# Is the S Matrix Unitary in One-Dimensional Scattering Problems?

• jaobyccdee
In summary, the conversation discusses a one-dimensional scattering problem with a localized potential and a wave-function that is expressed in terms of "incoming" and "outgoing" waves. The equations S11A + S12B = C and S21A + S22D = B are related by the matrix equation, and the S matrix is shown to be unitary using these equations. The conditions |S11|^2 + |S21|^2 = 1, |S12|^2 + |S22|^2 = 1, and S11S12* + S21S22* = 0 can be derived by matching the wavefunction and its derivative at x=a.
jaobyccdee
1. The problem statement, all variables and given/known
A general one dimensional scattering problem could be characterized by an
(arbitrary) potential V (x) which is localized by the requirement that V (x) = 0
for |x|> a. Assume that the wave-function is
ψ (x) =

Ae^(ikx) + Be^(-ikx) x < -a
Ce^(ikx) + De^(-ikx) x > a
Relating the \outgoing" waves to the \incoming" waves by the matrix equation

C=S11A+ S12B
B=S21A+ S22D


show that
|S11|^2 + |S21|^2 = 1
|S12|^2 + |S22|^2 = 1
S11S12* + S21S22* = 0
Use this to show that the S matrix is unitary.

## Homework Equations

I don't understand why C=S11A+S12B or B=S21A+S22D

## The Attempt at a Solution

I calculate the flux for the incoming beam and the outgoing beam and set them equal, i get 2A^2 ik-2ikB^2=2C^2ik-2D^2ik i don't see how C and B can be expressed with only two other variables.

jaobyccdee said:
I don't understand why C=S11A+S12B or B=S21A+S22D

You can derive these conditions by matching both the wavefunction and its derivative at $x=a$ (we generally require these functions to be continuous everywhere, when possible).

## What is a one dimension scattering problem?

A one dimension scattering problem is a type of problem in physics that involves studying the scattering of particles or waves in a one-dimensional space. This is typically done by considering a potential barrier or well that the particles or waves encounter.

## What is the difference between a one dimension scattering problem and a two or three dimension scattering problem?

The main difference between these types of scattering problems is the dimensionality of the space in which the particles or waves are scattering. In a one dimension scattering problem, the particles or waves are only moving in one direction, while in a two or three dimension scattering problem, they may be moving in multiple directions.

## What are some common approaches to solving one dimension scattering problems?

Some common approaches to solving one dimension scattering problems include using analytical methods such as solving the Schrödinger equation, as well as using numerical methods such as the transfer matrix method or the R-matrix method.

## What are some real-world applications of one dimension scattering problems?

One dimension scattering problems have many real-world applications, including studying the scattering of electrons in solid-state materials, the scattering of light in optical fibers, and the scattering of particles in nuclear and particle physics experiments.

## Are there any limitations to using one dimension scattering problems?

While one dimension scattering problems can provide valuable insights into the behavior of particles or waves in simple one-dimensional systems, they may not fully capture the complexities of real-world systems that often involve multiple dimensions and more realistic potentials.

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