Particle Wave duality

Photon

• Reflection and Refraction refraction phase shift only happens when $n_1<n_2$

• Probabilistic notation

  • (the photon becomes a “wave”)

    • We can label the notation for the vector is called “ket” vector
      • $\text{right}=|right>$ it is important we need to make sure that it is a linear combination of the superposition

      • to stop us from thinking these vectors are directions we use column vectors

        • $|right>=\left[\begin{array}{c}1\\ 0\end{array}\right],|left>=\left[\begin{array}{c}0\\ 1\end{array}\right]$
          • these two states are orthoganal, representing that they are incompatible

• Beam Splitters

  • When a photon goes through a beam splitter it enters superposition and it could go both ways

    • the photon becomes a linear combination
      • $a|right>+b|left>=\left[\begin{array}{c}a\\ b\end{array}\right]$ and when it goes through a beam splitter the coefficients are $\frac{1}{\sqrt{2}}$ because it is equally likely
        • remember it becomes a wave and can be constructive and destructively interfiled
    • the full matrix of a beam splitter is
      • $\frac{1}{\sqrt{2}}\left[\begin{array}{cc}-1& 1\\ 1& 1\end{array}\right]$ so we just multiply the dirac vector (the ket vector), we jsut add more per beam splitter

• Phase shifter

  • Its a piece of glass that adds a phase shift

    • $\text{phase shift}=\left[\begin{array}{cc}1& 0\\ 0& e^{i\varphi }\end{array}\right]$
      • $e^{i\varphi }$ is just the euler identity guy, we multiply it last so that way we get it in terms of it