Let |x > and |y > represent the usual horizontal and vertical polarization basis
ID: 2078647 • Letter: L
Question
Let |x > and |y > represent the usual horizontal and vertical polarization basis states for light traveling in the z direction. (a) Show that the following states also from an orthonormal basis: |phi > = cos phi|x > + sin phi|y >, |phi_ > = -sin phi|x > + cos phi |y >, and discuss their relationship to the states |x > and |y >. (b) Interpret (describe the physical meaning of) the polarization state represented by this ket: |phi > = 1/Squareroot 2 (|x > + e^3i pi/2|y >).Explanation / Answer
a) For orthonormal basis, we need to verify <|> = <|> = 1 and <|> = <|> = 0
as |x> and |y> forms an othonormal basis, then <x|x> = <y|y> = 1 and <x|y> = <y|x> = 0
<|> = cos2<x|x> + sin2<y|y> = 1. Similarly <|> = 1
<|> = cos sin<x|x> - cos sin<y|y> = 0. Similarly <|> = 0
Hence |> and |> forms an orthonormal basis.
b) The state |> represent the right circular polarized state. In this case, the state |y> has a phase shift of 3/2 with |x>. This state is normalized as <|> = 1.
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