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Wave-Particle Duality Dr. Rohit Narula Electromagnetic waves or photons? Wave-Particle Duality PYL100: Electromagnetic Waves and Quantum Mechanics (Fall 2016) Dr. Rohit Narula1 1 Department of Physics The Indian Institute of Technology, Delhi September 15, 2016 Wave-particle duality The principle of spectral decomposition Matter or de Broglie waves Outline Wave-Particle Duality Dr. Rohit Narula Electromagnetic waves or photons? Wave-particle duality Electromagnetic waves or photons? The principle of spectral decomposition Matter or de Broglie waves Wave-particle duality The principle of spectral decomposition Matter or de Broglie waves Is light wave- or particle-like? Wave-Particle Duality Dr. Rohit Narula Electromagnetic waves or photons? Wave-particle duality The principle of spectral decomposition Matter or de Broglie waves ? Refection of light from a mirror Wave-Particle Duality Dr. Rohit Narula Electromagnetic waves or photons? Wave-particle duality The principle of spectral decomposition Matter or de Broglie waves Figure: [Image from http://images.tutorvista.com] I On observing that light could bounce back upon reflection from a mirror (See Fig 1), Sir Isaac Newton deemed light to consist of a beam of particles. Interference of light Wave-Particle Duality Dr. Rohit Narula Electromagnetic waves or photons? Wave-particle duality The principle of spectral decomposition Matter or de Broglie waves Figure: [Image from Quantum Mechanics by Claude Cohen-Tannoudji] I I I (b) The intensity pattern for each of the slits open separately. (c) The observed intensity pattern when both slits are open simultaneuosly I 6= I1 + I2 . The pattern on the screen suggests a wave-like interpretation. Diffraction of light Wave-Particle Duality Dr. Rohit Narula Electromagnetic waves or photons? Wave-particle duality The principle of spectral decomposition Matter or de Broglie waves I Wave-like. Black-body radiation Wave-Particle Duality Dr. Rohit Narula Electromagnetic waves or photons? Wave-particle duality The principle of spectral decomposition Figure: Spectral radiance as a function of wavelength. I I I In 1900, the study of black body radiation (the EM radiation within or surrounding a body in thermodynamic equilibrium with its environment, or emitted by an opaque, non-reflective body) led Planck to suggest the hypothesis of the quantization of energy i.e., for an EM wave of frequency ν, the only possible energies are integral multiples of h: the Planck’s constant. Matter or de Broglie waves Wave-Particle Duality The photoelectric effect Dr. Rohit Narula Electromagnetic waves or photons? Wave-particle duality The principle of spectral decomposition Matter or de Broglie waves Figure: The photoelectric effect. [Image from Wikipedia.] I In 1905 Einstein explained the photoelectric (the emission of electrons from a material when light is shone on it). I KEmax = hν − φ, I quantization of energy of light. (1) The Compton effect Wave-Particle Duality Dr. Rohit Narula Electromagnetic waves or photons? Wave-particle duality The principle of spectral decomposition Matter or de Broglie waves Figure: The Compton effect. [Image from Wikipedia.] I I I I Twenty years later, the photon was finally shown to exist via the Compton effect that describes the inelastic scattering of a photon by a charged particle, usually an electron. The change in wavelength on scattering is given by: h (2) λ0 − λ = (1 − cos θ) me c Particle-like. Wave-Particle Duality Conclusions from the experiments suggesting particle-like behaviour? I I the interaction of an EM wave with matter occurs by means of elementary, indivisible processes in which radiation appears to be composed of particles, the photons. The particle parameters such as the energy E and the momentum p of a photon, and the wave parameters such as the angular frequency ω = 2πν and the wave vector k = 2π/λ are linked by the fundamental relation: E = hν = ~ω p = ~k I Dr. Rohit Narula (Planck-Einstein relations) (3) where ~ = h/2π is given in terms of Planck’s constant h = 6.62 × 10−34 J s. During each elementary process, energy and total momentum must be conserved! Electromagnetic waves or photons? Wave-particle duality The principle of spectral decomposition Matter or de Broglie waves Wave-Particle Duality Returning to Young’s double-slit experiment Dr. Rohit Narula Electromagnetic waves or photons? Wave-particle duality The principle of spectral decomposition Figure: [Image from Quantum Mechanics by Claude Cohen-Tannoudji] I Matter or de Broglie waves Should we abandon the wave theory? I I (x) 6= I1 (x) + I2 (x) I I (4) The wave theory provides a satisfying interpretation of the fringes. i.e., the fringes are due to the superposition of the electric fields emanating from each of the slits. E(x) = E1 (x) + E2 (x) (5) Returning to Young’s double-slit experiment Wave-Particle Duality Dr. Rohit Narula Electromagnetic waves or photons? Wave-particle duality The principle of spectral decomposition Matter or de Broglie waves I Since I (x) ∝| E(x) |2 we get that, I (x) ∝| E(x) |2 =| E1 (x) + E2 (x) |2 =| E1 (x) |2 + | E2 (x) |2 +2 | E1 (x) · E2 (x) | =| E1 (x) |2 + | E2 (x) |2 +2 | E1 (x) || E2 (x) | cos(δ) (6) I where 2 | E1 (x) · E2 (x) |2 is the interference term that depends on the phase difference δ between E1 (x) and E2 (x). Returning to Young’s double-slit experiment Wave-Particle Duality Dr. Rohit Narula Electromagnetic waves or photons? Wave-particle duality The principle of spectral decomposition Matter or de Broglie waves I What happens when the source S emits the photons practically one by one? I If we replace the screen E with a photographic plate and increase the exposure time, we find that the fringes develop eventually, I thus rejecting the hypothesis that the fringes are due to some interaction between the photons. Returning to Young’s double-slit experiment Wave-Particle Duality Dr. Rohit Narula Electromagnetic waves or photons? Wave-particle duality The principle of spectral decomposition Matter or de Broglie waves I If we shorten the exposure time of the photographic plate we find that instead of very faint smears (a weak interference pattern) on the photographic plate, I we get well-defined, discrete points indicating that each photon results in a localized impact I thus rejecting the hypothesis of a purely wavelike view. Returning to Young’s double-slit experiment Wave-Particle Duality Dr. Rohit Narula Electromagnetic waves or photons? Wave-particle duality The principle of spectral decomposition Matter or de Broglie waves I As we increase the integration time of the photographic plate from zero we find that the dots that correspond to individual photons initially appear to be randomly distributed I (hinting at the probabilistic nature of the process), I and slowly, as the integration time is increased, the image begins taking the shape of the fringe pattern. Returning to Young’s double-slit experiment Wave-Particle Duality Dr. Rohit Narula Electromagnetic waves or photons? Wave-particle duality I I I I Since we have eliminated the possibility of photon-photon interactions (by having the source S emit photons one by one only) we are presented with the intriguing paradox that the photon passing through one slit, seems to be aware of the presence of the other slit. By placing a photodetector, we immediately observe that the fringe pattern washes out, and we simply the sum of the intensities of the individual slits, i.e., I (x) = I1 (x) + I2 (x). Thus when one performs a measurement on a microscopic system, one disturbs it in a fundamental way. The principle of spectral decomposition Matter or de Broglie waves Wave-Particle Duality for EM Waves Wave-Particle Duality Dr. Rohit Narula The particle and wave aspects of light are inseparable. Light behaves simultaneously like a wave and a like a flux of particles, the wave enabling us to calculate the probability of the manifestation of a particle. Electromagnetic waves or photons? I Predictions about the behaviour of the photon can only be probabilistic. Matter or de Broglie waves I The information about a photon at time t is given by the wave E(r , t), which is a solution of Maxwell’s euqations. I This wave characterizes the state of the photon at time t. E(r , t) is interpreted as the probability amplitude of a photon appearing, at time t, at the point r . I This means that the corresponding probability is proportional to | E(r , t) |2 I Wave-particle duality The principle of spectral decomposition Wave-Particle Duality for EM Waves I I I I I I Since Maxwell's equations are linear and homogenous, we can use a superposition principle: if E1 and E2 are separately the solutions of the Maxwell's, then λ1 E1 + λ2 E2 is also a solution, where λ1 and λ2 are constants. It is this superposition principle which explains wave phenomena in classical optics (interference, diffraction). In quantum phyiscs, the interpretation of E(r , t) is of a probability amplitude. Quantum theory merely allows one to calculate the probability of occurence of an event. Experimental verification must thus be founded on statistics, or conducting a large number of identical trials. The interference pattern so obtained in the Young's double-slit experiment is a manifestation of these probabilities. Wave-Particle Duality Dr. Rohit Narula Electromagnetic waves or photons? Wave-particle duality The principle of spectral decomposition Matter or de Broglie waves Wave-Particle Duality The principle of spectral decomposition Dr. Rohit Narula Electromagnetic waves or photons? Wave-particle duality The principle of spectral decomposition Matter or de Broglie waves I Imagine a plane monochromatic wave, polarized in the p direction, and moving in the z direction: E(z, t) = E0 p̂e i(kz−ωt) . I After passing through an analyzer oriented along the Ox direction it becomes, E 0 (z, t) = E00 x̂e i(kz−ωt) , I (7) intensity I 0 is ∝ to | E00 |2 is given by Malus' law: (8) Wave-Particle Duality The principle of spectral decomposition Dr. Rohit Narula Electromagnetic waves or photons? Wave-particle duality The principle of spectral decomposition Matter or de Broglie waves I I I I Suppose we release the photons, one by one. In the above experiment, there are only two possible results: the photon crosses the analyzer or is stopped. In a sense, the possible results of measurement from the analyzer are quantized, which we shall call eigen results. To each of these eigen results, corresponds an eigenstate. The eigenstates are characterized by: p̂ = x̂ p̂ = ŷ . and, (10) The principle of spectral decomposition Wave-Particle Duality Dr. Rohit Narula Electromagnetic waves or photons? Wave-particle duality The principle of spectral decomposition Matter or de Broglie waves I If p̂ = x̂ then we know with certainty that the photon will traverse the analyzer; I if p̂ = ŷ , it will, on the contrary, definitely be stopped. I Thus, if the particle is, before the measurement, in one of the eigenstates, the result of this measurement is certain: it can only be the associated eigen result. Wave-Particle Duality The principle of spectral decomposition Dr. Rohit Narula Electromagnetic waves or photons? Wave-particle duality The principle of spectral decomposition Matter or de Broglie waves I When the state before the measurement is arbitrary, only the probabilities of obtaining the different eigen results can be predicted. I To find these probabilities, one decomposes the state of the particles into a linear combination of the various eigenstates. Thus, p̂ = cos θx̂ + sin θŷ (11) Wave-Particle Duality The principle of spectral decomposition Dr. Rohit Narula Electromagnetic waves or photons? Wave-particle duality The principle of spectral decomposition Matter or de Broglie waves I I I The probability of obtaining a given eigen result is then proportional to the square of the absolute value of the coefficient of the corresponding eigenstate. For Px̂ , | cos θ |2 Px̂ = (12) | cos θ |2 + | sin θ |2 For Pŷ , Pŷ = | sin θ |2 | cos θ |2 + | sin θ |2 (13) Wave-Particle Duality Matter or de Broglie waves Dr. Rohit Narula Electromagnetic waves or photons? Wave-particle duality The principle of spectral decomposition Matter or de Broglie waves Just as light has both wave-like and particle-like properties, electrons and other matter also have wave-like properties. Thus, in analogy with EM waves where we have, E = hν = ~ω p = ~k (14) Wave-Particle Duality Matter or de Broglie waves Dr. Rohit Narula Electromagnetic waves or photons? Wave-particle duality The principle of spectral decomposition Matter or de Broglie waves de Broglie hypothesized in Ph.D. thesis of 1924 that the wavelength λdB associated with an electron is: λdB = I I 2π h = k |p| (15) It is important to note that the very small value of Planck’s constant h explains why the wavelike nature of matter is very difficult to demonstrate on a macroscopic scale. de Broglie Waves and the Bohr model of the H atom Wave-Particle Duality Dr. Rohit Narula Electromagnetic waves or photons? Wave-particle duality The principle of spectral decomposition Matter or de Broglie waves Neil Bohr’s model of the hydrogen atom was successful in explaining the Rydberg formula: En = −13.6 n2 where En is the energy level (in eV ) of the nth energy transition of the hydrogen atom. (16) de Broglie Waves and the Bohr model of the H atom Wave-Particle Duality Dr. Rohit Narula Electromagnetic waves or photons? Wave-particle duality The principle of spectral decomposition I The Bohr model employed the use of the condition that the angular momentum L is an integer multiple of ~: L = mvr = I nh = n~ 2π (17) de Broglie's result led to a reinterpretation of this condition as a standing wave condition or a condition for constructive interference of an electron in a circular orbit, i.e., 2πr = nλn , (18) Matter or de Broglie waves de Broglie Waves and the Bohr model of the H atom Wave-Particle Duality Dr. Rohit Narula Electromagnetic waves or photons? Wave-particle duality The principle of spectral decomposition Exercise 1. Show that the de Broglie hypothesis leads to the quantization of the angular momentum L for a circular orbit in Bohr’s model of the hydrogen atom. Matter or de Broglie waves