1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Quantum Mechanics: Transformation Matrix

  1. Jan 31, 2015 #1
    1. The problem statement, all variables and given/known data

    Determine a ##2\times 2## matrix ##\mathbb{S}## that can be used to transform a column vector representing a photon polarization state using the linear polarization vectors ##|x\rangle## and ##|y\rangle## as a basis to one using the circular polarization vectors ##|R\rangle## and ##|L\rangle## as a basis.


    2. Relevant equations

    ##|R\rangle = \frac{1}{\sqrt{2}}\left(|x\rangle+i|y\rangle\right)##

    ##|L\rangle = \frac{1}{\sqrt{2}}\left(|x\rangle-i|y\rangle\right)##


    3. The attempt at a solution

    Not sure exactly what the question is asking.

    Is it asking to use ##|x\rangle## and ##|y\rangle## as a basis to find the transformation matrix that transforms ##|x\rangle## and ##|y\rangle## to ##|R\rangle## and ##|L\rangle##?

    If that is what it's asking, then would this be correct:

    ##\mathbb{S}=\left[{\begin{array}{cc} \langle R|x\rangle & \langle L|y\rangle \\ \langle R|x\rangle & \langle L|y\rangle \\\end{array}}\right]?##
     
    Last edited: Jan 31, 2015
  2. jcsd
  3. Jan 31, 2015 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Hello and welcome to PF!

    I think you have the right idea but your matrix for ##\mathbb{S}## is not correct (It has two identical columns).

    An arbitrary polarization state, ##|\psi \rangle = a|x \rangle + b|y \rangle##, can be written as a column matrix $$\binom{a}{b}_{\!xy}$$ where the subsrcipt tells us which basis we're using.

    The same state of polarization could be expressed in terms of the ##|R \rangle## and ##|L \rangle## basis states:
    ##|\psi \rangle = c|R \rangle + d|L \rangle##. Or, as $$\binom{c}{d}_{\!RL}$$ I believe the question is asking you to find a matrix ##\mathbb{S}## that will transform any ##\binom{a}{b}_{\!xy}## into the corresponding ##\binom{c}{d}_{\!RL}##.

    If, for example, the polarization state is ##|\psi \rangle = \frac{1}{\sqrt{3}}|x \rangle + \sqrt{\frac{2}{3}}e^{i\pi/4}|y \rangle##, then if you apply the matrix ##\mathbb{S}## to
    \begin{pmatrix}
    \frac{1}{\sqrt{3}} \\\sqrt{\frac{2}{3}}e^{i\pi/4}
    \end{pmatrix} you will get the corresponding column matrix that expresses the same state in the ##|R \rangle## and ##|L \rangle## basis.
     
  4. Feb 1, 2015 #3
    Thank you!

    I made a mistake. It should be ##\mathbb{S}^{\dagger}## instead of ##\mathbb{S}##.

    Can you elaborate please?

    So we need to find a matrix ##\mathbb{S}## such that ##\mathbb{S} \binom{a}{b}_{\!xy} = \binom{c}{d}_{\!RL}## is true?
     
  5. Feb 3, 2015 #4

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Yes, I believe that's what you are asked to do. For example, suppose you take a state that is linearly polarized in the x direction: ##\binom{1}{0}_{\!xy}##. After multiplying by ##\mathbb{S}##, what column vector should you get?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted