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Topic 9 (Matter Waves & Electrons In Atoms) Readings for Topic 9 (from Knight Jones Field) : Ch.28 + Ch.29

plan for Topic 9 Quiz to be Tue.Apr.09 - new equations are described below,

      but you will USE this full page of 203 eq'ns from now on

equations on the bottom line are for Topic 10 (atomic nuclei)

plan for Unit 2 mini-Exam (40 point) to be thRs.Apr.18

plan for the Phy.203 Common Exam to be Sat.Apr.20

Very brief history of humans understanding matter & light

categories of Elementary Things in Nature

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

Evidence: electron waves show interference patterns . . . after passing thru foil

We'll boil electrons off a hot filament at electric potential −6500 V ... their PE are qV = +6500 eV as they start
. . . if the foil is at V=0 , each electron's KE at the foil will be +6500 eV = 1.04E−15 J
      their momentum should be √{2mK} = √{2(0.911E−30)(1.04E−15)}[kg·m/s] ~ 4.35E−23[kg·m/s] at the foil.
      Their (DeBroglie) wavelength as they go thru the foil will be λ = h/p = 663E−36 Js / 4.35E−23 = 15.23 pico-meter . . .
. . . where the electrons hit the screen, 183mm past the foil, their energy makes the phosphor glow ;
      hexagon interference pattern shows first constructive interference at y= 15 mm from the center "n=0" spot.
      tan θ = 15mm / 183mm = .082 => θ = 4.7° . . . 1 λ / sin θ = d = 186 pm . . . consistent spacing for aluminum atoms
. . . more voltage would mean more momentum, so shorter wavelength ... and interference occurs in smaller angles.

NOTATION: electrons have mass , symbol "m_e" ... so we will use   "n"   as the symbol to count electron interference modes (no refractive index)

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 .

Flat-Bottom Potential Energy "Well" - - - - - - electron λ resonance must "fit"

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 tail of the previous 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 .

A "flat-bottom" Potential Energy well has uniform (+)Potential V, so a (− charge) electron       has uniform (−)PE there, so also uniform KE ,   & p ,   & λ in the well . . . n=1 "ground state" : 2 w = 1 λ , so   p = h/2w ... plug into   KE = p²/2m . . . . example: for an electron in a well that is 0.61 nm wide,   KE1 = 1.012 eV above the well bottom       the electron's total Energy   E = PE + KE   must be negative (from −e V), or it keeps going past . . . n=2 (1st "excited state") : 2 w = 2 λ , so   p = h n/2w ... KE = h²n²/4w²/2m ... goes as n²       for an electron in 0.61 nm-wide well,   E2 = −e V + 4.047 eV . . . (3.035 eV above n=1)       ... the Potential well has to be almost 5 Volt deep, for the n=2 excited state to be retained . . . . (3) the n = 3 electron in a 0.61 nm well has 9.107 eV Kinetic Energy ... 5.060 eV above n=2 A thermal electron "falling in" a 10 Volt-deep 0.61 nm-wide well must lose energy to get trapped.       not only its original KE (0.040 eV) but also 0.893 eV more to reach the n=3 condition . . . this radiates a 0.933 eV photon   ( = 1.493E−19 J = hc/λ)   ⇒ 1.332 μm wavelength (IR)

. . . to drop from the n=3 to the n=2 requires it to emit a 5.060 eV photon ⇒ hard UV
. . . to drop from the n=2 to the n=1 would emit a 3.035 eV photon ⇒ 0.409 μm (violet)

     a wider well would not need to be so deep : a 10 nm (wide) well has n=3 at 0.0339 eV above the bottom

Round-Bottom Potential Energy Wells - - - - - - λ varies with location x

smoother PE shape is more realistic than flat-bottom PE wells with "infinitely steep" sides
      the "easy math" had already been done for parabolic PE = ½ k_s x ²   wells ... quadratic harmonic oscillator.
Energy levels have uniform spacing . . . n =1 wave will have 1 antinode , n =2 wave will have 2 antinodes, etc ...
=> E_n = ( n − ½ ) h f . . . here , 2π f = ω = √(k_s / m
      . . . and we interpret k_s as the curvature of the PE graph (just like in physics 1 mechanics)

One result is that (again) the electron wave cannot sit still at the well bottom
. . . a parabolic well has zero width at zero energy - the electron wave can't have zero wavelength (would be infinite KE!)

The important new issue is that the wave doesn't end exactly at the turning point.
. . . outside the turning point, the wave curvature is proportional to the positive wave value
      curving up to stay away from zero ... as the wave value itself gets closer to zero
=> dies out "exponentially" , but penetrates past the turning-point roughly ¼ wavelength . . . sound familiar ?

Schroedinger: imagine where wave Value · wave Curvature is positive - - - - - -

Likelihood of finding an electron at some place is the square of the wave value there
      so we don't need to interpret what a negative part of the electron wave means.
. . . likelihood of the electron having some momentum value depends on the wave slope
. . . likelihood of the electron having some Kinetic Energy value depends on the wave curvature.

Spike-Bottom PE Wells : Atoms contain discrete electron wave-forms

Let's look at the simplest atom first ... Hydrogen is just one electron trapped in the PE well of a single proton.
. . . in 3-d, the n=1 electron sloshes back & forth , left & right , up & down - only reinforcing itself near the center (nucleus)
      the wave-form looks kinda like a 3-d gaussian distribution , with "standard deviation diameter" 2 × 52.92 pm
. . . that 52.9 pm = h²/(4π²m_ek_Ce² )   is called the "Bohr radius", symbol r₁
      occurs at the wave-form inflection (where the interior downward curve becomes an upward curve, at the turning-point)
. . . the n=1 electron's total Energy is −k_C e² / 2r₁ ≈ −13.6046 eV
      and its average speed is 2.2 Megameters/second, so its pacing frequency is 6.5E15 oscillation/second

Because the n=2 electron wave-form is zero at the center, its average radius is way out where the nucleus' Potential is much less negative
. . . it is 4 times as far as the n=1 radius , so its total Energy is only ¼ as deeply negative.
      with larger distance between reflections (turning points, where KE→0) , the momentum is actually less ! (~ 1/√r )
. . . because the n=2 wave-form has "lumps" away from the origin, it can have orbital anglar momentum , L
      but these vectors must have magnitude h/2π , so that the electron will do 1h of Action during an orbit.
. . . the pattern continues for n=3 being 9× as big , with only ¹/₉ as much (negative) Energy , and allowing angular momentum

Other atoms have nuclei containing Z protons , not just 1 ... Z is their "atomic number"
. . . a single electron will experience their PE well being Z times as deep ; the electron wave will be only ¹/_Z as large
      the electron average speed will be Z× as fast , and they will have Z² as much energy (much more negative)

=> 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 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 half 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, estimate the effective charge Z by proton number Z − 2 ... unless an n= 1 electron is missing)

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 (from n=2) into the n=5 condition.
. . . it is attracted to its nucleus (Z = 3 protons), but repelled by the 2 inner electrons (n=1) => 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 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 ... in a grad level course)

emitted Photon Energy Comes From Decrease in the source (emitter) 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 - - - - - - -

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])÷4÷π÷(3.8E−15[m] = 1.5E−20[kg·m/s] .
. . . since their average momentum was zero, the range must be from −¾ to +¾ (E−20[kg·m/s]) , oscillating
      their oscillation   K = p·p/2m = (0.75E−20)²÷2÷(.911E−30)[J] = 4E−9 [J] ≈ 200 [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.
this is just like we heard with 2 tuning-forks with different frequency, making "beats" time-wise ... the beat's frequency was the difference between the individual fork frequencies.
      say, a 256 Hz and a 255 Hz beating at 1 Hz ... if a third fork at 254 Hz was synchronized at number 0's center, it could cancel a lot of the sound in beat #1
      but we'd need a fourth fork at 252 Hz to cancel the sound in beat #2 ... and a fifth fork
. . . as the range in frequencies (Energies) gets bigger, the time for the sound decreases
      but the product   ΔE·Δt   is minimized with a single pulse ("Gaussian" speaker-shot)
      which has a "Gaussian" distribution of its frequencies.
same with 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.