PH253 Lecture 12: its waves all the way down de Broglie waves P. LeClair Department of Physics & Astronomy The University of Alabama Spring 2020 LeClair, Patrick (UA) PH253 Lecture 12 February 7, 2020 1 / 23
Outline de Broglie’s Hypothesis 1 LeClair, Patrick (UA) PH253 Lecture 12 February 7, 2020 2 / 23
Last time: Double slit experiment - waves or particles? 1 Yes. 2 Depending on scale and details of experiment, e − and photons 3 can look like either Because they are neither! 4 Arrive as particles, distribution of particles is wave-like 5 Can have interference, but not if you watch . . . 6 Next: how to explain waviness of an electron? 7 LeClair, Patrick (UA) PH253 Lecture 12 February 7, 2020 3 / 23
Outline de Broglie’s Hypothesis 1 LeClair, Patrick (UA) PH253 Lecture 12 February 7, 2020 4 / 23
Nature is discretized Photons (light) and electrons are discrete 1 Energy states of atoms must also be discrete 2 Follows that any observable energy difference will be 3 Slit experiments: waves and particles behave very differently 4 Photons and electrons look a bit like both (but are neither) 5 But how does this work for matter like electrons? 6 LeClair, Patrick (UA) PH253 Lecture 12 February 7, 2020 5 / 23
What makes photons so special? Relativity: nothing, just lack of mass. 1 Modern view: matter acquires mass by interactions 2 Photon happens to have zero rest mass, requiring v = c always 3 p 2 c 2 + m 2 c 4 � General case: E = 4 Photon: v = c , m = 0, = ⇒ E = pc = h f 5 e − : if p = 0, E rest = mc 2 ; if p ≫ mc , E ≈ pc 6 Only rest mass distinguishes electron. 7 High enough energies: KE ≫ E rest - photon-like 8 LeClair, Patrick (UA) PH253 Lecture 12 February 7, 2020 6 / 23
What makes photons so special? If only rest mass distinguishes e − (for now) . . . 1 Why should it not also have wave properties? 2 Dynamical properties still explainable 3 By analogy with photon, p sets length scale 4 Photon: λ = h / p , p related to E 5 e − : why not λ = h / p = h / γ mv ? 6 What is the scale? Must be tiny to escape notice so long 7 LeClair, Patrick (UA) PH253 Lecture 12 February 7, 2020 7 / 23
What is the length scale? Calibrating ourselves first . . . 1 Visible light: λ ∼ 400 − 700 nm 2 Circuit features: ∼ 10 nm 3 Atoms: ∼ 0.1 nm 4 Clearly we can’t see the waviness ordinarily. 5 Let’s say our scale is 100 nm. For light, λ = hc / E 6 This gives E ∼ 12 eV, hard UV light 7 LeClair, Patrick (UA) PH253 Lecture 12 February 7, 2020 8 / 23
What is the length scale? For e − , if λ = h / p ≈ h / mv ≈ 100 nm, v ∼ 7000 m/s 1 2 mv 2 = 3 2 k b T , v ∼ 10 5 m/s Thermal speed at RT? 1 2 Actually hard to slow down the electron enough to observe! 3 At atom spacing? v ∼ 10 7 m/s, K ∼ 150 eV - doable 4 Electron wavelengths are tiny at everyday energies 5 This was de Broglie’s big idea: treat matter like photons 6 Borne out by experiments like double slits 7 1924: de Broglie publishes PhD thesis. 1927: experimental 8 confirmation. 1929: Nobel. 9 LeClair, Patrick (UA) PH253 Lecture 12 February 7, 2020 9 / 23
Why was it hard to figure out? e − beams need to be in vacuum 1 “Lenses” are harder - E and B fields 2 Still need regular atomic scale features to see 3 E.g., a perfect crystal and surface 4 Long story short: 5 λ = h h h p = ( v ≪ c ) γ mv ≈ (1) mv LeClair, Patrick (UA) PH253 Lecture 12 February 7, 2020 10 / 23
Wave-particle? As with photons, probe size matters! 1 λ ≪ probe size: wave behavior can’t bee seen. Lumps/particles 2 λ ≫ probe size: can see wave effects, e.g., interference 3 Basically: m is tiny for e − , and so is λ 4 Never see this in everyday life. 5 100 mph baseball, λ ∼ 10 − 35 m 6 Proton diameter ∼ 10 − 15 m . . . 7 This is what allows electron microscopes. 8 (There are several right above us.) 9 LeClair, Patrick (UA) PH253 Lecture 12 February 7, 2020 11 / 23
Visualizing Same Gaussian wave packet ( y ∼ e − x 2 cos x ). Just zooming out on length ( x ) axis. LeClair, Patrick (UA) PH253 Lecture 12 February 7, 2020 12 / 23
Uncertainty? Bad news: this is weird. Matter has to be treated like photons 1 Both wave and particle aspects 2 Good news: we already figured out the math 3 Scale is unobserveably small most of the time 4 Interesting new effects to exploit 5 We need this for cell phones and computers 6 Bad news: we know enough now to expect unsavory new things 7 LeClair, Patrick (UA) PH253 Lecture 12 February 7, 2020 13 / 23
Time and frequency If waves are the right mathematical tool, consequences? 1 Forget spookiness, think more like signal processing 2 Measure frequencies? Need to watch wave fronts go by 3 Longer you measure, more accurate. Shorter? Less accurate 4 Short pulse? Only a few wave fronts to measure, not accurate 5 As time spread ↓ , frequency spread ↑ 6 LeClair, Patrick (UA) PH253 Lecture 12 February 7, 2020 14 / 23
Time and frequency This is a general thing and has nothing to do with quantum 1 “Benedicks’s theorem” - cannot be both time & band limited 2 Can’t sharpen in both time and frequency - dual variables 3 Narrow in time = broad in frequency 4 Perfectly periodic in time = single frequency 5 Pulse: too short to measure f very well, spread out 6 ∆ f ∆ t = ( bandwidth )( duration ) ≥ 1/4 π 7 LeClair, Patrick (UA) PH253 Lecture 12 February 7, 2020 15 / 23
Time and frequency Time and frequency pictures related by Fourier transformation 1 Basic property of waves: trade off in resolution 2 Optics: diffraction limit of microscope ∆ x ∼ λ 3 How does this apply to quantum particles? 4 Let’s think about measuring an e − position with a photon 5 Better photon resolution = smaller λ , but then higher p 6 Better resolution = more invasive experiment 7 LeClair, Patrick (UA) PH253 Lecture 12 February 7, 2020 16 / 23
Measurement Making photon λ smaller makes p higher 1 Photon momentum kicks the e − , alters its position 2 e − acquires p proportional to what photon has 3 ∆ p e − ∼ p photon,i = h / λ 4 So as λ ↓ , better resolution . . . 5 . . . but in the process we messed up e − position more, randomly 6 Uncertainty in resolution and position are antagonistic 7 LeClair, Patrick (UA) PH253 Lecture 12 February 7, 2020 17 / 23
Measurement resolution uncertainty means momentum uncertainty 1 Works against position resolution/uncertainty 2 In the end: ∆ x ∆ p � ¯ h /2 3 There is a limit to how well you can measure p or x 4 Minimum exists, but tiny due to size of ¯ h 5 Comes out of any wave mechanics (e.g. signal processing, optics) 6 If you know where you are, you don’t know how fast you’re going 7 LeClair, Patrick (UA) PH253 Lecture 12 February 7, 2020 18 / 23
Measurement Shorter pulse = ill-defined frequency (FTIR FTW) 1 Long/continuous signal = well defined frequency 2 Wave needs to “hang around” long enough to measure well 3 e − and photons: more localized x = ill-defined p 4 Uncertain x = well-defined p 5 Along each axis separately x , y , z 6 Similar: ∆ E ∆ t ≥ ¯ h /2, ∆ θ ∆ L ≥ ¯ h /2 7 LeClair, Patrick (UA) PH253 Lecture 12 February 7, 2020 19 / 23
Uncertainty Only on tiny scales! 1 10 g ball at 100 m/s, know ∆ v to ± 0.01 m/s? 2 ∆ x ∆ p = ∆ x ∆ ( mv ) = m ∆ x ∆ v ≥ ¯ h /2 3 h /2 m ∆ v ∼ 10 − 30 m - not a problem! ∆ x ≥ ¯ 4 e − at 100.00 ± 0.01 m/s? ∆ x ≥ 1 cm - fuzzy! 5 e − at 10 7 m/s, 1% uncertainty? ∆ x ≥ 6 × 10 − 10 m - 2-3 atoms! 6 Clearly particle-like for most cases. But tiny λ = electron 7 microscopy! LeClair, Patrick (UA) PH253 Lecture 12 February 7, 2020 20 / 23
Size of an atom Can get a ballpark estimate from uncertainty. 1 But what does size really mean now? 2 Classical orbiting charge model doesn’t work. 3 Quantum: if we know position too well, don’t know speed 4 e − must be “spread out” around proton to satisfy ∆ x ∆ p ≥ ¯ h /2 5 I.e., minimum approach, maximum extent for e − 6 From x-ray diffraction, know rough size of atom ∆ x = a 7 LeClair, Patrick (UA) PH253 Lecture 12 February 7, 2020 21 / 23
Size of an atom Then ∆ p ∼ ¯ h /2 ∆ x 1 Or, minimum p must be p min ∼ ¯ h /2 a 2 p spread is set by size of atom! 3 2 mv 2 = p 2 /2 m = ¯ h 2 /8 ma 2 ∼ h 2 / a 2 K = 1 4 h 2 /8 ma 2 − ke 2 / a Total energy? E = K + U = p 2 /2 m − ke 2 / a = ¯ 5 Atom will minimize its energy. PE wants closer, uncertainty limits 6 Need ∂ E / ∂ a = 0 7 LeClair, Patrick (UA) PH253 Lecture 12 February 7, 2020 22 / 23
Size of an atom h 2 4 ma 3 + ke 2 ∂ E ∂ a = − ¯ a 2 = 0 (2) 4 kme 2 ∼ 10 − 11 m h 2 ¯ a ∼ 1 Basically right (from experiments)! 2 Implies E min ≈ − 10 eV 3 Negative = bound state, stable 4 Implies ionization energy ∼ 10 eV - about right! 5 (For H: − 13.6 eV) 6 Atoms are stable! But still hand-wavy . . . more details yet 7 LeClair, Patrick (UA) PH253 Lecture 12 February 7, 2020 23 / 23
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