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Class Meets Mon + Wed + Fri   11:00 - 11:50pm in Science 277 ... Thurs at 11 in Sci.100
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Topic Nine - Electron Waves & Atoms

Quiz 9 was Mon.Dec.04 . . . here is my grading key

Exam 4 will be Tue.Dec.12 @ 10:15 am (or later day by appointment)

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Readings for Topic 9: Chs.39 − 41 Home-Work PRACTICE QUESTIONS for topic 9       (NOT for grading; numbers refer to Young & Freedman's University Physics, 14th ed.)

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Physics II Topic 9 Summary

brief history of understanding matter and light

categories of Elementary Things in Nature

Idea #18: (deBroglie says) electrons (matter) travel as waves: λ = h/p ... p is momentum

A moving electron wave partially reflects when it encounters a difference in the Electric Potential (and/or δB) ;
. . . reflecting from a higher PE region (where it would be slower) "flips" its wave amplitude => add ½λ
      of course, transmitted wave doesn't flip its amplitude
      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 usual 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 interfere constructively 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)
=> electrons are found in negative PE "wells", with enough (+) KE so that   2 w = n λ . . . w is the well width .

KE = p²/2m , so "flat-bottom" potential Energy wells (made to be at uniform (+)potential V)
. . . (1) "ground state" : 2 w = 1 λ , so   p = h/2w ... KE = h²/4w²/2m .
      for an electron in 1 nm wide well, at potential V , this puts   E₁ = −e V + 0.3765 eV .
      ... that total Energy must be negative, or the electron escapes the well ... so V must be positive
. . . (2) "excited states" : 2 w = n λ , so   p = h n/2w ... KE = h²n²/4w²/2m .
      for an electron in 1 nm wide well, at potential V , this puts   E₂ = −e V + 1.506 eV . . .
      ... if the Potential well is 1.600 Volt deep, the n=2 excited state has Energy −0.094 eV .
. . . (3) the n = 3 electron in a 1 nm well is 3.388 eV above the well bottom . . . 3.012 eV above the ground-state (n=1)
      a wider well would not need to be so deep : 10 nm (wide) well has n=3 at 0.0339 eV above the bottom

Recall (from Topic 6) that a photon must do Work (transfer Energy) to an electron, to eject it from a conductor surface.
      (the minimum Energy needed being called the "Work function" for the metal surface)
. . . these are usually about 1-3 eV photons, for most metals (with clean surface)
      that frees an electron from the metal's conduction level ... shared by all the atoms in the metal:
=> PE_{initial electron} + KE_{initial electron} + E_{initial photon} = E_{final electron}  
. . . initial electron Energy is negative in a metal, so even after absorbing the photon's positive Energy , its final Energy can be zero ... or even negative still!
. . . other electrons are at lower Energies, but they are in one particulat atom (not shared freely by all atoms)
      this means that their wavelengths must be much shorter than λ for a conduction electron
      ... so their momentums and their KE's are much more ... but they are trapped in one atom rather than free
=> most electrons in an atom have very deeply negative PE .

Implication 1 : Atoms contain discrete electron wave-forms

Caution : this cute little formula for electron Energy in an atom IGNORES electrical screening ("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 an 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, similar to n=5 wave-form but beginning to interfere (slightly) destructively .
. . . as the shape of its charge density changes, the E-field and B-field change, and an Electro-Magnetic Wave propagates 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 the main absorber to absorb the photon (by a few %).

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

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
      Schrodinger re-wrote Huygen's construction in terms of the curvature of the De Broglie wave excess above the Potential function ... A exp(p·x/h)

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 .

We might think of a moving particle (electron or photon) that is N wavelengths long , as being made from two waves with slightly different wavelengths
      these interfere constructively at the center of the particle (middle of its "envelope")
      but interfere destructively at the front end and at the back end of the particle.
. . . then, the envelope's length , where they interfere constructively , would be (N − ½ ) λ_long = ( N + ½ ) λ_short ...
      these two wavelengths would have different momentum : the shorter the particle's length   Δx , the more different the wavelengths (hence momentums) need to be.