# Delta force

1. Apr 28, 2010

### Petar Mali

1. The problem statement, all variables and given/known data
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

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

2. Relevant equations

3. 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.

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

Why I don't get same solution different then zero like in case

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

?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Apr 28, 2010

### 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

No you cant. The wavefunction $$\psi(x)=\left\{\begin{array}{lr}Ae^{kx}, & x<0 \\ Ae^{-kx}, & x>0\end{array}\right.[/itex] is undefined at $x=0$ (as it should be for a delta function potential). The wavefunction $\psi(x)=Ae^{-k|x|}$ is defined at $x=0$; the two wavefunctions are not equivalent. I don't see how you are getting zero for that limit. Show the rest of your steps. Last edited: Apr 28, 2010 3. Apr 29, 2010 ### Petar Mali [tex] lim_{\epsilon \rightarrow 0} \int^{\epsilon}_{-\epsilon}\frac{d^2\psi}{dx^2}dx=lim_{\epsilon \rightarrow 0} \int^{0}_{-\epsilon}\frac{d^2\psi}{dx^2}dx+lim_{\epsilon \rightarrow 0} \int^{\epsilon}_{0}\frac{d^2\psi}{dx^2}dx=0$$

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

$$\frac{d\psi}{dx}=kAe^{kx}$$ for $$x<0$$

$$\frac{d\psi}{dx}=-kAe^{-kx}$$ for $$x>0$$

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

4. Apr 29, 2010

### BerryBoy

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