Greens Function Solution of Schrodinger Equation

[chill_factor]

Homework Statement

Using the Greens function technique, reduce the Schrodinger Equation for the following potential:

V = V0 , 0<x<a
V = infinite, elsewhere.

Homework Equations

The Attempt at a Solution

I have no idea what "reduce" means. The professor did not go over this and internet searches revealed nothing about this.

Does this mean put the Schrodinger equation in the form Hψ(x) = Eψ(x), then setting a Greens function G(x) such that HG(x) = δ(x-s) with the boundary conditions 0<s<a and G(0)=G(a)=0, and using the integral formula u(x) = ∫ G(x,s)f(s)ds for Lu(x) = f(x) where L is a linear differential operator?

In our case L = H, f(x) = Eψ(x)

So the final equation would be ψ(x) = ∫ G(x,s)Eψ(s)ds

But the question does not ask to solve for ψ(x), it said REDUCE... is REDUCE the same as solve?

 

[mark726]

Dear student,

Thank you for your question. "Reduce" in this context means to simplify or transform the equation into a more manageable form. In this case, we want to use the Greens function technique to simplify the Schrodinger equation for the given potential.

Your approach is correct. By using the Greens function, we can transform the Schrodinger equation into an integral equation, as you have shown. This integral equation is a simplified form of the original equation, making it easier to solve.

However, the question does not specifically ask you to solve for ψ(x). It is asking you to reduce the equation using the Greens function technique. So your final answer should be the integral equation you have derived, not the solution for ψ(x).

I hope this helps clarify the task at hand. Please let me know if you have any further questions or need further clarification.