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Unformatted text preview: January 7, 2008 minor correction January 16, 2008 Physics 225/315 J. Rothberg Outline: Introduction to Quantum Mechanics 1 Polarization of Light 1.1 Classical Description Light polarized in the x direction has an electric field vector E = E ˆx cos( kz ωt ) Light polarized in the y direction has an electric field vector E = E ˆy cos( kz ωt ) Light polarized at 45 degrees (call this direction x ) has an electric field vector E ˆx cos( kz ωt ) = E ˆ 1 √ 2 ˆx cos( kz ωt ) + 1 √ 2 ˆy cos( kz ωt ) ! Light which is right circularly polarized has an electric field vector E R = E ˆ 1 √ 2 ˆx cos( kz ωt ) + 1 √ 2 ˆy cos( kz ωt + π 2 ) ! There are other sign conventions for circularly polarized light; see French and Taylor[1] Chapter 6 and Le Bellac[2] p. 65 and Sakurai[3] p. 9,10. E x and E y are 90 degrees out of phase and the electric field vector in right circularly polarized light rotates. The fields may be represented using complex notation; the actual electric field is the Real part of the complex electric field. The real part of the exponential is the cosine function cos φ = Re( e iφ ) and recall i = e i π 2 . E R = Re ˆ E ˆ 1 √ 2 ˆx e i ( kz ωt ) + 1 √ 2 ˆy e i ( kz ωt + π 2 ) !! = Re ˆ E ˆ 1 √ 2 ˆx e i ( kz ωt ) + i √ 2 ˆy e i ( kz ωt ) !! 1.2 Polaroid Filters The component of the electric field along the transmission direction of the filter is transmit ted. After the filter the light is polarized along the transmission direction. Polaroid filters “prepare” light in a state of polarization. The intensity of light is proportional to the square of the electric field; therefore after the filter the intensity is proportional to the square of the cosine of the angle between the incoming electric field and the Polaroid transmission direction. See French and Taylor[1] p. 234 1 1.3 State Vector Description We can represent the polarization states by “state” vectors in a two dimensional vector space. Light polarized in the x direction can be represented by  x and in the y direction by  y . These vectors like  x are called “kets”, a name proposed by Dirac; it is part of the word “bracket”. This way of writing vectors in Quantum Mechanics is called Dirac Notation. Polarization at 45 degrees can be represented by the state  θ = 45 o = 1 √ 2 (  x +  y ) Polarization at an angle θ in the x, y plane, the direction ˆ n θ , can be described by  θ = cos θ  x + sin θ  y Right and Left circular polarization can be described by  R = 1 √ 2 (  x + i  y ) and  L = 1 √ 2 (  x i  y ) Notice that the state vectors have complex coefficients in this case. In the state notation the x , y , θ , R , and L in the brackets are labels for the states....
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 Spring '10
 ROTHBERG
 mechanics, Light, Polarization, Hilbert space

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