J. Rothberg April 2, 2014 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~ = E0 ˆx cos(kz ωt) − Light polarized in the y direction has an electric field vector E~ = E0 ˆy cos(kz ωt) − The wave number k = 2π/λ where λ is the wavelength. ω = 2πf where f is the frequency. fλ = c.x ˆ = ˆi andy ˆ = ˆj are unit vectors. Light polarized at 45 degrees (call this direction x′) has an electric field vector ′ 1 1 E0 ˆx cos(kz ωt)= E0 ˆx cos(kz ωt)+ ˆy cos(kz ωt) − √2 − √2 − ! 1.1.1 Polarization Analyzers The component of the electric field along the transmission direction of a Polaroid filter is transmitted. After the filter the light is polarized along the filter transmission direction. Polaroid filters “prepare” light in a state of linear 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. 2 Ex = E cos θ and intensity is proportional to cos θ. See| | French| | and Taylor[?] p. 234 An ideal Polaroid filter has these properties: a) The outgoing light leaving a Polaroid sheet is polarized along the transmission direction of the Polaroid. b) The Polaroid sheet transmits or selects the component of the incoming electric field which is along its transmission direction. For transmission along x Ex = ˆx E~ c) An ideal Polaroid transmits 100% of the light polarized in its transmission· direction and no light polarized in the perpendicular direction. A Polaroid sheet prepares light in a state of linear polarization. A second Polaroid sheet analyzes the polarization state. 1.1.2 Polarization States 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 light polarized in the y direction by y . | i | i 1 Polarization at 45 degrees can be represented by the state 1 θ = 45o = ( x + y ) | i √2 | i | i Polarization at an angle θ in the x, y plane, the directionn ˆθ, can be represented by θ = cos θ x + sin θ y | i | i | i Right and Left circular polarization can be described by 1 1 R = ( x + i y ) and L = ( x i y ) | i √2 | i | i | i √2 | i − | i Notice that the state vectors have complex coefficients in this case which correspond to a 90 degree phase shift. In the state notation the x, y, θ, R, and L in the brackets are labels describing the states. Basis states can be chosen to be x and y and all states can be written as linear combinations of the two orthogonal basis states.| i | i The most general polarization state is Φ = λ x +µ y where λ and µ are in general complex with λ 2 + µ 2 = 1. Then Φ is normalized.| i | i | i In the| case| | of| linear polarization| i λ and µ are real and can be written as cos θ and sin θ. sin2 θ + cos2 θ = 1 The basis states, x and y , form an orthonormal basis. x x = 1 and x y = 0 The basis is said to| i be “complete”| i if any state can be writtenh as| ai linear combinationh | i of basis states. Another orthonormal basis can be written as θ = cos θ x + sin θ y | i | i | i θ⊥ = sin θ x + cos θ y . | i − | i | i For circular polarization we can use the basis states R and L . | i | i 1.1.3 Preparing and Analyzing Polarized Light When light reaches a Polaroid sheet the outgoing state is the state vector of the incoming light projected along the state representing the Polaroid transmission direction. The amplitude of the outgoing state is the inner product of the state representing the incoming light and the state representing the analyzer selection. Inner products are written as y θ where y is the dual of y . The inner product is a scalar. h | i h | | i Compare with ordinary vectors, Ex = ˆx E~ · If a Polaroid filter has its transmission axis in the y direction; we represent it by y . If the incoming light is polarized at angle θ and the analyzer transmission directionh | is in the y direction then the outgoing amplitude is the inner product of the state representing the Polaroid and the state of the incoming light y θ = y (cos θ x + sin θ y ) = cos θ y x + sin θ y y = sin θ h | i h | | i | i h | i h | i We use an orthonormal basis: x x = 1 y y = 1 x y = 0 h | i h | i h | i 2 After the analyzer the polarization direction is y and the amplitude is sin θ. The state of the outgoing light is described by y . | i Classically the intensity of light is proportional to the square of the electric field so the outgoing intensity is proportional to the magnitude squared of the amplitude y θ 2. 2 |h | i| In this example the intensity after the Polaroid is I = I0 sin θ. 1.1.4 Examples A polaroid sheet has its transmission axis at an angle θ with respect to the x direction. The state representing the Polaroid is θ = cos θ x + sin θ y h | h | h | To find the amplitude for light transmission through a Polaroid we must project the state of the incident light beam on the state representing the Polaroid. If the incident light is polarized in the x direction it is represented by x . If we send this light through a Polaroid filter whose transmission direction is at angle θ|theni the amplitude passing the Polaroid is θ x = cos θ using x x = 1 y y = 1 x y = 0 h | i h | i h | i h | i Incident Light Polarized at 45 degrees For 45 degree Linear Polarization the amplitude is 1 1 θ 45o = (cos θ x + sin θ y ) ( x + y ) θ 45o = (cos θ + sin θ) h | i h | h | √2 | i | i ! h | i √2 The magnitude squared of the amplitude (the intensity) is 1 1 θ 45o 2 = (1 + 2 sin θ cos θ)= (1 + sin 2θ) |h | i| 2 2 o o 1 If θ = 45 we get intensity = 1 and if θ =0 or θ = 90 we get intensity = 2 . If θ = 45o we get intensity = 0 since sin ( π )= 1. The analyzer Polaroid is perpendicular − − 2 − to the Polaroid which prepared the light (the states are orthogonal). 1.2 Photons We expect to find the same Energy (intensity) relationships with the photon picture as with the classical wave picture. From the analysis of the photoelectric effect each photon carries a quantum of energy E = hf = hc/λ. The total energy will be proportional to the number of photons. The light intensity is proportional to the number of photons. Approximate numerical values are 1 eV 1.6 10−19 J hc 1240 eV nm 1 nm = 10−9 m. ≈ × ≈ − 3 E 1240/λ where E is in eV and λ is in nanometers. For≈ visible light λ is of order 500 nm so E 2.5 eV ≈ In the photon description of light we associate a polarization state with a beam of photons which has passed a Polaroid filter or has been prepared by atomic transitions. We can assign a polarization state x , y , θ , R , etc. to a beam of photons. | i | i | i | i An unpolarized beam can be viewed as a mixture of polarizations states, for example a mixture of 50% of the photons in the state x and 50% in the state y . Be careful - this is completely different from| ai polarized beam described| i as a linear combination of states 1 ( x + y ). 2 | i | i If a beam of photons is in the state x and it reaches a Polaroid filter whose transmission | i direction is y (represented by y ) then no photons will get through the filter. If the filter were x then (forh an| ideal filter) all the photons should get through. For a filter at 45h degrees| half the photons are transmitted. A photon either goes through the Polaroid filter or it does not. There are no fractional photons. The probability that a photon is transmitted through a Polaroid sheet depends on the polarization state in which it was prepared. We prepare a beam of photons in a state θ by sending the light through a Polaroid filter whose transmission axis is at angle θ measured| i with respect to the x axis. Any photon observed after the Polaroid filter will be in the state θ . | i 1.2.1 Amplitude and Probability Suppose we have a beam of photons prepared in the state θ . θ = cos θ x + sin θ y | i | i | i | i The Amplitude for finding a photon after a Polaroid analyzer is given by the above inner product rules for finding amplitudes. For example for a Polaroid whose transmission axis is x the amplitude is x θ . The amplitude is cos θ in this case. The “amplitude” ish sometimes| i called the “probability amplitude”. The Probability for finding a photon after the analyzer is the magnitude squared of the amplitude; the probability is x θ 2. The probability would be cos2 θ in this case.