PH253 Lecture 11: photons vs. electrons de Broglie waves P. LeClair Department of Physics & Astronomy The University of Alabama Spring 2020 LeClair, Patrick (UA) PH253 Lecture 9 February 5, 2020 1 / 28
Outline Better proof for photons? 1 Double slit experiment 2 LeClair, Patrick (UA) PH253 Lecture 9 February 5, 2020 2 / 28
Last time: Compton scattering: e − -photon scattering 1 Light behaved like particles . . . 2 . . . so long as h f ∼ m e − c 2 , or λ ∼ λ c 3 Implications for measuring position on small scales - uncertainty 4 Next: better proof for photons? 5 Next: why should e − and photons behave differently? 6 LeClair, Patrick (UA) PH253 Lecture 9 February 5, 2020 3 / 28
Open problems according to Einstein, ca. 1905 1 why does the appearance of a photochemical reaction depends 1 only on the color of light, and not on its intensity? why is short wavelength radiation generally more active 2 chemically than long wavelength radiation? why is the kinetic energy of cathode rays (electrons) produced by 3 the photoelectric effect independant on the light intensity? energy given to a light particle when it is emitted is not spread out 4 in infinite space, but remains available for an elementary absorption process i.e., light remains in “bundles” 5 All explained by photon model! 1 Adapted from P. Grangier, Séminare Poincaré 2 , 1-26 (2005) LeClair, Patrick (UA) PH253 Lecture 9 February 5, 2020 4 / 28
Outline Better proof for photons? 1 Double slit experiment 2 LeClair, Patrick (UA) PH253 Lecture 9 February 5, 2020 5 / 28
Single photon scattering Why not use a single photon source? 1 Coherent beam of individual, well-separated photons 2 Atom emits 2 photons of 2 frequencies a few ns apart 3 First one triggers detector to measure second one 4 Second one goes through a beam splitter 5 Which way does it go, or does it split? 6 Figure: P. Grangier et al , Europhysics Letters 1 , 173 (1986) LeClair, Patrick (UA) PH253 Lecture 9 February 5, 2020 6 / 28
Single photon scattering Source S ejects pairs of photons. 1 First ν 1 triggers measurement, counts how many emitted ( N 1 ) 2 Second photon ν 2 encounters beam splitter BS 3 First photon triggering ensures timing is good 4 Wave: both paths (coincident detection, N c ). 5 Particle: has to take one or the other ( N r or N t ) 6 Particle: never see both detectors fire at once 7 Figure: P. Grangier et al , Europhysics Letters 1 , 173 (1986) LeClair, Patrick (UA) PH253 Lecture 9 February 5, 2020 7 / 28
Single photon scattering “Anti-bunching”: never see simultaneous hits on both detectors 1 Photon can’t be split: either reflected or transmitted, 50/50 2 chance, never both at once Scan time delay between detectors τ 3 At zero delay (simultaneous detection), intensity → zero 4 Light is photons, individual particles! 5 Figure: Modern version. P. Grangier, Séminare Poincaré 2 , 1-26 (2005) LeClair, Patrick (UA) PH253 Lecture 9 February 5, 2020 8 / 28
Single photon scattering Interference with single photons? (Mach-Zehnder interferometer) 1 Vary path difference of the two arms = vary phase difference 2 Waves: expect interference. (Broadly similar to double slit) 3 One detector sees constructive, other destructive interference 4 If particles, same for either - 50/50 chance 5 Figure: P. Grangier et al , Europhysics Letters 1 , 173 (1986) LeClair, Patrick (UA) PH253 Lecture 9 February 5, 2020 9 / 28
Single photon scattering Observe: one is low when the other is high! 1 Exactly what one expects for waves! 2 Light does split?!? 3 Clearly light is neither particle nor wave exactly 4 path di ff erence LeClair, Patrick (UA) PH253 Lecture 9 February 5, 2020 10 / 28
Outline Better proof for photons? 1 Double slit experiment 2 LeClair, Patrick (UA) PH253 Lecture 9 February 5, 2020 11 / 28
Back to the drawing board? No, we just need to be more careful and open-minded 1 Look at interference for particles and waves separately 2 Contrast results for photon, e − with wave/particle 3 Should electrons and photons be different? 4 LeClair, Patrick (UA) PH253 Lecture 9 February 5, 2020 12 / 28
An experiment with particles Gun sprays particles randomly, large spread 1 Shoot at wall with two particle-sized holes 2 Detect hits on far wall. Probability one hits at x ? 3 Has to be probability - can’t say for certain 4 May bounce off slit, large angular spread 5 Presume constant rate of fire 6 Particles all identical, can’t split in two 7 LeClair, Patrick (UA) PH253 Lecture 9 February 5, 2020 13 / 28
An experiment with particles What does pattern look like? 1 P 1 =prob. particle came through slit 1 with slit 2 blocked 2 P 2 =prob. particle came through slit 2 with slit 1 blocked 3 P 12 =prob. through either with both open at same time 4 If we collect at the screen with both open, only P 12 is meaningful 5 Close hole 2, get P 1 ; close 1, get P 2 6 Both open: clearly P 12 = P 1 + P 2 7 P ’s add, no interference – clearly particles 8 LeClair, Patrick (UA) PH253 Lecture 9 February 5, 2020 14 / 28
An experiment with waves Try the same with waves! 1 Waves can propagate around holes . . . 2 Difference here: intensity on screen can have any value! 3 Not discrete like particles 4 Intensity ∝ ( amplitude ) 2 – height of wave squared 5 I 1 = | h 1 | 2 , I 2 = | h 2 | 2 with one at a time 6 With both open, I 12 = | h 1 + h 2 | 2 7 Meaning: I 12 � = I 1 + I 2 ! 8 LeClair, Patrick (UA) PH253 Lecture 9 February 5, 2020 15 / 28
An experiment with waves I 12 depends on relative phases of waves at any point! 1 Can write wave as a complex exponential: 2 h 1 ( t ) = h 1 e i ω t , h 2 ( t ) = h 2 e i ( ω t + δ ) 3 δ = phase difference based on path difference to screen 4 Energy at detector ∼ | h i | 2 for one slit i open 5 Both holes open? 6 h tot ( t ) = e i ω t � h 1 + h 2 e i δ � 7 Energy ∝ | h tot ( t ) | 2 = | h 1 | 2 + | h 2 | 2 + 2 | h 1 || h 2 | cos δ 8 LeClair, Patrick (UA) PH253 Lecture 9 February 5, 2020 16 / 28
An experiment with waves Or, I 12 = I 1 + I 2 + 2 √ I 1 I 2 cos δ 1 Sum of intensities plus interference term 2 Since waves take any height, interference shows up 3 Just what you see with water waves. 4 What about photons, or electrons? 5 Figure: https://www.feynmanlectures.caltech.edu/III_01.html LeClair, Patrick (UA) PH253 Lecture 9 February 5, 2020 17 / 28
An experiment with electrons or photons How about photons or electrons? 1 Both behave the same way! 2 But: see both wave and particle aspects 3 Depends on the details . . . 4 Probability of going through a single slit is the square of a 5 complex number P 1 = | ϕ 1 | 2 , P 2 = | ϕ 2 | 2 , so P 12 = | ϕ 1 + ϕ 2 | 2 6 Figure: https://www.feynmanlectures.caltech.edu/III_01.html LeClair, Patrick (UA) PH253 Lecture 9 February 5, 2020 18 / 28
An experiment with electrons or photons Detector “clicks” when e − hits. 1 Only hear full clicks - no “half clicks” 2 Discrete events. Rate erratic, but well-defined average 3 All clicks have same intensity = all events same 4 Try 2 detectors at once? Only one fires at a time 5 Like previous experiment - come through as clumps of definite 6 size, like particles Figure: https://www.feynmanlectures.caltech.edu/III_01.html LeClair, Patrick (UA) PH253 Lecture 9 February 5, 2020 19 / 28
An experiment with electrons or photons Electrons & photons clearly discrete, like particles 1 But interference is clearly observed! 2 Probability an e − or photon hits detector at x ? 3 Proposition 1 : each e − or photon goes through hole 1 or hole 2, 4 not both Is it true? Has to be for particles. 5 Figure: https://www.feynmanlectures.caltech.edu/III_01.html LeClair, Patrick (UA) PH253 Lecture 9 February 5, 2020 20 / 28
An experiment with electrons or photons If true, two types of particles: 1 Those going through hole 1, those going through hole 2 2 If so, observed curve must match superposition of single slit 3 results Close 1, measure P 2 , close 2, measure P 1 4 Both P 1 & P 2 look like particle result 5 But when both slits open? Interference like waves! 6 P 12 � = P 1 + P 2 like waves! 7 Figure: https://www.feynmanlectures.caltech.edu/III_01.html LeClair, Patrick (UA) PH253 Lecture 9 February 5, 2020 21 / 28
An experiment with electrons or photons How can this be true? 1 Complex paths back & forth? 2 No - some spots have higher intensity with both open! 3 Split in half? No - only full “clicks” heard 4 Worse: at center, P 12 > P 1 + P 2 5 As if closing one hole decreased intensity through other! 6 Figure: https://www.feynmanlectures.caltech.edu/III_01.html LeClair, Patrick (UA) PH253 Lecture 9 February 5, 2020 22 / 28
An experiment with electrons or photons More mysterious as you look closer 1 Math is like water waves. Amplitude for each slit φ i 2 P 1 = | ϕ 1 | 2 , P 2 = | ϕ 2 | 2 , so P 12 = | ϕ 1 + ϕ 2 | 2 3 Conclusion: e − or photons arrive in lumps, like particles 4 But, probability of arriving is like wave interference 5 Proposition 1 is false: not true that e − or photon takes only 1 hole 6 a particle, it takes both like a wave! Figure: https://www.feynmanlectures.caltech.edu/III_01.html LeClair, Patrick (UA) PH253 Lecture 9 February 5, 2020 23 / 28
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