MP 11.pdf

Grade 3 MODERN PHYSICS Prof Azeez A. Barzinjy (PhD at Leicester University) PHYS 302/P, Fall Term, Week 12, 17-21/12/2023 Course content: √ √ √ √ √ √ √ √ √ √ Week 12 The Double-slit Experiment The objectives of this lecture: i. ii. iii. iv. v. Unifications in physics What is light? W...

Grade 3 MODERN PHYSICS Prof Azeez A. Barzinjy (PhD at Leicester University) PHYS 302/P, Fall Term, Week 12, 17-21/12/2023 Course content: √ √ √ √ √ √ √ √ √ √ Week 12 The Double-slit Experiment The objectives of this lecture: i. ii. iii. iv. v. Unifications in physics What is light? Wave How fast does the wave move? Superposition—a Characteristic of All Waves vi. Interference: Result of the Superposition of Waves vii. Optical path length difference viii.Young’s Double Slit Experiment ix. The Visible Spectrum and Dispersion x. Taylor experiment Unifications in physics Mechanical theory Electromagnetic theory (Newton, 1687) Entities: particles, inertia, force that lies along a line between interacting particles. (Maxwell, 1864) Entities: electric & magnetic fields, force that is perpendicular to both field & motion. · · · celestial motions terrestrial motions, in 3-D heat (kinetic theory) · · · magnetism electricity optics Relativity theory (Einstein, 1905 & 1916) unites all phenomena above Quantum theory (Planck 1900; Einstein 1905,16,17; Bohr & Heisenberg 1925; Schrodinger 1926; Dirac 1927; Bethe, Tomonaga, Schwinger, Feynman & Dyson 1940s …) · atoms and nuclei Dissolves the classical distinction between point particles & non-local fields/waves. Quantum objects manage both at once. What is light? In the late 1600’s Newton explained many of the properties of light by assuming it was made of particles. ”Tis true, that from my theory I argue the corporeity of light; but I do it without any absolute positiveness…” Because of Newton’s enormous prestige, his support of the particle theory of light tended to suppress other points of view. “The waves on the surface of stagnating water, passing by the sides of a broad obstacle which stops part of them, bend afterwards and dilate themselves gradually into the quiet water behind the obstacle. But light is never known to follow crooked passages, nor to bend into the shadow.” In 1678 Christian Huygens argued that light was a pulse traveling through a medium, or as we would say, a wave. I’m thinking waves. Wave •Variation (disturbance) of physical quantity that propagates through space • Often: oscillation in space and time y(x,t) = A sin (kx − ωt) . y • Phase of this wave θ(x,t) = kx − ωt . • if you are moving with the wave, phase is constant (for =/2, you sit at the maximum) x How fast does the wave move? • if  is constant with time dθ dx 0= =k −ω . dt dt • phase velocity: dx  vp = = . dt k y x Imagine yourself riding on any point on this wave. The point you are riding moves to the right. The velocity it moves at is vp. If the wave is moving from left to right then /k must be positive. Superposition—a Characteristic of All Waves When waves of the same nature arrive at some point at the same time, the corresponding physical quantities add. Example: If two electromagnetic waves arrive at a point, the electric field is the sum of the (instantaneous) electric fields due to the two waves. Implication: Intensity of the superposed waves is proportional to the square of the amplitude of the resulting sum of waves. Interference: Result of the Superposition of Waves Constructive Interference: If the waves are in phase, they reinforce to produce a wave of greater amplitude. Destructive Interference: If the waves are out of phase, they reinforce to produce a wave of reduced amplitude. Optical path length difference • two sources emit waves in phase • waves travel different distances L1 and L2 to point of interest • optical path difference L = L1 - L2 determines interference 4 5 L = m  In phase—constructive L = (m+1/2)  Out of phase—destructive Young’s Double Slit Experiment • famous experiment, demonstrates wave nature of light • single light source illuminates two slits, each slit acts as secondary source of light • light waves from slits interfere to produce alternating maxima and minima in the intensity Young’s double slit experiment In 1803 Thomas Young’s double slit experiment showed that, much like water waves, light diffracts and produces an interference pattern. Light must be waves! =2dsin Example “…it seems we have strong reason to conclude that light itself is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws.” In the 1860’s Maxwell, building on Faraday’s work, developed a mathematical model of electromagnetism. He was able to show that these electromagnetic waves travel at the speed of light. Double Slit Experiment Double-slit Experiment How does this work? At some locations on the screen, light waves from the two slits arrive in phase and interfere constructively. At other locations light waves arrive out of phase and interfere destructively. Interference: Young’s Double-Slit Experiment We can use geometry to find the conditions for constructive and destructive interference: Interference: Young’s Double-Slit Experiment Between the maxima and the minima, the amplitude varies smoothly. The Visible Spectrum and Dispersion ◦ Wavelengths of visible light: 400 nm to 750 nm ◦ Shorter wavelengths are ultraviolet; longer are infrared Conditions for Interference Why the double slit? Can't I just use two flashlights? • sources must be coherent maintain a constant phase with respect to each other • sources should be monochromatic - contain a single wavelength only Conditions for Interference in the Double Slit Experiment. Constructive Interference: L1 L = d sin  = m, m=0, 1, 2...  d  L2 L = L2 –L1 = d sin  Destructive Interference: 1  L = d sin  =  m+  , m=0, 1, 2... 2  The parameter m is called the order of the interference fringe. The central bright fringe at  = 0 (m = 0) is known as the zeroth-order maximum. The first maximum on either side (m = ±1) is called the first-order maximum. Conditions for Interference in the Double Slit Experiment. For small angles: y = R tan   R sin  L1 S1 y  L2 Bright fringes: m = d sin   d S2 L y tan  = R R P m = d y= y R R m d This is not a starting equation! Do not use the small-angle approximation unless it is valid! Conditions for Interference in the Double Slit Experiment. For small angles: y = R tan   R sin  L1 S1 y  L2 1  m +    = d sin  2   d S2 L tan  = Dark fringes: y R P 1 y  m +  = d   2 R  y= R R  1 m +   d  2 This is not a starting equation! Do not use the small-angle approximation unless it is valid! Example Example: a viewing screen is separated from the double-slit source by 1.2 m. The distance between the two slits is 0.030 mm. The second-order bright fringe (m = 2) is 4.5 cm from the center line. Determine the wavelength of the light. y = R tan   R sin  S1 Bright fringes: m = d sin  m = d yd = Rm  S2 = ( 4.5  10 -2 )( m 3.0  10-5 m (1.2 m)( 2 ) ) = 5.6  10 −7 m = 560 nm L y L2  d y R L1 tan  = R y R P Example Example: a viewing screen is separated from the double-slit source by 1.2 m. The distance between the two slits is 0.030 mm. The second-order bright fringe (m = 2) is 4.5 cm from the center line. Find the distance between adjacent bright fringes. y = R tan   R sin  Bright fringes: S1 m = d sin  y m = d R y m+1 -y m = R R R = ( m + 1) − m = d d d  S2 (5.6  10 m) (1.2 m) = 2.2  10( 3.0  10 m) L2 L tan  = R −7 -5 y  d R y= m d L1 2 m = 2.2 cm y R P Example Example: a viewing screen is separated from the double-slit source by 1.2 m. The distance between the two slits is 0.030 mm. The second-order bright fringe (m = 2) is 4.5 cm from the center line. Find the width of the bright fringes. Define the bright fringe width to be the distance between two adjacent destructive minima. y dark 1  m +  = d sin  = d   2 R  y dark S1 R  1 = m +  d  2 y dark,m+1 -y dark,m =  y dark,m+1 -y dark,m = S2 (5.6  10 m) (1.2 m) = 2.2 cm ( 3.0  10 m) −7 -5 L y L2  d R  1  R  1  R  ( m + 1) +  − m +  = d  2 d  2 d L1 tan  = R y R P Taylor experiment “It would seem that the basic idea of the quantum theory is the impossibility of imagining an isolated quantity of energy without associating with it a certain frequency.” In 1909 G.I. Taylor experimented with a very dim light source. His work, and many modern experiments show that even though only one photon passes through a double slit, over time, an interference pattern is still produced - one “particle” at a time. Louis de Broglie, in 1923, reasoned that if light waves could =h/p behave like particles then particles should have a wavelength. Soon after, an experiment by C. J. Davisson and L. H. Germer showed that electrons could produce interference patterns just like those produced by light. n=2dsin Heisenberg’s Uncertainty Principle Heisenberg’s Uncertainty Principle helps us examine the dual nature of light, electrons, and other particles. p=/h If we add together many wavelengths... If we know the wavelength we are certain about the momentum. But, because a wave is spread out in space, we are uncertain about its position. { We are uncertain about the momentum. { But, we are now more certain about position. Heisenberg’s Uncertainty Principle A similar experiment can be done with beam splitters and mirrors. Photons striking a double slit, one at a time, produce interference. If we observe which slit the photon chooses… the interference pattern disappears. Quantum Keys ZUP MOS How can Alice and Bob know that their communications will remain private? Eve may measure a classical signal without detection. We do not know how much Eve has learnt about the key! When evolving freely, quantum systems exhibit wave character. When measured, character. quantum Measurements that quantum system. acquire systems exhibit information particle perturb the Eve’s measurements of a quantum signal causes perturbation and can be detected. Electron diffraction de Broglie h h l= = me v p polycrystalline target (a 2-dimensional grating) The double-slit experiment with increasing numbers of electrons: a: 10 electrons b: 200 c: 6000 d: 40 000 e: 140 000 Double Slit Experiment ◦ What happens if we close one of the slits? ◦ No interference pattern. Just the diffraction pattern from the single open slit. Double Slit Experiment ◦ What happens if we monitor which slit the single photon entered? Detector on No interference pattern Detector off Interference pattern 35 Double Slit Experiment ◦What happens if we use electrons? ◦No difference in any of the preceding discussion 36 Double Slit Experiment ◦ The act of observing the electron has changed the experiment pelectron  p photon  h electron h  photon h ~ ; where d is the slit separation d h  d ◦ Thus the photon momentum is at least as large as that of the electron and will change the direction of the electron (destroying the interference pattern) 37 Double Slit Experiment There is a lot going on here! Consider the example where the photon or electron is measured in order to determine through which slit it passed If the photon through slit 1 is detected with a photodetector it is removed (and equivalent to blocking slit 1) If an electron through slit 1 is measured using light (e.g. Compton scattering), again the interference pattern vanishes 38 Problem set 9 Q1. At what angle is the first-order maximum for 450-nm wavelength blue light falling on double slits separated by 0.0500 mm? Q2. Calculate the angle for the third-order maximum of 580-nm wavelength yellow light falling on double slits separated by 0.100 mm. Q3. What is the separation between two slits for which 610-nm orange light has its first maximum at an angle of 30.0º? Q4. Find the distance between two slits that produces the first minimum for 410-nm violet light at an angle of 45.0º. Q5. Calculate the wavelength of light that has its third minimum at an angle of 30.0º when falling on double slits separated by 3.00 μm. Q6. What is the wavelength of light falling on double slits separated by 2.00 μm if the third-order maximum is at an angle of 60.0º? Q7. At what angle is the fourth-order maximum for the situation in Question 1?