The Successes of Classical Physics Classical Mechanics: For Objects – Given: Positions, Momenta,...

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/√(ε0 μ0) 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 Relativity for objects that move 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 v2

MAX = 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 )

Photoelectric Effect (cont’d)

• Einstein (1905; Nobel in 1921) explained results:

KEMAX = hf - φ

Remaining Energy of the electron = eV0

Total Energy gained by electron from absorbed photon.

Energy used to dislodge electron from surface

=-

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)

Photoelectric Effect (cont’d)

• Einstein (1905; Nobel in 1921) explained results:

KEMAX = hf - φ

Remaining Energy of the electron = eV0

Total Energy gained by electron from absorbed photon.

Energy used to dislodge electron from surface

=-

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

a. The maximum kinetic energy of the emitted electrons?

b. The maximum speed of the electrons?c. The stopping potential?d. 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)

• 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.

•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.

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)/r2

n ; 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 circle2πr = 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”.