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More filters = ...less filtering?Another important physical example of a two-dimensional Hilbert space is photon polarization. A photon travelling in the +z direction has two linear polarizations, corresponding to an electric field that oscillates purely along the x direction or purely along the y direction. Ill denote these states as |X and |Y , respectively. They satisfy X|Y = 0. Linear polarizations at other angles are real superpositions of these states. In particular, consider linear polarizations along axes x, y which are related to the x, y directions by a counterclockwise rotation about the +z direction. These are related to the old polarization states by|X = cos |X + sin |Y |Y = sin |X + cos |Y .(a) Suppose unpolarized light is passed through a sequence of two filters with their polarization axes arranged in the sequence: x, y. What fraction of the photons will pass through both filters?(b) Now suppose unpolarized light is passed through a sequence of three filters with polarization axes y, then x + y, then x. (Here x + y represents a direction +45 from the x axis). What fraction of photons will pass through all three filters?(c) What is the probability that an incident photon is transmitted by the array?(d) Evaluate the probability of transmission in the limit of a large number of filters, N. This is one way to rotate a photon.

          More filters = ...less filtering?Another important physical example of a two-dimensional Hilbert space is photon polarization. A photon travelling in the +z direction has two linear polarizations, corresponding to an electric field that oscillates purely along the x direction or purely along the y direction. Ill denote these states as |X and |Y , respectively. They satisfy X|Y  = 0. Linear polarizations at other angles are real superpositions of these states. In particular, consider linear polarizations along axes x, y which are related to the x, y directions by a counterclockwise rotation  about the +z direction. These are related to the old polarization states by|X = cos |X + sin |Y  |Y  =  sin |X + cos |Y .(a) Suppose unpolarized light is passed through a sequence of two filters with their polarization axes arranged in the sequence: x, y. What fraction of the photons will pass through both filters?(b) Now suppose unpolarized light is passed through a sequence of three filters with polarization axes y, then x + y, then x. (Here x + y represents a direction +45 from the x axis). What fraction of photons will pass through all three filters?(c) What is the probability that an incident photon is transmitted by the array?(d) Evaluate the probability of transmission in the limit of a large number of filters, N. This is one way to rotate a photon.

Added by Ian G.

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