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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 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 )
Photoelectric Effect
(cont’d)
• Einstein (1905; Nobel in 1921) explained results:
KEMAX
Remaining
Energy of the
electron = eV0
=
=
hf
-
Total Energy
gained by
electron from
absorbed
photon.
-
φ
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
Remaining
Energy of the
electron = eV0
=
=
hf
-
Total Energy
gained by
electron from
absorbed
photon.
-
φ
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)
•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)
•
–
Lyman Series (Ultraviolet (UV))
•
–
(1/λ) =R((1/12)-(1/n2)); n = 2, 3, 4, 5, ….
Paschen Series (Infrared (IR))
•
•
(1/λ) =R((1/22)-(1/n2)); n = 3, 4, 5, 6, ….
(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”.