_{0}\frac{d^2\psi}{dx^2}dx=0Why I don&#039;t get same solution different then zero like in case </div> </div> </blockquote><br /> I don&#039;t see how you are getting zero for that limit. Show the rest of your steps.

 

[Petar Mali]

<br /> lim_{\epsilon \rightarrow 0} \int^{\epsilon}_{-\epsilon}\frac{d^2\psi}{dx^2}dx=lim_{\epsilon \rightarrow 0} \int^{0}_{-\epsilon}\frac{d^2\psi}{dx^2}dx+lim_{\epsilon \rightarrow 0} \int^{\epsilon}_{0}\frac{d^2\psi}{dx^2}dx=0<br />

lim_{\epsilon \rightarrow 0} \int^{0}_{-\epsilon}\frac{d^2\psi}{dx^2}dx+lim_{\epsilon \rightarrow 0} \int^{\epsilon}_{0}\frac{d^2\psi}{dx^2}dx=lim_{\epsilon \rightarrow 0}\frac{d\psi}{dx}|^{0}_{-\epsilon}+lim_{\epsilon \rightarrow 0}\frac{d\psi}{dx}|^{\epsilon}_{0}<br />

\frac{d\psi}{dx}=kAe^{kx} for x&lt;0

\frac{d\psi}{dx}=-kAe^{-kx} for x&gt;0

lim_{\epsilon \rightarrow 0} \int^{0}_{-\epsilon}\frac{d^2\psi}{dx^2}dx+lim_{\epsilon \rightarrow 0} \int^{\epsilon}_{0}\frac{d^2\psi}{dx^2}dx=<br /> kA-lim_{\epsilon \rightarrow 0}kAe^{k\epsilon}-lim_{\epsilon \rightarrow 0}kAe^{-k\epsilon}+kA=2kA-2kA=0

 

[BerryBoy]

Double check your exponentials (hint hint) and maybe expand them out ignoring terms higher than \epsilon^1