» Syllabus

Topic 8 : Interference & Atoms (Ch.26 & 28)

Home-Work Set for Topic 8 :

PRACTICE QUESTION SUGGESTIONS ...( NOT for grading ...) from Young 9th edition
format codes :
      bold : Basic Bread-and-butter ... you'd better know how to do `em
      Italic : Important Ideas In here ... at least read `em
      Underline : Understand the Underlying structure ... not just the answer

ch.26 conceptual quests : 1,3,4,5,7,10,11,12,13,14,15 ;
      multiple-choice : 1,2,3,4,5,6,7,9,8,10,11,12,13,14,15 ;
      problems 1,3,5,7,8,9,11,13,35,37,11,41,42,43,44 ; 25,27,29,31,32,33,45,47,49,50,51(number?) ;
      . . .  21,22,15,16,17,18(sunlight?),19(bad #'s),20(bad #'s) ; 53,55,56,57,58,59,60,62

ch.28 conceptual quests : 2,3,4,6,7,8,10,11,12,13,14,15,16
      multiple-choice : 1,2,3,4,5,(electron?),6,7,8,9,10,11,12,13,14,15,16 ;
      problems ch.28 1,2,3,4,5,7,8,9,11,29,31,35,36,37,39 ; 16,20,21,23,25,26,27,28,29 ; 41,43,45,51 ;
      . . . . . . 46,47,49,53,56,58,64,68,69  ;

ch.29 conceptual quests 1,8,9,10,11,12,13,15
      multiple-choice : 4,6,8,9
      problems : 1,2,3,7,11,12,13,15 ; 21,25,27,33

Physics II Topic 8 Summary

Idea #16 : Two Waves can Interfere with each other - - - - - - - -

Treating a photon of Light as a wave , originally with synchronized E-field undulations - - - - - - - - - - - - -
There might exist two paths from the original source to some detector (screen)
Phase : If the light from the different paths arrives "in phase" , they add Constructively (bigger than either) ;
      if they arrive "out of phase" (by more than ¼λ) , they add Destructively (smaller than either)
C: If the difference in effective path length is an integer number of wavelengths
. . . called = = = = >> m << = = = = usually is a count of bright regions , starting with the "Huygens path" (with m=0).
      the Electric fields are "again" synchronized, so they add UP => constructive interference => bright Intensity detected there .
D : If the difference in effective path length is a half-wavelength , or other odd # ½λ
      the Electric fields are opposite so they "tend to cancel" => destructive interference => minimal (dim) intensity detected there .
=> δL = m λ ... constructive . . . or   = (m + ½) λ ... destructive

Wave-Front Splitting - (light thru openings) - - - - - - - - - - - - -

Wave-Amplitude Splitting - (light partially reflected) - - - - - - - - - - -

a) thin film ... light reflecting from 2nd surface travels "2 hypotenuses" after it separates from light reflecting from 1st surface
. . . (down + up) geometric length L = 2t/cosθ ... relate with shorter λ IN that film material = n L = n 2 t
b)  thin film ... light transmitted after 2 reflections travels "2 hypotenuses" farther than light transmitted with no reflections .
. . . supressed reflection is same as enhanced transmission
      the wave form inside the film is a resonant mode ... integer number of ¼ λ fit ... (organ pipe sound wave, guitar string wave)

by putting another piece of glass very near a total internal reflection surface , we can get light to transmit thru the gap . . . (yes, ½λ)

c) Fabry-Perot ... "thick film" is vacuum or air ... n = 1 => regular λ ... t ~ thousand or million λ
      . . . as surfaces are separated, the count of dim-to-bright "fringes" gives the distance, N ½λ ... (3,301,527½ fringes in Kr⁸⁶ orange is 1[m])

d) Michelson ... "thick film" where the reflected part is tilted ... (usually 90°) ... from the transmitted part
      . . . one runs past the park's West side and back , one runs past its North side and back ; are they in-step when they get back together?
            what if ONE path ("arm") effective length is modified slightly? => ½λ's are counted
e) Mach-Zender ... like Michelson, but one goes clockwise around the block while the other goes ccw around it ...
      . . . very sensitive to changes in rotation rate (e.g, Earth spinning faster in Winter)

Holography & Inverting the Point-Spread Function

Idea #17 : Heisenberg's Uncertainty Principle

There is more to light than just a disturbance in the ElectroMagnetic field.
. . . Most light sources (ie, atoms) bundle a small packet of “nearly-one-frequency” waves into an envelope.
      Each packet of light, called a photon, carries Energy and momentum and Angular momentum as it travels.
. . . Reminder: the Energy contained in each photon depends on its center wavelength : E = h c / λ .
A typical photon contains many waves , caused by a few thousand oscillations of charges at the photon's source ;
. . . the duration it takes to make the photon depends on how many oscillations it contains, and the oscillation rate (frequency)
=> Δt = N /f   [(waves/photon duration)/(waves/sec)] = [sec/photon duration]
We might think of a photon as a "beat" phenomenon (recall from Physics 1) , with two slightly different frequencies
. . . At the very beginning of the photon, they're out of phase (so they cancel to zero amplitude)
      ... as they become not so perfectly out-of-phase, the amplitude grows until perfectly in-phase ;
      as they get out-of-phase again, the amplitude dies away to zero ... tail-end of photon .
The beat frequency is the difference between the two individual contributions' frequencies
      (that is, a 256 [cycle/s] tuning fork and a 250 [cycle/s] tuning fork will constructively interfere 6 times per second)
. . . a bit of algebra shows : Δt = 1/f_beat = 1/(f₁ − f₂) ≈ 1/(E₁/h − E₂/h)
      associating the range of different frequencies with a range of Energies   δE :
=> δE Δt ≈ h   ... a smaller range in Energies means that the photon takes longer to make (or to pass by)
. . . with a Gaussian-shape amplitude envelope and a range of frequencies with Gaussian-shaped intensity ,
      δE Δt can be as small as h/2π ... that's Heisenberg's contribution .

Waves in ElectroMagnetic fields do not act exactly the same as sound waves in “ponderous” (inertia-based) materials ;
Whenever a photon is made or detected, the entire packet is emitted or absorbed,
. . . so they seem to interact with matter as if they were particles.
While they are traveling, though, they exhibit diffraction (spreading out) and interference
. . . so they seem to travel as if they were waves. This is called the wave-particle duality .

Idea #18: electrons travel as waves : λ = h/p (deBroglie)

So electron beams diffract and interfere after aperatures and scattering centers
  a)  "Thermal-Energy Electrons" have Kinetic Energy K ≈ (3/2) k_BT . . . k_B = 1.38E−23[J/K] = 86¼ [μeV/K]
      so thermal K ~ 40 [milli-eV] ( = 6E−21 [J])   at (300[K]) room Temperature.
      since K = p²/2m ,   p = √{2mK} = √{2(0.911E−30)(6E−21)}[kg·m/s] ~ 0.1E−24[kg·m/s]
      . . . so   λ = h/p = .6626E−33[J·s]/0.1E−24[kg·m/s] ≈ 6 [nm]
. . . about 20 atom diameters => conducting electrons reflect (scatter) from lumps of impurities bigger than this,
      but not from smaller defects (called "doping" when tiny defects put in on purpose, spread uniformly)
      thermal electrons scatter from magnetic domains, and from crystal grain boundaries ; these cause resistivity
  b)  "Atomic-Energy Electrons" ... have Kinetic Energy K ≈ 4 [eV] ... 100× the thermal Kinetic Energy,
      so their momentums are   √100× = 10× thermal p ,   and their wavelengths are (1/10)× thermal electrons' ... ~ 0.6 [nm]
. . . about the size of an atom ... no, it is not a coincidence ... the Kinetic Energy determines the atom's size.
      these will reflect and diffract from big (organic) molecules ... "electron microscope" wavelengths
  c)  "Inner-atom Electrons" ... have wavelengths λ ≈ 10 pico-meter . . . so   p = h/λ = 6.626E−23[kg·m/s]
      . . . so their Kinetic Energy   K = p·p/2m = (6.626E−23[kg·m/s])²÷(2)÷(.911E−30) = 2.4E−15[J] .   → ÷(1.6E-19) → 15 [keV] .
. . . there isn't anything else (besides the inner electrons of heavy atoms) that are this size ... an electron this small can make an x-ray
      an old electron microscope that shoots such high K will damage its specimen quickly (metal-plate it).
  d)  "Low-Nuclear-Energy Electrons" ... say, beta rays with K ≈ 9 [MeV] ... more than their "rest mass" = .511 [Mev/c²]
      . . . so they are relativistic . . . from Ch.26 :   p²c² + (mc²)² = E² = 9.511² => pc ≈ K , at this high of Energy !
      p ≈ (9E6)(1.6E−19[J])÷(3E8[m/s]) = 4.8E−21[kg·m/s]   so wavelengths λ = h/p ~ 140 [femto-meter]
. . . way too big to fit in the nucleus, or form diffraction patterns from it ... (so KE ≠ p²/2m anymore)
  e)  "Nuclear-Probe Electrons" ... say, 900 [MeV] ... have wavelengths 1/100 the low-Nuclear Energy size , ~ 1 [fm]
. . . finally small enough to get clear diffraction patterns from the atomic nucleus, and individual protons & neutrons (etc.)
      man-made in accelerator laboratories (CEBAF, near NewportNews VA, shoots 6.6 [GeV] electrons => λ ~ .14 [fm])
=> p·λ = h   . . . h = "Planck's constant" = 0.6626E−33 [J·s] = 4.136E−15[eV]

A moving electron wave partially reflects when it encounters a difference in the Electric Potential (and/or δB) ;
      as its momentum changes , the wavelength also changes - changing wavelength always includes some reflection (recall sound waves & light waves)
. . . this behavior is reinforced when a reflection from a second surface would constructively interfere with that first reflection.
      the paths (drawn as rays) look just like thin-film constructive interference on reflection.
. . . this behavior is supressed if the extra path distance would have the two partial electron waves ½λ out of step.
      just as for light, a supressed reflection is the same as an enhanced transmission .

An electron might become "trapped" in a PE well , by having negative Energy (with the convention that V ≈ 0 , at ∞) .
. . . it can only do this if it "fits" in the width of this PE well with some achievable momentum
      - remember , it had some momentum even where its PE was higher ... ↓ PE means ↑ KE and ↑ p , so ↓ λ ...
. . . if the electron wave is too short to fit exactly . . . if it doesn't interfere constructively with itself "perfectly"
     (there & back again must be an integer wavelength to "resonate" in the well)
      then the electron often radiates away some of its KE until it does fit
. . . usually an electron makes many reflections from both sides of the PE well in the process of radiating a photon
      the better the electron "fits" (2nd reflection reinforcing the original wave),
      the less each new echo changes the E-field and B-field of the new electron wave (which is being established)

Implication 1 : Atoms contain discrete electron wave-forms

The −kQ/r Electric Potential due to an Atom's Nucleus - - - - - - - - - - - - - -
means that the "n"=2 electron wave-form does NOT have twice the momentum (nor half the λ) as the "n"=1 wave-form
      with larger distance between reflections (turning points, where KE→0) , the momentum is actually less ! (~ 1/√r )
. . . we call the electron interference order number "n", rather than "m" like it was for photons ... to not confuse it with mass (or magnetic level-splitting)
=> E_n = −13.6[eV] Z²/n² . . . is KE + PE_{by nucleus} for each Energy Level in an atom.

Caution : this cute little formula for electron Energy in an atom IGNORES screening (or "shielding") by "inner" electrons
. . . outer electrons actually feel a much shallower PE well than what the nucleus would cause by itself.
the "rule-of-thumb" is to treat all the inner electrons as screening, and none of the same-n electrons as screening.

a) for innermost electrons , in n=1 , you replace the nuclear charge number Z by the effective charge number Z − 1 ...
      in heavier atoms, these are responsible for extra-intense x-rays that are characteristic for that atom
. . . (for n=2 electrons, it is a useful estimate to replace the nuclear charge Z by the effective charge number Z − 2 )

b) for lone outer-shell electrons , you replace the Z by the "interior" charge number ≈1
      an electron that has been excited is usually alone in that excited level ; there are (Z − 1) electrons screening the nucleus.
Consider the highest-energy electron in a Lithium atom; suppose it has been excited into the n=5 condition.
. . . it is attracted to its nucleus (Z = 3 protons), but repelled by the 2 inner electrons => Z_effective ≈ 1 .
=> Lithium's E₅ ≈ −13.6[eV] (1)² / 5² = − 0.544 [eV]

c) for an electron in a shell that is also occupied by other electrons, the other ones in that level screen less depending on their distances ;
      (after the complete screening caused by all the inner electron shells)
. . . an n=2 electron in Flourine (Z=9) should have   E₂ ≈ −13.6[eV] ( 9 − 2_n=1 − 2_n=2s − 4_n=2p/√2 )² / 2² = −16[eV] .
      it is measured as ionization Energy to actually be −17.5 [eV] , so there's another effect not accounted for yet.

The formula also IGNORES the electron's magnetic attraction to the other electrons (and to the nucleus).
. . . electron Energy depends on whether its magnetic moment is aligned with local B , or opposite B
      this Energy excess or deficit causes one electron Energy level to "split" into 2 or 3 (or more) levels , different by a few percent.
For the first electron in any angular momentum orbit, the magnetic Energy can be typically −½[eV] due to being attracted magnetically to the nucleus (if its nucleus has magnetic moment).
For a second electron in the same angular momentum orbit, the second electron pairs up and lowers its Energy by typically 2[eV] .
. . . so the second n=2 electron in Beryllium (Z=4) would have E₂ ≈ −13.6[eV]×( 4 − 2 − 1/√2 )² /2² − 2[eV] = −7.7 [eV]
      you can see that this is still not quite right, but it is the right amount more than the Lithium ionization Energy.
. . . that n=2 electron in Flourine (Z=9) has   paired 2s, and paired 2p₀ and paired 2p₁ ... but a single (unpaired) 2p₋₁ .
      to ionize it, you need to split up one of the pairs, which takes another 2[eV], before you move it to ∞ distance (at ∞ n)

To calculate these energy levels "correctly" involves multiple nasty calc.III equations to estimate the wave form for the each of the other electrons first ...
      . . . (start with 3 eq'ns for the two n=1 electrons , then 48 eq'ns for the seven n=2 waves , for Flourine ...)

Photon Energy Comes From Decrease in the source electron's Energy - - - -

if we wait for a while, that excited electron in the Lithium atom will eventually become slightly disturbed from its n=5 resonance,
      so its wave will shift, to start a slightly destructive interference with the n=5 wave-form.
. . . as the shape of its charge changes, the E-field and B-field change, and an Electro-Magnetic Wave starts to propagate outward.
      As it loses Energy, the electron falls toward its nucleus, gaining speed and momentum, so its wavelength shortens as it falls in. Eventually it "fits" again.
Suppose it gently shifts to now fit in the n=3 condition ; it now has − 1.511 [eV]
. . . so the photon carried away 0.967 [eV] ... 1.55E-19[J] ... at frequency 234E12[Hz] ... wavelength 1.28[μm] (InfraRed)
Whatever atom absorbs that photon, must absorb its energy ... not just 0.95[eV] of it, all 0.967[eV] of it.
. . . In a gas, atoms are isolated - so it is somewhat unusual to find an isolated gas atom that can do this, for some energies.
      (actually not hard for this energy, because its source was hydrogen-like);
. . . in solids and liquids, neighbor atoms can help (a few %) the main absorber to absorb it .

Idea #19 : Nature does the Least Action - - - - - - -

The wavelength associated with matter waves satisfies   p·λ = h (where p = momentum , and h is Planck's constant).
. . . So the propagation is still perpendicular to the wave front, and we can still use superposition in interference.
The old "optical path length" formula has to be replaced, though ;
. . . instead of   = n L /λ_o   to count wavelengths along a path,
=> # waves = p·δx / h     . . . momentum vector ( | wave fronts) is along the path being traveled

to interfere constructively with itself, in the forward direction , the number of waves along neighboring paths has to be the same
      so, for example, an electron will tend to travel straight along the bottom of a PE valley
. . . that way , the electron does the least Action
. . . Action is a process done when anything carries momentum for a ways ... Σ p·Δx  
      a ½[kg] lab cart travelling ¼[m/s] for 1½[m] does 0.75[J·s] of Action ... 1.1E33 h's of Action
      an electron in a hydrogen atom, in n=1 level, does p·λ = 1 h of Action each oscilllation
. . . this is not a coincidence! ... every repeating process (like an oscillation) has to do at least 1 h of action
      Feynman's contribution was writing Huygen's wave construction like this (in powerful math).

For other waves, including light, we treat the frequency as something determined by the source when the wave is formed.
. . . It’s not so clear for matter waves. If the momentum is increased (by Force thru a duration) to 10× what it was,
      then the wavelength becomes 1/10 .... what must happen to the wave’s frequency? ... if   f = v/λ = 10/(1/10) = 100× ??
=> no . . . the consensus is that the wave velocity does not match the particle velocity
. . . the wave is not the electron, nor is the electron a wave in some ethereal substance.
      Rather, the wave is contained within the electron's "wave packet envelope", but can move through it
. . . so the speed of the wave fronts is not the same as the speed of the electron.
In a sense, new wave fronts are made in the electron as it moves along
      but the wave fronts die out as they leave the “envelope” that contains the electron.

Heisenberg's Uncertainty Principle - - - - - - - -

One early application of this was to test the expectation that the nucleus was made of protons and electrons (neutron had never been seen yet in 1929)
. . . using the range of locations Δx ≈ 3.8E−15[m] as a nuclear diameter (Helium-4),
      the range of momentum would need to be   Δp ≥ .6626E−33[J·s])÷2÷π÷(3.8E−15[m] = 3E−20[kg·m/s] .
. . . since their average momentum was zero, the range must be from −1.5 to +1.5 (E−20[kg·m/s]) , oscillating
      their oscillation   K = p·p/2m = (1.5E−20)²÷2÷(.911E−30)[J] = 1.2E−10 [J] ≈ 800 [MeV] (at n=1 !)
      nuclear transitions emit about ¼ MeV to 8 MeV , so they could not be transitions between electron levels
. . . whatever does those transitions must have a mass similar to the proton's mass .