The Successes of Classical Physics

The Successes of Classical Physics Classical Mechanics: For Objects Given: Positions, Momenta, Applied Forces We Predict: Future Motion (Trajectories and Rotations) Classical Electrodynamics: For Light Given: Maxwell’s Four Equations We Predict: Wave Nature of Light (E, B, and propagation) Geometric Optics Physical Optics

Loose Ends and Mysteries of Classical Physics • From Maxwell’s Equations: • c = 1/√(ε0μ0) implies a mysterious relationship between light and space since no inertial reference frame is specified. (i.e. How does “relative motion” fit in?) • Leads to Theory of Special Relativityfor objects thatmove very fast. • Also From Maxwell’s Equations: • Predicted patterns of black body radiation that are not quite observed • First step in a series of discoveries leading to Quantum Theory for objects that are very small.

Evolution of Quantum Theory • Three scientific directions fueled the development of Quantum Theory: • Investigating the structure of the atom • Observations contradicting classical physics leading to new drastic hypotheses • Experiments designed to test the hypotheses in various ways. • The atom was the test case of the hypotheses and experiments.

Discovery of the Electron • JJ Thomson (1897) • Cathode ray tube: boiled off electrons (“cathode rays”) • Deflection of rays using a B-field was counteracted by an applied E-field giving a charge to mass ratio: • (e/m) = (v/Br) = (E/B2r) (methods of mass spectrometry) • See related homework problems • Led to the “Plum Pudding Model” of the atom • Light negative charges were boiled off from a heavier (+) background charge. • “negative plums in a positive pudding”

Charge and Mass of Electron • Robert Millikan (1913): • Oil Drop Experiment • Charged oil drops • Subjected to opposing Electric and Gravitational Forces. • Terminal velocity of drops measured. • Determined charge on Electron • Demonstrated quantization of charge • With e/m ratio; the mass of the electron was found.

How Black Body Radiation Led to the Quantum Revolution • “Black Bodies” are perfect absorbers and emitters of light. They radiate according to • Q/Δt = AσT4 (Stefan-Boltzmann Radiation Law) • Classical E&M gave two theories: • Wien’s Law: λpeak T= 2.9*10-3m·K (fails at long λ) • Rayleigh-Jeans Theory: fails at short λ • Max Planck (1900) (Nobel 1918): • Worked on a theory to explain observed curves • Drastic Hypothesis: Had to assume light came in • packets (“quanta”) • Later called “photons” by Einstein • Eph=hf; h = 6.63*10-34 J·s; f=frequency • h = Planck’s Constant; a fundamental constant of nature.

Intensity of Light • If Energy is radiated • in individual packets of light, • Each with energy: Eph=hf • Then How is Intensity measured? • We classically measured it by square of E-field. • Energy is given off by vibrating charges in amounts: E = nhf (n = # photons) • Intensity (power) is proportional to the number of photons emitted per second: (ΔE/Δt) = (n/Δt) hf ; I ~ # photons emitted per unit time.

Young’s Double Slit Experiment(Revisited) • What do bright fringes mean? • Many photons hit per second. • What do the dark fringes mean? • No or few photons hit. • What if we send one photon through. Where will it hit? • We don’t know for sure; most likely the central max, • We know some places where it won’t go. • By the way, through which slit did the photon pass? • Planck’s Wave-Particle Duality of Light: leads to a probabilistic view of light.

How Was the Particle Nature of Light Confirmed? • Photoelectric Effect: • Heinrich Hertz (1887): Light shining in a metal surface dislodges negative charges. • Philipp Lenard (1902): Identified particles as electrons. • Electrons • Absorb an amount of energy (φ) to dislodge them from the metal. (φ = “Work Function” for the metal”) • Any additional energy absorbed is excess kinetic energy (KEMAX).

Photoelectric Effect(cont’d) • Experiment: • Shine light on a plate with intensity (I) and frequency (f). • Electrons (with energy= KEMAX) are dislodged causing a current • Just enough opposing voltage (V0 =“stopping potential”) is applied to stop the current so : • ½ m v2MAX = KEMAX = e V0

Photoelectric Effect(cont’d) • Wave Prediction: • Higher Intensity would cause greater KEMAX and need greater V0 to stop the current. • Frequency would not affect KEMAX (or V0 ) • Actual Results: • Higher Intensity caused more electrons to leave plate (higher current measured), but at same KEMAX (V0 did not change) • Higher frequency caused greater KEMAX (found by measuring a higher V0 )

KEMAX = hf - φ Total Energy gained by electron from absorbed photon. Energy used to dislodge electron from surface Remaining Energy of the electron = eV0 = - Photoelectric Effect(cont’d) • Einstein (1905; Nobel in 1921) explained results: A plot of KEMAX vs. f (and experiment) shows a threshold frequency below which no photocurrent is seen: h fthreshold = φ (φ=work function of the material)

Example 1 • The work function of a metal is φ = 2.00 eV. If the metal is illuminated by light of wavelength λ=550nm, what will be • The maximum kinetic energy of the emitted electrons? • The maximum speed of the electrons? • The stopping potential? • What is the threshold frequency? • If a different wavelength of light is used such that the stopping potential is V0=.50V, what is its wavelength?

Compton Effect • Arthur Compton (1923) (Nobel 1927)- • “scattered” x-rays off a stationary carbon target. • Will the photons “collide” with the electrons in the carbon? • The x-ray photons behaved like particles in collision. • The incoming photons lost energy and momentum to the initially stationary electrons just like the carts of last term. • Special Relativity (SR) gives expressions for energies and momenta of high speed electrons and photons. • Eph= hf also holds for the photon by quantum theory. • Energy and Momentum are conserved. • Some SR things you should know (on AP ref sheets): • For a photon, momentum and energy are related: • Eph=hf = pc • λ = h/p

Compton Effect (cont’d) • Wave Prediction: electron would absorb and re-emit at same frequency (did not happen). • In colliding, the photon was actually absorbed and a lower energy photon emitted. • Amount of lowering depends on “scattering angle” (θ) • Emitted photon: same speed (c), longer wavelength (λ) • Δλ = λf – λi = (h/mec) (1-cos θ) = λC (1-cos θ) • Where: Δλ= “Compton Shift” and λC = “Compton wavelength” • Electron may or may not be ejected.

Compton Effect (cont’d) • Δλ = λf – λi = (h/mec) (1-cos θ) = λC (1-cos θ) • For θ=0o , Δλ=0; photons have scattered off inner (very tightly bound electrons). No energy is absorbed. • For θ=180o, Δλ=2 λC, = max. Most energy absorbed. • What is the difference between the Compton effect and the Photoelectric Effect? • In the photoelectric effect, the photon is completely absorbed.

Investigation into the Atom • Henri Bequerel (1896): • Radioactivity: A new tool with which to probe the atom. • Ernest Rutherford: passed beam from sample between charged plates. Deflections showed: • Alpha Particles (He nuclei) (charge +2e) (α2+) • Beta Particles (high speed electrons emitted by nuclei) (charge –e) (β-) • Also Undeflected (uncharged), hence undetected: • Gamma Rays (uncharged high frequency E&M) (γ)

Rutherford’s Gold Foil Experiment • Shot a beam of α-particles at a gold foil target. • α-particles very dense • Plum-Pudding (Thomson) Gold atom not so dense • Expectation: All α-particles will pass through foil. • Results: • Most α-particles did go through the foil unhindered. • A very few were deflected away. • A very, very, few were turned back the way they came. Rutherford Model for the Atom: Electrons in orbit about a positive nucleus. (Two problems with this model)

Atomic Spectra • Radiation from Black Bodies • Continuous spectrum emitted due to interactions of atoms and molecules with their neighbors. • Radiation from Individual Atoms • Only discrete wavelengths are emitted. • Balmer (1885) found four visible wavelengths from Hydrogen and fitted them to a formula • (1/λ) =R((1/22)-(1/n2)); • n = 3, 4, 5, 6 • R = “Rydberg Constant” • = 1.097*107/m

Atomic Spectra (cont’d) • Further Study: Other wavelengths are emitted outside the visible range. • Balmer Series (partly visible) • (1/λ) =R((1/22)-(1/n2)); n = 3, 4, 5, 6, …. • Lyman Series (Ultraviolet (UV)) • (1/λ) =R((1/12)-(1/n2)); n = 2, 3, 4, 5, …. • Paschen Series (Infrared (IR)) • (1/λ) =R((1/32)-(1/n2)); n = 4, 5, 6, 7, …. • Problems with Rutherford Model: • Accelerating charges radiate E&M so electrons should lose energy spiraling into the nucleus. • Resulting emission should be continuous (not discrete)and the atom; unstable (but matter is stable).

The Bohr Model • Niels Bohr • Used quantum theory, and atomic spectra to fix problems with the Rutherford model. Proposed: • An electron can only occupy certain allowed orbits without radiating • Each nth orbit has a radius (rn) and an energy (En). • An electron can make a transition between two orbits through • Absorbing a Photon (ELOWER EHIGHER) • Emitting a Photon (EHIGHER  ELOWER) • Where energy gained or lost by the electron is: |ΔE| = Eph = hf = hc/λ = |EHIGHER – ELOWER|

The Bohr Model (cont’d) • How did Bohr find the allowed orbits and energies? He used Balmer as a guide. • Drastic Hypothesis: The angular momentum of the electron is quantized • L = mvrn = n [h/(2π)]; n = 1, 2, 3, …. • Where r1 is the smallest orbit. • “n” is called the “radial” or “principal” quantum number. Has the most influence on Energy. • How did Bohr find the allowed radii and energies of the atom?

The Bohr Model (cont’d) • How did Bohr find the allowed radii and energies of the Hydrogen atom (Z=1= number protons) ? • By equating electric force on electron to its centripetal force (See related homework problem) • And adding in the condition: L = mvrn = n [h/(2π)] • Felect = kZ(e)(e)/r2n ; k = Coulomb’s constant • Fcent = mv2/rn • From quantization: v = n[h/(2π)]/(mrn) • Results: • rn = n2[h/(2π)]2/(Ze2mk) • Lowest Orbit: n=1, Z=1, r1= .53*10-10m = Bohr Radius • Higher Orbits: rn = (n)2 r1

The Bohr Model (cont’d) • Now let’s find the Quantized Energies • Total Energy of an Orbit: En = P.E. + K.E. • PE = qV = (-e)(kZe/rn) (from hwk34, probs18-19) • KE = (1/2)mv2 = (1/2)m {n[h/(2π)]/(mrn)}2 (see last slide) • If you add PE + KE and substitute for rn: rn = n2[h/(2π)]2/(Ze2mk) • You will get: En= - {(Z2e4mk2)/(2[h/(2π)])}(1/n2) • Lowest Orbit(Ground State): n=1, Z=1: E1= -13.6 eV= 1 Rydberg • Higher Orbits: En = En/n2 ; n = 1, 2, 3, …. • Why are the energies negative? • Negative Energy means the electron is bound.

The Bohr H-Atom: The Energy Levels

The Wave Nature of Matter • Louis DeBroglie (1923) – postulated a wave nature to particles. The wavelength of a particle is: • λ = h/(mv); m, v are the mass and velocity • Calculate the wavelength of a .20 kg ball traveling at 15 m/s. • λ = h/(mv) = 2.2*10-34 m • too small detect; a “classical particle” • Calculate the wavelength of an electron accelerated across a voltage of 100V. • v = √(2K.E./m) = √(2eV)/m = 5.8*106m/s • λ = h/(mv) =1.2*10-10m = .12 nm • Detectable; a “quantum particle”

The Wave Nature of Matter (cont’d) • What makes a wave, a wave? • Does it diffract? • Experimental Confirmation • Davisson and Germer (1927)- • passed a beam of electrons through a metal • Spaces between atoms acted as slits in a diffraction grating • The electrons showed a diffraction pattern on the screen. • G.P. Thomson(1927): • electrons diffracted through aluminum foil. • He shows electrons are waves • His father showed electrons were particles. • All concerned got Nobels.

The Wave Nature of Matter (cont’d) • So if you send one electron through a crystal, where will it land? • “most likely” at the central max. • DeBroglie’s Wave-Particle Duality of Matter: leads to a probabilistic view of Matter. • Newton’s Second Law has been overturned. • But didn’t we use it for the Bohr Atom? • How can we reconcile the Bohr Atom with DeBroglie’s postulate?

Matter Waves • How does DeBroglie’s postulate agree with Bohr’s hypothesis? • DeBroglie: λ = h/(mv) • Bohr: L = mvrn = n [h/(2π)] Recall Standing Waves: L = n (λ/2) (integer half λ) “Reflective boundary condition” To fit standing waves on a circle 2πr = n λ (integer whole λ) “Periodic boundary condition”

Matter Waves (cont’d) • DeBroglie required that the electron in the Bohr atom have a wavelength so that an integral number of them would fit on a Bohr Orbit: 2πrn = n λ ; n=1, 2, 3, …. • Then with λ = h/(mv) • We get: 2πrn = n h/mv • Or: mvrn = n [h/(2π)] • Which is Bohr’s Quantization Condition.

Matter Waves(cont’d) • DeBroglie’s waves are standing waves for Bohr Orbits: • Classical Mechanics is overturned for small “quantum” particles. • Quantum particles are described by wave functions (like sines and cosines) • The Squares of their amplitudes (intensities) give probabilities of where the particles may be found. • This is called “Quantum Mechanics”.

The Successes of Classical Physics