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 = E0 ˆx cos(kz − ωt) ~ Light polarized in the y direction has an electric field vector E = E0 ˆy cos(kz − ωt) Light polarized at 45 degrees (call this direction x0) has an electric field vector à ! ~ 0 1 1 E0 ˆx cos(kz − ωt) = E0 √ ˆx cos(kz − ωt) + √ ˆy cos(kz − ωt) 2 2 Light which is right circularly polarized has an electric field vector à ! ~ 1 1 π ER = E0 √ ˆx cos(kz − ωt) + √ ˆy cos(kz − ωt + ) 2 2 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. Ex and Ey 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 iφ i π cos φ = Re(e ) and recall i = e 2 . à à !! ~ 1 i(kz−ωt) 1 i(kz−ωt+ π ) ER = Re E0 √ ˆx e + √ ˆy e 2 2 2 à à !! 1 i(kz−ωt) i i(kz−ωt) = Re E0 √ ˆx e + √ ˆy e 2 2 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 |xi and in the y direction by |yi. These vectors like |xi 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 = 45oi = √1 (|xi + |yi) 2 Polarization at an angle θ in the x, y plane, the directionn ˆθ, can be described by |θi = cos θ |xi + sin θ |yi Right and Left circular polarization can be described by 1 1 |Ri = √ (|xi + i |yi) and |Li = √ (|xi − i |yi) 2 2 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. Basis states can be chosen to be |xi and |yi or |Ri and |Li or |θi and |θ⊥i and all states can be written as linear combinations of two orthogonal basis states. What does the State Vector mean? “A state vector is not a property of a physical system, but rather represents an experimental procedure for preparing or testing one or more physical systems.” F. Lalo¨e[4]quoting A. Peres.[5] There are other interpretations of the state vector: it may refer to an ensemble of identically prepared systems or even to a single system. See Le Bellac[2] page 97 footnote. 1.4 Inner Product of State Vectors With ordinary vectors the inner product can be written as A · B or B · A but with state vectors we define for each vector a “dual” vector. There is a one-to-one correspondence between a vector and its dual vector. From a more mathematical perspective the dual vector provides a mapping from the ket vector onto com- plex numbers (the inner product). see Cohen-Tannoudji[6] page 110. The dual vector corresponding to c|θi is c∗hθ|. Notice the complex conjugate of the constant c. The inner products of the basis states are hx|xi = 1 and hy|yi = 1 and hy|xi = 0. Suppose |ψi = λ|xi + µ|yi hψ| = λ∗hx| + µ∗hy| |φi = α|xi + β|yi hφ| = α∗hx| + β∗hy| 2 where |xi and |yi are orthogonal and normalized. Show that hψ|φi = hφ|ψi∗ and from this we see that hψ|ψi is real. 1.4.1 Polarization Analyzers An ideal Polaroid sheet has these properties: a) When light goes through a Polaroid sheet (analyzer) the outgoing light is polarized in the transmission direction of the analyzer. b) The Polaroid sheet transmits, selects (or projects) the component of the incoming polar- ization state which is along its transmission direction. The outgoing state is the component of the state vector of the incoming light projected along 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. ~ Recall the analogy with ordinary vectors, Ex = ˆx · E For example if the incoming light is polarized at angle θ and the analyzer transmission direction is in the y direction then the outgoing amplitude is hy|θi = hy| (cos θ |xi + sin θ |yi) = cos θhy|xi + sin θhy|yi = sin θ After the analyzer the polarization direction is y and the amplitude is sin θ. 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. 2 In this example the intensity I = I0 sin θ. Question: One light beam is linearly polarized at 45 degrees; a second beam is Right Circu- larly polarized. Each of these beams is sent through a Polaroid whose transmission direction is y. How do the intensities compare? How can you distinguish between those two beams of light using Polaroid filters? If we multiply a state vector by eiα does the intensity change? 1.5 Photons In the photon description of light we have to associate a polarization state with a beam of photons which has passed a Polaroid filter or has been prepared in some way. We can assign a polarization state |xi, |yi, |θi, |Ri, etc. to a beam of photons. We prepare a beam of photons in a state |θi by sending the light through a Polaroid filter whose transmission axis is at angle θ. Any photon observed after the Polaroid filter will be in the state |θi. The angle θ is measured with respect to the x axis. 3 We expect to find the same intensity relationships using the photon picture as with the wave picture. Recall that each photon carries a quantum of energy (E = hν) and that the intensity will be proportional to the number of photons. The probability that a photon in the state |θi will be transmitted by a Polaroid sheet whose transmission axis is in the x direction is cos2 θ. This is expected since the result must agree with the classical picture. (The energy is proportional to the square of the Electric field.) Suppose we have a beam prepared in the state |θi. 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 y the amplitude is hy|θi. The “amplitude” is sometimes called the “probability amplitude”. The Probability for finding a photon after the analyzer is the magnitude squared of the amplitude; the probability is |hy|θi|2. If the states are normalized then the probability will be ≤ 1. Of course after the analyzer any photon that we observe will be in state |yi but we know (almost) nothing about the state of the photon before the analyzer. We must not conclude that the photon was in state |yi before the analyzer. “quantum theory is incompatible with the proposition that measuremements are processes by which we discover some unknown and pre-existing property.” F. Lalo¨e[4]quoting A. Peres.[5] 1.6 Polarization States The most general polarization state is |Φi = λ|xi+µ|yi where λ and µ are in general complex with |λ|2 + |µ|2 = 1. Then |Φi is normalized. In the case of linear polarization λ and µ are real and can be written as cos θ and sin θ. The basis states, |xi and |yi, form an orthonormal basis. The state of a system can be expanded in a linear combination of basis states. The basis is said to be “complete” if any state can be written as a linear combination of basis states. Another orthonormal basis can be written as |θi = cos θ |xi + sin θ |yi |θ⊥i = − sin θ |xi + cos θ |yi. In the case of elliptical polarization (circular is a special case) the coefficients are complex and can be written as λ = cos θ and µ = sin θ eiδ. π π For circular polarization θ = 4 and δ = 2 . The overall phase is not significant; a state can be multiplied by eiα with no physical conse- quence. However if we form a superposition of states the relative phase is certainly impor- tant. 4 2 Principles of Quantum Mechanics 2.1 Postulates - Preliminary Version 1. The state of a system is represented by a vector |Φi. This is chosen to be normalized and |hΦ|Φi|2 = 1.