## Elementary Wave Optics

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Category Archives: Waves & Wave Mechanics

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Also in 1927, Heisenberg made another major contribution to quantum physics. He was not attracted to the idea of a military or diplomatic career. We are doing this so that you can have some qualitative feeling for some quantum phenomena before we get into the mathematical details of quantum mechanics. For instance, in the case of the figure above, where the momentum space wave function was localized around the pink-red momentum, while the wave function was spread all over the spectrum.

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Imagine them as two quantities that, when added together, cannot equal more than 1. The experiment that we're interested in is a "slit experiment," and it goes something like this. According to Physics 2000 (a site from the University of Colorado), Bohr proposed that electrons were restricted to "special" orbits around an atom's nucleus. The location of a subatomic particle is not defined until it is observed (such as striking a screen). Thermal expansion: increase of length or volume of object due to change in temperature.

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The white indicates where the displacement is large and positive, while the black indicates where it is large and negative. In 1749, the wave equation was improved upon by Leonhard Euler; he began to apply d'Alembert's theories to all wave forms, not just strings. Bohm's work, in its turn, was carried on and kept alive in Refs.[16-19], but remained almost ignored for many years, and began to bear fruit in chemical physics and nanoscale systems just before the beginning of the present century, raising a widening trend [20-24].

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For instance, consider the dot product of a four-vector A which resolves into (Ax, At ) in the unprimed frame. Absolute temperature, 7 Acceleration, 109, 110, 116, 143 centripetal, 112, 113 in relativity, 114 intrinsic, 114, 118 of wave packets, 143 Angle of incidence, 58 Angle of reﬂection, 58 Angular momentum, 160, 173, 197 conservation of, 201 deﬁnition, 200 location of origin, 200 orbital, 172 spin, 160, 175, 203 vector, 174 quantized length, 174 angular momentum orbital, 203 Aristotle, 142 Atom, 125 Barrier penetration, 159, 170, 171 and nuclear ﬁssion, 172 and radioactive decay, 172 Beam, 14, 15, 36, 38–41, 44, 57, 70 diameter, 40 orientation, 42, 43 spread, 46 stationary, 38 width, 40 Beats, 14 beat frequency, 14 Black hole, 109 Bose, Satyendra Nath, 176 Boson, 176 Bragg diﬀraction, 125, 129 by a powder, 127 by a single crystal, 127 of electrons, 125, 129, 131 of X-rays, 125 Bragg’s law, 125, 127 test of quantum mechanics, 136 Calcite crystal, 57, 60, 61 Center of mass, 202 no relativistic generalization, 204 position, 202 velocity, 202 Centrifugal force, 117 Charge conjugation, 167 Circular motion, 111 Classical mechanics, 138 and geometrical optics, 147 Clocks in relativity, 85 Closed system, 188 Collisions, 184 elastic, 184 inelastic, 184, 187 Collocation, 81 reference frame dependence, 81 Compatible variables, 166, 174 and conservation, 167 227 228 Complex number, 160 absolute square, 162 complex conjugate, 162 graphical representation, 161 real and imaginary parts, 160 Conﬁnement, 159 and potential energy, 168 in classical mechanics, 159 in quantum mechanics, 159 Contour plot, 36, 37 Conveyor belt, 190 Coordinate system, 31 Cartesian, 31, 32 choice of, 33 rotated, 32, 33 Crystal, 125 scattering of X-rays, 125 Davisson, Clinton, 129 De Broglie, Louis, 131 Nobel Prize address, 147 Derivative, 17 partial, 141, 149 Diﬀraction, 29, 125, 143 pattern, 29, 48 Diﬀraction grating, 29, 48, 125, 127 resolution, 50 Dispersion relation, 19, 35 anisotropic, 41, 57, 60, 61 dispersive, 6, 20, 22, 23 isotropic, 38, 42, 60 non-dispersive, 6, 21 relativistic wave, 95, 100 Doppler shift, 95, 101, 120 light, 101 sound, 102 Double slit, 47, 130 Dynamics, 141 INDEX Newtonian, 141, 142 and geometrical optics, 149 pre-Newtonian, 141, 142 quantum mechanical, 141 Einstein, Albert, 75, 80, 109, 117, 129, 131, 143 Electromagnetic radiation, 125 Electromagnetism, 75 Electron, 125 as matter wave, 125, 129 particle and wave properties, 130 Energy, 125, 131 collisions, 184 conservation of, 144 and conservative forces, 144 kinetic, 144 non-relativistic, 135 of free particle, 163 potential, 143 total, 144 Energy level diagram, 169 Equivalence principle, 109, 117 Euler’s equation, 161 inverse, 162 Event, 76 Event horizon, 109, 121 Exhaust velocity, 189 Fermat’s principle, 66, 68–70 maximum time, 68 minimum time, 68 Fermi, Enrico, 176 Fermion, 176 Feynman, Richard, 69, 70, 169 view of quantum mechanics, 130 Force, 109, 116 central, 200 angular frequency, 216 angular momentum conservation, diﬀerential equation, 215 202 conservative, 141, 143, 149 energy analysis, 214 forced, 216 inertial, 116 mass-spring system, 213 internal and external, 182 Newton’s laws analysis, 215 short range in collisions, 184 quantum, 218 Four-momentum, 133 quantum frequencies, 219 Four-vector, 95, 97 resonance, 217 direction, 97 Heisenberg uncertainty principle, 125, dot product, 97 136 magnitude, 97 and wave packets, 136 position, 97 position-momentum, 138 spacelike component, 97 proper time-mass, 138 timelike component, 97 time-energy, 138 wave, 97, 98, 101 Hooke’s law, 213 Free particle, 163 spring constant, 213 in quantum mechanics, 163 wave function for, 163 Image, 64 Frequency, 3, 42, 95, 131 real, 65, 66 angular, 3, 4, 35 virtual, 65, 66 of wave, 3 Inclined ramp, 151 rotational, 3 Index of refraction, 7, 16, 58, 59, 62 spatial variations, 147 Galilei, Galileo, 75 Inertial force, 117 Geiger counter, 129 Interference, 8, 29, 143 Geometrical optics, 17, 57, 66 constructive, 8, 16, 38, 42, 47, 70, and classical mechanics, 138, 141 127 Germer, Lester, 129 destructive, 8, 16, 38, 48, 70 Gravitational ﬁeld, 119, 145 order, 48, 127 Gravitational red shift, 109, 119 Interferometer, 14 Gravity, 109 Michelson, 14, 15 as a conservative force, 145 optical, 15 as inertial force, 117 thin ﬁlm, 14, 16 Group velocity, 19, 22, 131 Interval in spacetime, 84 and wave packets, 19, 101 meaning, 85 of matter waves, 148 spacelike, 84 timelike, 84 Harmonic oscillator, 213 230 Invariance, 166 and deﬁniteness, 165 displacement in time, 165 under displacement, 163 under rotation, 174 up to a phase factor, 164 slope, 82, 88 Lorentz contraction, 87, 88 Mass, 109, 116, 131 and rest frequency, 133 in inelastic collision, 188 physical meaning, 132 Matter wave, 125 Kinematics, 141 dispersion relation, 147 of waves, 1 displacement of, 129 Kinetic energy, 134, 144, 148 example of relativistic wave, 95 inelastic collision, 187 refraction, 151 internal, 203 theory of, 147 non-relativistic, 135 two and three dimensions, 150 translational, 203 Michelson, Albert, 15 Kinetic frequency, 134 Mirror, 58 Law of reﬂection, 67 concave, 66 Law of refraction, 67 convex, 66 Lens, 63 curved, 63, 66 focal length, 64, 66 ellipsoidal, 69 negative, 65 focal length, 66 positive, 63, 64, 66 plane, 58 Light, 2 Moment of inertia, 205, 206 and photoelectric eﬀect, 132 for solid bodies, 206 and soap bubbles, 17 Momentum, 125, 131, 132 and thin ﬁlms, 16 and group velocity, 144 dispersion relation, 35, 147 and Newton’s second law, 181 Doppler shift, 101 collisions, 184 electromagnetic radiation, 2 conservation of, 183 in telescopes, 47 kinetic, 154 no special reference frame, 100 and particle velocity, 154 particle and wave properties, 129, non-relativistic, 135 130 of free particle, 163 slope of world line, 79 total (or canonical), 154 speed in matter, 7 and wave vector, 132, 154 speed in vacuum, 7, 75, 96 N¨ther’s theorem, 159 o transverse wave, 2 N¨ther, Emmy, 159 o Line of simultaneity, 77, 82, 118 INDEX Newton’s ﬁrst law, 142 Newton’s second law, 116, 142 accelerated reference frame, 117 and accelerating wave packets, 149 conservative force, 143 for open systems, 188 relativistic form, 181 rotational version, 200 Newton’s third law, 182 and Newton’s second law, 182 Newton, Isaac, 75 Nobel prize winners, 125, 126 Non-relativistic limit, 135 Open system, 188 conveyor belt, 190 mass equation, 189 Newton’s second law, 189 rocket, 188, 189 Parity, 167 Particle in a box, 159, 168 boundary condition, 172 non-relativistic, 169 ultra-relativistic, 169 Phase shift, 16 Phase speed, 4, 22, 23 Photoelectric eﬀect, 132 Photon, 129, 132 Planck’s constant, 132, 148 Planck, Max, 131 Plane wave, 29, 34, 36 Potential energy, 143, 144, 148, 165 barrier, 172 gravitational, 145 Power, 141, 147 in three dimensions, 153 total, 147 231 Principle of relativity, 77, 80, 184 and Newtonian dynamics, 142 Prism, 62 Probability, 130 in quantum mechanics, 131 Probability amplitude, 131 Proper time, 84 meaning, 85 relation to spacetime interval, 84 Pulse, 36 isolated, 38 Quantization, 169 angular momentum, 173 energy, 169 Quantum mechanics, 14, 95, 125, 129, 138, 143 bizarreness of, 129 eﬀect of measurement, 130 geometrical optics limit, 141 particles and waves, 135 probabilistic theory, 129 sense and nonsense, 130 two-slit interference, 130 uncertainty principle, 136 Quantum number, 169 angular momentum, 173, 175 energy, 169 orientation, 175 spin, 175 spin orientation, 175 Quark theory, 167 Ray, 57, 59, 61 Reference frame, 77, 78 accelerated, 89, 109, 113, 116, 118, 119 center of momentum, 184–187 232 for collisions, 184 inertial, 77, 80, 113 preferred, 77 lack of, 77 Reﬂection, 16, 143 in Bragg diﬀraction, 125 law of, 57 Refraction, 17, 143 law of (see Snell’s law), 57 Relativistic wave, 100, 133 dispersion relation, 100, 134 group velocity, 100, 134 phase speed, 100 Relativity, 75 Einsteinian, 80, 143 Galilean, 75, 77, 78, 80 general, 75, 109, 117 postulates, 81 special, 75, 78, 95 Rest energy, 133, 148 Rest frequency, 133 Rigid body, 206 Rocket, 189 acceleration, 190 exhaust velocity, 189 Scalar, 33 in spacetime, 98 Scattering, 125 Simultaneity, 81 Einsteinian relativity, 83 Galilean relativity, 81 reference frame dependence, 84 Sine wave, 2, 3, 8, 9, 29, 34, 36, 39 superposition, 13 Single slit, 46 Snell’s law, 59, 60, 62, 67, 68 and matter waves, 151 world line, 96 Wave function, 129, 131 collapse, 130 is complex, 129, 160 Uncertainty principle, 14 probability as absolute square, 129 Wave packet, 9, 22, 23, 29, 36, 46, 57 Vector, 29 acceleration of, 149 absolute value, 31 and group velocity, 19 addition, 31 and Newtonian dynamics, 143 components, 31, 32 isolated, 12, 42 cross product, 32, 197 of light, 147 component form, 198 speed of movement, 23 does not commute, 198 Wave period, 3 magnitude, 198 Wave phase, 4, 8, 15, 35, 96 right-hand rule, 198 Wave trough, 3, 22 direction, 31 Wave types, 4 displacement, 29 gravitational waves, 36 dot product, 32, 197 gravity waves, 36, 37 component form, 32, 34 light waves, 1, 7, 35, 37, 39 cosine form, 32, 34 ocean waves, 4, 37 head, 31 deep water, 5, 35 magnitude, 31 shallow water, 5 tail, 31 sound waves, 1, 6 unit vector, 31 vibrations in solids, 1 Velocity, 110, 112 constant intrinsic acceleration, 118 Wave vector, 34, 35, 151 central, 44, 57 Velocity addition direction, 36 Galilean, 77 magnitude, 36 relativistic, 95, 104, 113 perpendicular, 97 Wave-particle duality, 131 Wave, 1 Wavelength, 3, 15 longitudinal, 1 Wavenumber, 3, 4, 9, 95, 131 transverse, 1 central, 13, 19 Wave amplitude, 2, 7, 9, 44, 95 diﬀerence, 9 Wave crest, 3, 22 imaginary, 171 speed of movement, 23 spread, 12, 14 Wave displacement, 2, 14, 22, 35, 45, Work, 141, 146 143 by conservative force, 146 Wave front, 34, 43, 57, 96

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Even for the high temperatures in the center of a star, fusion requires the quantum tunneling of a neutron or proton to overcome the repulsive electrostatic forces of an atomic nuclei. Stand in front of a stereo or hi-fi loudspeaker on at full volume, and you can feel some of the vibrations from the music. So much so that he has adopted Bearden's view and given it a different name, "hyperdimensional physics." The medium is the material that’s in the wave.

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Because he could imagine constructing a continuous chain of interconnected mathematical wave-functions, from observed particle through actively-observing device to passively-observing human, he concluded that anything composed of quantum-matter cannot “collapse the wave-function” but human consciousness can do this. So does this persuade proponents of Mystical Physics? Even a locked room was not a problem since Jesus appeared to his disciples inside as locked room and then He then disappeared again.

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The assessment in this route is through a one off opportunity of a practical activity. This conﬁrms that the oscillation frequency and period are independent of amplitude. The wavelength, λ, of a wave is the distance from any point on one wave to the same point on the next wave along. (The symbol is a Greek letter, 'lambda'.) To avoid confusion, it is best to measure wavelength from the top of a crest to the top of the next crest, or from the bottom of a trough to the bottom of the next trough.

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That it is not Nature which is strange, but our incorrect conceptions of Nature! In addition to his extensive work in optics, Newton is perhaps best known for his theory of universal gravitation. A single clock being in a superposition of two locations allows probing quantum interference effects in combination with general relativity. 2 = 4 sin2 (kx). (9.18) Which of the two possible values of the momentum the particle takes on is unknowable, just as it is impossible in principle to know which slit a particle passes through in two slit interference.

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In spite of the parallels, there is a crucial difference between QED and QCD. For example, the angular momentum of an electron bound to an atom or molecule is quantized and can only have values that are multiples of the reduced Planck constant. On the other hand, determination of the stable motion of electrons in the atom introduces integers, and up to this point the only phenomena involving integers in physics were those of interference and of normal modes of vibration.

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You could be forgiven for thinking that we haven’t got much closer to writing down the Schrödinger equation, but in fact there isn’t much more to do. It is not a theory accepted by modern physicists. Examples of longitudinal waves: Sound waves, tsunami waves, earthquake ,P - waves, ultra sounds, vibrations in gas, and oscillations in spring, internal water waves, and waves in slink etc. Attach the stand with the clip to demonstrate fixed boundary conditions. So four conditions-- continuity of psi and psi prime, continuity of psi and psi prime, four conditions.