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.

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

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

?

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 [tex]\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.

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

I don't see how you are getting zero for that limit. Show the rest of your steps.

 

[Petar Mali]

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

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

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

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

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

 

[BerryBoy]

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

 

[paranormal]

I would like to provide some clarification and additional information on the topic of solving the delta potential barrier problem using the Schrodinger equation.

Firstly, it is important to note that the Schrodinger equation is a fundamental equation in quantum mechanics that describes the behavior of quantum systems. It is a partial differential equation that describes the evolution of a wave function, denoted by \psi, which represents the state of a quantum system.

In the delta potential barrier problem, we are interested in finding the solution to the Schrodinger equation in the presence of a delta function potential barrier. This type of potential barrier is a mathematical construct that represents an infinitely thin and infinitely high potential barrier.

In the given solution attempt, the function \psi(x)=Ae^{-k|x|} is only valid for x>0. For x<0, the correct solution is \psi(x)=Ae^{kx}. Additionally, the solution should also include a constant term for x=0, as the potential barrier is discontinuous at this point. Therefore, the complete solution for the delta potential barrier problem is given by:

\psi(x) = \begin{cases} Ae^{kx} + B & \text{for } x < 0 \\ Ce^{-kx} + D & \text{for } x > 0 \\ \end{cases}

where A, B, C, and D are constants that can be determined by applying boundary conditions.

Now, to address the question regarding the difference in the two solutions for the integral of the second derivative of \psi. The first solution uses the fundamental theorem of calculus to solve the integral, while the second solution uses the fact that the derivative of \psi is discontinuous at x=0. Both methods are valid and result in the same solution, as long as the boundary conditions are properly applied.

In conclusion, solving the delta potential barrier problem using the Schrodinger equation requires careful consideration of boundary conditions and understanding of the fundamental concepts of quantum mechanics. It is important to note that this is a simplified example and in real-world scenarios, the potential barrier may not be an ideal delta function, thus requiring more complex solutions.