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Chapter 10 - Duality

P. J. Grandinetti

Chem. 4300

P. J. Grandinetti Chapter 10: Wave-Particle Duality There is nothing new to be discovered in now. All that remains is more and more precise . – Lord , 1900 P. J. Grandinetti Chapter 10: Wave-Particle Duality Why do objects glow when they get hot?

P. J. Grandinetti Chapter 10: Wave-Particle Duality Why do objects glow when they get hot? As is heated to higher it begins to glow, first as red, then orange, yellow, and finally a blue white, as keeps increasing. Heating an object increases of all its microscopic constituents. Since these constituents have electric (, cations, anions), or electric , etc, then their motion generates electromagnetic , i.e., . All kinds of microscopic motion can give off electromagnetic : accelerating charge, rotating , vibrating dipole, ...

+ _ + _ e–

tumbling vibrating dipole dipole changing

P. J. Grandinetti Chapter 10: Wave-Particle Duality radiation Spectrum of radiation emitted is plot of light intensity or light as function of light or . Spectrum depends somewhat on object’s composition. A class of hot bodies, called black bodies, have universally the same thermal spectrum. Black bodies are non-reflective–they absorb all light incident upon them–and thus appear black. ▶ Light absorptivity, 훼휆, is fraction of incident light () absorbed by real body when it is emitting and absorbing in thermodynamic equilibrium. ▶ Light emissivity, 휀휆, ratio of the emissive light power of real body to emissive power of an otherwise identical black body. ▶ Kirchoff’s law say 훼휆 = 휀휆 at .

Perfect black body has 훼휆 = 휀휆 = 1 for all . Only when black bodies are hot enough do they give off visible light–becoming self-luminous. Engineers often make the gray body assumption that 휀휆 is independent of wavelength. For most non-metallic surfaces we find 휀휆 ≈ 1.

P. J. Grandinetti Chapter 10: Wave-Particle Duality Emissivity of selected surfaces in gray body assumption.

Material Emissivity Charcoal: powder 0.96 3M, black velvet coating 9560 series optical black 1.00 Skin, 0.98 Polyethylene Black Plastic 0.92 Oxide White Paint 0.90 Snow 0.80 Copper, Thick Oxide Layer 0.78 Water 0.95 Glass, Smooth 0.94 Cement 0.54 Copper, Polished 0.023 Silver, Polished 0.020-0032 Aluminum Foil 0.03

P. J. Grandinetti Chapter 10: Wave-Particle Duality Black bodies Black bodies at equilibrium with surroundings emit characteristic spectrum. with increasing temperature... most intensity light frequency shifts to higher (color changes) 1.5 2000 K area under curve increase (gets brighter).

1.0 Definition 1500 K ⋆ 0.5 The radiant emittance, j , comes from area under curve: 1000 K ∞ 0.0 ⋆ 휋 휈, 휈, 0 1 2 3 4 5 j (T) = ∫ B휈( T)d 0 B휈(휈, T) is spectral of black body. i.e., the power emitted per unit area of black body surface

P. J. Grandinetti Chapter 10: Wave-Particle Duality Stefan-Boltzmann law

Experimentally found that j⋆(T) of black body increases rapidly with temperature,

j⋆(T) = 휎T4 Stefan-Boltzmann law,

휎 is Stefan-, 휎 = 5.670367 × 10−8 W/(m2⋅K4).

Real objects are not perfect black bodies and total light intensity emitted is

⋆ 4 j (T) = 휀휆휎T ,

휀휆 is emissivity of object’s surface, which varies between 0 and 1.

P. J. Grandinetti Chapter 10: Wave-Particle Duality Wien law

Familiar color change with increasing T comes from shifting of distribution mode (maximum) according to 휆 maxT = b휆 Wien displacement law

b휆 is Wien displacement constant, b휆 = 0.0028977729 m⋅K.

Color temperature of light bulb is temperature where most intense wavelength of ideal black body radiator matches most intense wavelength of light emitted by bulb.

Optical Pyrometry: For example, Wien displacement law is used by astronomers to 휆 determine surface temperature of from their max.

▶ 휆 has max = 510 nm, so T = 5682 K.

▶ 휆 North has max = 350 nm, so T = 8279 K.

P. J. Grandinetti Chapter 10: Wave-Particle Duality Stamp commemorating Fraunhofer’s measurement of Sun’s spectrum in 1817. Intensity as function of 휆 follows black body distribution for T = 5682 K.

Line Assignment wavelength/nm Other interesting aside...  H훼 656.2849  Na 589.5923 Sun’s spectrum also reveals number of dark lines. 1  2 Na 588.9953 Fraunhofer saw that the solar  lines were in same  Ca 527.0276 location as bright yellow doublet in  Fe 526.9541  dispersed light of a salted (NaCl) flame. H훽 486.1327  Fe 430.7906 Other solar lines have been assigned...  Ca 430.7741  Ca 396.8468

P. J. Grandinetti Chapter 10: Wave-Particle Duality Ideal blackbody

Good model for ideal blackbody is small into cavity with walls opaque to radiation.

Any light that enters hole is absorbed by inner walls and never escapes, i.e., hole absorbs all light and reflects none. At thermal equilibrium walls emit and fill cavity. Hole also acts as ideal emitter of thermal radiation at T and has blackbody spectrum.

P. J. Grandinetti Chapter 10: Wave-Particle Duality Raleigh & Jeans’ attempt to predict blackbody distribution

Assume walls of ideal black body cavity are metallic.

Classic E&M says electric field of light must be zero at walls.

Assume standing electromagnetic waves inside box. Recall the normal modes in 3D box.

How many normal modes in frequency interval 휈 to 휈 + d휈?

Raleigh and Jeans showed that number of normal modes in this interval is given by 8휋V N(휈)d휈 = 휈2d휈 3 c0 V is volume of box.

P. J. Grandinetti Chapter 10: Wave-Particle Duality Raleigh-Jeans’ attempt to predict blackbody distribution

To predict B휈(휈, T) &J needed average energy contained in each normal mode with frequency 휈.  Classical E&M theory says energy of wave is proportional to of amplitude, i.e., 0. Taking each normal mode as harmonic with 2 degrees of freedom, R&J assumed thermal equilibrium between normal modes and walls and used — 1 k T to 2 B 휖 each motional degree of freedom, or = kBT for each normal mode. This to the Raleigh-Jeans formula for black body radiation,

N(휈)d휈 8휋휈2d휈 u(휈, T)d휈 = k T = k T V B 3 B c0 where u(휈, T) is the energy per unit volume or the spectral energy density.

Spectral radiance, B휈(휈, T), is related to spectral energy density, u(휈, T), by

2 c0 2휈 d휈 B휈(휈, T) = u(휈, T) = k T 4휋 2 B c0

P. J. Grandinetti Chapter 10: Wave-Particle Duality The Plot of R&J expression with black body curve at T = 1500K

1.5 Raleigh-Jeans 1500 K

1.0

0.5 Experiment

0.0 0 1 2 3 4 5

R&J formula was an epic failure. Called the ultraviolet catastrophe as formula predicts cavity would emit ultraviolet and higher frequency light at room temperature. No one at the , 1900, could find anything wrong with derivation. What prevents higher frequency standing waves from filling cavity?

P. J. Grandinetti Chapter 10: Wave-Particle Duality Max started thinking about problem in 1894.

Max Planck (1858 -1947)

P. J. Grandinetti Chapter 10: Wave-Particle Duality Planck black body distribution Planck thought something wrong with equipartition of energy theorem. Somehow average energy of each normal mode depended on its frequency. At low frequencies, where R&J formula worked 휖 , lim = kBT 휈→0 but somehow at high frequencies lim 휖 = 0. 휈→∞ Planck could achieve these limits if he assumed that each electromagnetic oscillation produced could only have discrete energy values of 휖(휈) = nh휈 where n = 0, 1, 2, … h was proportionality constant, now called ,

h = 6.62607004 × 10−34J⋅s

P. J. Grandinetti Chapter 10: Wave-Particle Duality Planck black body distribution With this assumption Planck followed Boltzmann’s theory of statistical and got 휈 휖 휈 h ( ) = 휈 eh ∕kBT − 1

This expression had correct limiting behaviors in the low frequency limit with ( ) 휈 h휈 h휈 lim eh ∕kBT = lim 1 + + ⋯ ≈ 1 + 휈→ 휈→ 0 0 kBT kBT so that 휈 휖 h lim = ( ) = kBT 휈→0 휈 1 + h ∕kBT − 1 and in the high frequency limit with 휈 lim eh ∕kBT = ∞ 휈→∞ so that h휈 lim 휖 = = 0 휈→∞ ∞ − 1

P. J. Grandinetti Chapter 10: Wave-Particle Duality Planck black body distribution Back to Raleigh-Jeans’ expression, instead of N(휈)d휈 N(휈)d휈 ( ) u(휈, T)d휈 = 휖 = k T → Raleigh-Jeans V V B Planck came up with ( ) 휈 휈 휈 휈 휈 휈, 휈 N( )d 휖 N( )d h → u( T)d = = 휈 V V eh ∕kBT − 1 And then using N(휈)d휈 8휋 = 휈2d휈 V 3 c0 Planck finally obtained ( ) 휋휈2 휈 휈, 8 h u( T) = 휈 Planck black body distribution 3 eh ∕kBT − 1 c0 He published this expression in 1900 as it perfectly matched all observed curves for black body radiation. ...and so began the revolution of physics. P. J. Grandinetti Chapter 10: Wave-Particle Duality The Photoelectric Effect

P. J. Grandinetti Chapter 10: Wave-Particle Duality Photoelectric Effect In 1887 discovered the photoelectric effect—electric discharge occurred more readily when UV light is shined on a metal .

Cathode e–

current meter 0 –V +V

Voltage Adjust

P. J. Grandinetti Chapter 10: Wave-Particle Duality Photoelectric Effect In 1887 Heinrich Hertz discovered the photoelectric effect—electric discharge occurred more readily when UV light is shined on a metal electrode.

vacuum Quartz glass Anode e–

0 11 Light Light current Intensity Frequency meter 0 –V +V

Voltage Adjust

It wasn’t until the electron was discovered by J. J. Thomson in 1897 that more systematic investigations of the photoelectric effect could be undertaken.

P. J. Grandinetti Chapter 10: Wave-Particle Duality Photoelectric Effect

Monochromatic light is shined through quartz window onto clean surface of metal cathode and ejects electrons.

vacuum Quartz glass Cathode Anode e–

0 11 Light Light current Intensity Frequency meter 0 –V +V

Voltage Adjust

When voltage on anode is positive relative to cathode, photo-ejected electrons are attracted to anode and photoelectric current is measured.

P. J. Grandinetti Chapter 10: Wave-Particle Duality Photoelectric Effect

current UV light with intensity I As voltage, 휙, increases current eventually reaches limiting value with all ejected UV light with intensity I/2 photo-electrons collected at anode cup. As light intensity increases, current also increases.

If voltage, 휙, is reversed—anode is negative relative to cathode—current doesn’t drop to zero 휙 until certain stopping voltage, 0, is reached. That is, light causes electrons to be ejected. 휙 0 is required to stop fastest (highest ) electrons from reaching anode. Maximum 휙 kinetic energy of electrons is qe 0.

Classical wave theory says light energy is related to light intensity. But changing light intensity does not change kinetic energy of ejected electrons. With brighter light more electrons are emitted (higher current) but all emitted electrons had same 휙 maximum kinetic energy, requiring same stopping voltage, 0. P. J. Grandinetti Chapter 10: Wave-Particle Duality Photoelectric Effect Only by varying incident light frequency does ejected electron kinetic energy change.

slope of this line Rb K Na is the planck constant 휈 0, is critical frequency, below which no electrons are emitted.

E = h휈 − h휈 kinetic ⏟⏟⏟0 Φ

휈 Φ = h 0, is called function of metal. Φ varies from metal to metal. Φ represents energy that goes toward releasing electron from metal while remainder goes into electron’s kinetic energy. Based on classical wave theory this explanation makes no .

P. J. Grandinetti Chapter 10: Wave-Particle Duality In 1905 came up with a simple explanation...

Einstein interpreted the results of the photo-electric effect as consistent with the hypothesis that light is composed of bundles of energy, later named , whose energy is

휈 Ephoton = h

For this interpretation Einstein received the in 1921.

Albert Einstein (1879-1955)

P. J. Grandinetti Chapter 10: Wave-Particle Duality

Einstein further reasoned, if light acts like a particle, then what is the photon’s momentum? His own theory of says

2 m0c m v⃗ E = √ 0 and p⃗ = √ 0 2 2 2 2 1 − v ∕c0 1 − v ∕c0

For photons with v = c0 both expressions give unphysical value of ∞

unless m0 = 0, in which case both expressions become undefined.

Let’s proceed with assumption that photon has zero rest and look for another approach to determine its momentum.

P. J. Grandinetti Chapter 10: Wave-Particle Duality Photon Momentum

We can rearrange special relativity expressions to get ( ) ( ) 2 2 2 2 E 1 pc v ∕c = and 0 = 0 2 2 2 2 2 2 m0c0 1 − v ∕c0 m0c0 1 − v ∕c0

Subtracting these expressions gives ( ) ( ) 2 2 2 2 E pc 1 v ∕c − 0 = − 0 = 1 2 2 2 2 2 2 m0c0 m0c0 1 − v ∕c0 1 − v ∕c0

leaving us with 2 2 2 2 E = (m0c0) + (pc0)

P. J. Grandinetti Chapter 10: Wave-Particle Duality Photon Momentum

Assumption that photon has zero rest mass, m0 = 0, leads to √ ¨*0 ¨¨2 2 2 Ephoton = ¨(m0c0) + (pc0) = pc0

휈 Equating with Einstein’s relation, Ephoton = h gives 휈 휆, Ephoton = pc0 = h = hc0∕

and finally photon momentum as 휆 pphoton = h∕

P. J. Grandinetti Chapter 10: Wave-Particle Duality Photon Momentum Vector Classical E&M theory tells us light waves carry momentum directed along direction of ⃗ propagation (ek) ⃗ E ⃗ p = ek c0 Since light energy comes in bundles (quanta) of h휈 then momentum ought to come in units of h휈∕c , or 0 휈 ⃗ h ⃗ h ⃗ h ⃗ p = ek = 휆ek = 휋 k c0 2 2휋 Recall that ⃗ ⃗ . k = 휆 ek It is common to define the reduced planck constant, ℏ, as ℏ = h∕(2휋) often called “h-bar”, and write photon momentum as ⃗ ℏ⃗ pphoton = k

P. J. Grandinetti Chapter 10: Wave-Particle Duality Einstein’s statistical interpretation of light intensity In wave picture of light the intensity of radiation, I, (power transferred per unit area) is 1 I = c 휖 2, i.e., independent of frequency, 휈 2 0 0 0 But in particle picture of light, we take intensity of radiation as I = ⟨N⟩ h휈 ⟨N⟩ is average number of photons per unit time per unit area (perpendicular to the propagation direction). 2 It was Einstein who suggested that 0 be interpreted as measure of average number of photons per unit volume. This implies statistical view of light intensity with ⃗ and B⃗ waves regarded as guiding wave for photons. In this interpretation it is not wave that carries energy but photons. 2 In this interpretation 0 , i.e., the light wave intensity, is probability that photon is absorbed or emitted. P. J. Grandinetti Chapter 10: Wave-Particle Duality While Einstein and others were trying to understand the true of light, ... Light is composed of , now called photons. 휈 The energy of a photon is proportional to its frequency: Ephoton = h 2 The light wave intensity, i.e. 0 , is probability that photon is absorbed or emitted.

...others were attempting to make sense of a number of experimental results on at the atomic level. What is the structure of the ? How do we explain the light emission and absorption spectra of ?

P. J. Grandinetti Chapter 10: Wave-Particle Duality Meanwhile back in … 1814 Joseph Frauenhofer’s spectroscope invention led to discovery of dark features in sun light spectrum. 1860 Kirchoff and Bunsen note that several Frauenhofer lines coincide with emission lines in spectra of heated elements. 1885 Balmer and Rydberg found emission line follows ( ) 1 1 1 = R − where n = 3, 4, 5, … 휆 H 22 n2 . 8 RH is , RH = 1 097 × 10 /m. 1896 saw D-lines broaden in flame inside magnetic field–clue about magnetic properties of unknown atom constituents. 1897 J. J. Thomson discovers electron; measures its charge to mass ratio. ▶ He knew electrons were constituents of atoms. ▶ He knew atoms were neutral with diameters of ∼ 10−10 m. ▶ He proposed “plum pudding” model : atom as positively charged ∼ 10−10 m diameter cloud with electrons fixed and distributed inside. P. J. Grandinetti Chapter 10: Wave-Particle Duality Meanwhile back in … 1895 Roentgen discovers -rays: produced by electron beam fired at metal anode. 1896 Becquerel discovers radioactivity (in minerals). 1899 Rutherford studies gas using radiation from uranium. ▶ He identifies 2 types of radiation: 훼 and 훽 rays ▶ He later identified 훼 as nucleus ▶ 훽 rays were composed of electrons (훽−) and (훽+) 1911 Rutherford observed large angle of alpha particles by matter (metal foil) and disproves Thomson’s . ▶ Only conclusion: all mass concentrated at center of atom, and electrons somehow positively charged nucleus, like earth around sun. ▶ Problem with Rutherford’s model–which he was well aware–is Maxwell’s equations say accelerating charges (i.e., electron) radiate light waves and lose energy. So orbiting electrons should spiral into positively charged nucleus at center of Rutherford’s model.

P. J. Grandinetti Chapter 10: Wave-Particle Duality Meanwhile back in … 1913 proposed an atomic model where 1 electron orbits the nucleus, 2 electron doesn’t radiate light while it’s in an orbit—Bohr intentionally gives no reason why this does not happen. 3 electron gains or loses energy by jumping between orbits, absorbing or emitting light at 휈 h = Ef − Ei, where E is orbit energy. Model predicts light wavelength of ( ) ( ) ( ) 2 4 1 1 meq 1 1 = e Z2 − where n , n = 1, 2, 3, … 휆 휋휖 휋ℏ3 2 2 i f 4 0 4 c0 n n ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ f i

perfect agreement with RH

That Bohr was able to obtain Rydberg constant from fundamental physical constants was important achievement and indicated that he might be on right track for model of atom. Unfortunately, fails for multi-electron atoms.

P. J. Grandinetti Chapter 10: Wave-Particle Duality Which brings us to Paris in the 1920s.

P. J. Grandinetti Chapter 10: Wave-Particle Duality In his 1923 PhD thesis hypothesized that if light—which everyone thought for so long was a wave—could also behave like a particle,

then particles, like electrons, nuclei, and atoms might also behave like waves.

De Broglie imagined a particle as a .

Louis de Broglie (1892 -1987)

P. J. Grandinetti Chapter 10: Wave-Particle Duality De Broglie’s matter waves 휕휔 p He equated a particle’s to wave packet : ≡ vg = 휕 ⏟⏟⏟k ⏟⏟⏟m wave particle velocity velocity E 휕휔 1 휕E From Einstein’s light quanta hypothesis, 휔 = , he obtained v = = ℏ g 휕k ℏ 휕k 휕E p2 휕E p 휕p To get for a particle with only kinetic energy he writes E = and obtains = 휕k 2m 휕k m 휕k 휕휔 1 휕E 1 p 휕p p Putting these 3 expressions together gives v = = = = g 휕k ℏ 휕k ℏ m 휕k m 휕p which becomes = ℏ, 휕k and upon integration yields ... p = ℏk or p = h∕휆

P. J. Grandinetti Chapter 10: Wave-Particle Duality The De Broglie relation packet of

p = ℏk or p = h∕휆

In this derivation de Broglie also finds the relationship for matter wave packet of free particle, E 1 p2 ℏk2 휔 = = = ℏ ℏ 2m 2m At the time it was well known that x-rays would give a diffraction pattern when shined on a . What about matter waves? According to de Broglie an electron moving with velocity v = 5.97 × 106 m/s has wavelength

h h h 1 휆 = = = = 0.122 nm 6 p mev me 5.97 × 10 m/s Similar to x-rays, but are not x-rays—they are de Broglie’s matter waves.

P. J. Grandinetti Chapter 10: Wave-Particle Duality Davisson and Germer’s experiment In 1927 Davisson and Germer fired slow moving electrons at crystalline and observed a diffraction pattern, as if they had used x-rays.

De Broglie’s was fully consistent with Davisson and Germer’s diffraction pattern. P. J. Grandinetti Chapter 10: Wave-Particle Duality Low Energy Electron Diffraction (LEED) on Si (100) surface

The diffraction spots are generated by of elastically scattered electrons onto a hemispherical fluorescent screen. Also seen is the which generates the primary electron beam. It covers up parts of the screen. – Wikipedia P. J. Grandinetti Chapter 10: Wave-Particle Duality Wave-particle duality

By 1927 it was established that both light and matter can be observed as particles or as waves. This is called wave-particle duality.

This behavior is only observed on the subatomic where the are small enough to get large enough wavelengths to see wave properties.

Example Calculate the wavelength of a 16 lb bowling ball down an alley at 20 mi/h.

h h h 휆 = = = ≈ 10−36 m p mv (16 lb)(20 mi/h)

That’s pretty small.

P. J. Grandinetti Chapter 10: Wave-Particle Duality Wave-particle duality Example How slow would a 16 lb bowling ball need to roll for it to diffract significantly when passing through a 3 foot doorway? For diffraction the wavelength would need to be on the order of the slit width, in this case the door width, 휆 = 3 feet. This would require a momentum of p h h v = = = ≈ 4 × 10−33 in/s m m휆 (16 lb)(3 ft)

To move one inch at this would take ∼ 1024 . That’s greater than the age of the : 13.8 × 109 years. Wave-particle behavior is only observed for particles with atomic scale or smaller masses. In any given measurement only one behavior applies—both behaviors (wave and particle) are not observed together in same measurement. Which behavior we observe depends on nature of measurement.

P. J. Grandinetti Chapter 10: Wave-Particle Duality Wave Packet Dispersion Example Suppose a free electron wave packet is confined originally to 10−8 cm. How long will the packet take to spread to twice its original width?

Recall expression for width of Gaussian wave packet undergoing dispersion. √ 훼2t2 |Δx| = Δx 1 + 0 4 Δx0 ℏ 2 From dispersion relationship for free electron 휔 = k we calculate 2m 휕2휔(k ) ℏ 훼 = 0 = 휕k2 m

and obtain √ ( ) ℏ 2 t2 Δx(t) = Δx 1 + 0 m 4 Δx0

P. J. Grandinetti Chapter 10: Wave-Particle Duality Wave Packet Dispersion Example Suppose an free electron wave packet is confined originally to 10−8 cm. How long will the packet take to spread to twice its original width?

With √ ℏ2t2 Δx(t) = Δx 1 + 0 2 4 m Δx0 Solve for t √ mΔx t = 0 Δx2 − Δx2 ℏ 0

Simplify by defining ratio of widths, s = Δx∕Δx0 √ 2 √ mΔx mΔx t = 0 sΔx2 − Δx2 = 0 s2 − 1 ℏ 0 0 ℏ −8 With Δx0 = 10 cm and s = 2 for a doubling of width m (10−8 cm)2 √ e . −16 s . fs t = ℏ 3 = 1 5 × 10 = 0 15 P. J. Grandinetti Chapter 10: Wave-Particle Duality Wave Packet Dispersion Example Suppose an free electron wave packet is confined originally to 10−8 cm. How long until it exceeds the width of the solar system?

Taking from Sun to Neptune as 30 ua (1 ua = 149,597,870.691 km) we obtain ratio

Δx 60 ua s = = = 8.98 × 1022 −8 Δx0 10 cm

Plugging into expression for time

m (10−8 cm)2 √ e . 22 2 months t = ℏ (8 98 × 10 ) − 1 ≈ 3

As a comparison Voyager 1 left Earth on September 5th, 1979 and left the Solar system in August of 2012 (35 years later).

P. J. Grandinetti Chapter 10: Wave-Particle Duality Heisenberg

When electron (or bowling ball) is observed it is always found to be within region of that is as well-defined as we choose to make it.

As with light quanta-waves, matter wave intensity must be regarded as only giving probability that particle is found in given position.

For Gaussian wave packet we know that product of position and wave number uncertainty is ΔxΔk = 1∕2.

For de Broglie’s matter wave we use relationship between momentum and wave number, p = ℏk, and find uncertainty product for a Gaussian matter wave packet

ΔxΔp = ℏ∕2

Recall, Gaussian wave packet has smallest uncertainty product.

P. J. Grandinetti Chapter 10: Wave-Particle Duality Heisenberg

For arbitrary shaped matter wave packet

ΔxΔp ≥ ℏ∕2

This is the Heisenberg uncertainty principle of .

Similarly, recall uncertainty relation Δ휔Δt ≥ 1∕2. With Einstein’s light quanta hypothesis, E = ℏ휔, it becomes ΔEΔt ≥ ℏ∕2

Werner Heisenberg (1901 - 1976) P. J. Grandinetti Chapter 10: Wave-Particle Duality Schrödinger’s To go any further with de Broglie’s matter wave we need a wave equation.

When Schrödinger first wrote it down, he gave a kind of derivation based on some heuristic arguments and some brilliant intuitive guesses. Some of the arguments he used were even false, but that does not matter; the only important thing is that the ultimate equation gives a correct description of nature.—

Erwin Schrödinger (1887-1961) P. J. Grandinetti Chapter 10: Wave-Particle Duality The Schrödinger Equation In 1925 Schrödinger set about finding wave equation for de Broglie waves. As matter waves undergo constructive and destructive interference like light, he assumed it to be linear differential equation so applied. He knew it must be consistent with de Broglie–Einstein postulates,

p = ℏk and E = ℏ휔

It must allow , V(x, t), to depend on position and time, consistent with total energy p2 E = + V(x, t) 2m Combining these 2 expressions he obtains ℏ2k2 E = ℏ휔 = + V(x, t) 2m General non-linear for particle wave in potential, V(x, t).

P. J. Grandinetti Chapter 10: Wave-Particle Duality The Schrödinger Equation Consider a particle that is a sinusoid Ψ(x, t) = Aei(kx−휔t) Taking 1st with respect to time 휕Ψ(x, t) = −i휔Aei(kx−휔t) = −i휔Ψ(x, t) 휕t Rearranged to i 휕Ψ(x, t) 휔 = Ψ(x, t) 휕t

Taking 1st and then 2nd derivative with respect to position gives 휕Ψ(x, t) 휕2Ψ(x, t) = −ikAei(kx−휔t) and = −k2Aei(kx−휔t) = −k2Ψ(x, t) 휕x 휕x2 Rearranged to 1 휕2Ψ(x, t) k2 = − Ψ(x, t) 휕x2 P. J. Grandinetti Chapter 10: Wave-Particle Duality The Schrödinger Equation Substituting expressions for 휔 and k2 back in energy (dispersion) relation

ℏ2k2 E = ℏ휔 = + V(x, t) 2m

gives ( ) ( ) i 휕Ψ(x, t) ℏ2 1 휕2Ψ(x, t) E = ℏ = − + V(x, t) Ψ(x, t) 휕t 2m Ψ(x, t) 휕x2 which rearranges to the Schrödinger equation

ℏ2 휕2Ψ(x, t) 휕Ψ(x, t) E Ψ(x, t) = − + V(x, t)Ψ(x, t) = iℏ 2m 휕x2 휕t

It was his best guess for the wave equation for De Broglie’s matter waves. It’s not derived from 1st like classical wave equation. Its validity lies in giving correct result.

P. J. Grandinetti Chapter 10: Wave-Particle Duality The Schrödinger Equation

Schrödinger equation differs from classical wave equation in 2 key ways.

(1) Partial derivative with respect to time is 1st order instead of 2nd order, √ (2) It contains i = −1, meaning physical solutions to Schrödinger wave equation can be complex.

Complex quantities cannot be measured by any physical instrument—measured quantities must be real.

Quantum wave functions don’t have same kind of physical as classical waves - Quantum waves are not mechanical waves.

What is physical meaning of quantum wave function?

P. J. Grandinetti Chapter 10: Wave-Particle Duality What is physical meaning of quantum wave function? Copenhagen interpretation of Neils Bohr and - (1925 to 1927) Copenhagen interpretation says product of wave function with its , Ψ∗(x, t)Ψ(x, t), i.e., |Ψ(x, t)|2, is probability of finding particle at given instance in given location. As with any probability density, its between limits gives actual probability,

x2 , , ∗ , , P(x1 x2 t) = ∫ Ψ (x t)Ψ(x t)dx x1 , , P(x1 x2 t): probability that particle is observed between x1 and x2 at time t. We require wave function to be normalized

∞ ∗ , , ∫ Ψ (x t)Ψ(x t)dx = 1 −∞ Only accept wave functions where both Ψ(x, t) and 휕Ψ(x, t)∕휕x are finite, single valued, and continuous functions. P. J. Grandinetti Chapter 10: Wave-Particle Duality Fifth , Brussels, October 1927

A. Piccard, E. Henriot, P. Ehrenfest, E. Herzen, Th. de Donder, E. Schrödinger, J.E. Verschaffelt, W. Pauli, W. Heisenberg, R.H. Fowler, L. Brillouin P. Debye, M. Knudsen, W.L. Bragg, H.A. Kramers, P.A.M. Dirac, A.H. Compton, L. de Broglie, M. Born, N. Bohr I. Langmuir, M. Planck, M. Curie, H.A. Lorentz, A. Einstein, P. Langevin, Ch.-E. Guye, C.T.R. Wilson, O.W. Richardson

P. J. Grandinetti Chapter 10: Wave-Particle Duality