# Particle moving in a delta potential

A particle of mass m, with energy E>0, is moving in the potential

V(x)=g[delta(x+a) + delta(x-a)]

Assuming that the particle is incident from the left, what is the solution of the Schrodinger equation in all three regions (x<a, -a<x<a, x>a) for this situation? Also, what are the appropriate continuity conditions at x=+a and x= -a?

#### Solution Preview

See attached files.

---------------------------

documentclass[a4paper]{article}

usepackage{graphicx,amsmath,amssymb}

newcommand{haak}[1]{!left(#1right)}

newcommand{rhaak}[1]{!left [#1right]}

newcommand{lhaak}[1]{left | #1right |}

newcommand{ahaak}[1]{!left{#1right}}

newcommand{gem}[1]{leftlangle #1rightrangle}

newcommand{gemc}[2]{leftlangleleftlangleleft. #1right | #2

rightranglerightrangle}

newcommand{geml}[1]{leftlangle #1right.}

newcommand{gemr}[1]{left. #1rightrangle}

newcommand{haakl}[1]{left(#1right.}

newcommand{haakr}[1]{left.#1right)}

newcommand{rhaakl}[1]{left[#1right.}

newcommand{rhaakr}[1]{left.#1right]}

newcommand{lhaakl}[1]{left |#1right.}

newcommand{lhaakr}[1]{left.#1right |}

newcommand{ket}[1]{lhaakl{gemr{#1}}}

newcommand{bra}[1]{lhaakr{geml{#1}}}

newcommand{brak}[2]{leftlangle #1vphantom{#2}right | !left.vphantom{#1}#2

rightrangle}

newcommand{braket}[3]{gem{#1lhaak{vphantom{#1}#2vphantom{#3}}#3}}

newcommand{floor}[1]{leftlfloor #1rightrfloor}

newcommand{half}{frac{1}{2}}

newcommand{kwart}{frac{1}{4}}

newcommand{bfm}[1]{mathbf{#1}}

renewcommand{imath}{text{i}}

renewcommand{dfrac}[2]{#1/#2}

begin{document}

title{Particle moving in a delta potential}

date{}

maketitle

The Schr"odinger equation is:

begin{equation}label{sch}

-frac{hbar^{2}}{2m}frac{partial^{2}}{partial ...

#### Solution Summary

We explain how to solve the Schrödinger equation for the potential V(x)=g[delta(x+a) + delta(x-a)]. We consider the case of an unbound particle with positive energy.