Quantum Mechanics

1
Quantum Mechanics

• Walker, Chapter 30 31

2
Particles are Waves

• Material particles
• Electrons, atoms, Buckyballs (C60), Soccer balls
• Described by waves Y(r,t) amplitude of wave as
  a function of position in space and time.
• Localized wave-packet (pulse on jump rope)
  particle
• Wavelength l h / momentum,
• h Plancks constant 6.63..10-34 Js
  (measured)
• Frequency f w/(2p) Energy / h
• For soccer balls in ordinary life, wavelength is
  so short and frequency so high that we are not
  directly aware of the wave (geometrical optics
  limit).
• Soccer ball p (0.5 kg)(2m/s) (1/2)mv2 1J
• l (6.610-34 Js)/(1 kg m/s) 6.610-34 m f
  1J/ (6.610-34 Js)1.5 1033 /s

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deBroglie Wavelength I

• What is the deBroglie wavelength l h/p h/(mv)
  of a 0.5 kg soccer ball travelling at 2m/s?
• h 6.610-34 Js
• 6.61034 m
• 1.0 m
• 6.6 10-34 m
• 0.15 1034 m

4
deBroglie Wavelength II

• What is the deBroglie wavelength l h/p h/(Mv)
  of a 60C Buckyball traveling at 1 m/s?
• h 6.610-34 Js
• M 60 0.012 kg / (6.0 1023 ) 1.2 10-24 kg
• 5.510-10 m
• 1.0 m
• 6.610-34 m
• 1.8 109 m

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deBroglie Wavelength III

• What is the deBroglie wavelength l h/p h/(mv)
  of an electron (mass m 10-30 kg) traveling at 2
  m/s?
• h 6.610-34 Js
• 3.0103 m
• 6.6 10-34 m
• 1.0 m
• 3.3 10 -4 m

6
Quantum Frequency I

• What is the oscillation frequency f K/h of a
  0.5 kg soccer ball traveling at 2m/s?
• h 6.610-34 Js
• 6.610-34 Hz
• 1.0 Hz
• 1.5 1033 Hz

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Quantum Waves Probabilities

• Y(r,t)2 Probability of finding the particle
  at position r and time t.
• Particle wave is localized Total area of graph
  of Y(r,t)2 vs position (at a fixed moment t)
  is unity.

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Waves are Particles

• Light waves of frequency f w/(2p) are absorbed
  or emitted in discrete packets
• Energy (Light wave of frequency f) n h f, n
  1, 2,
• Plancks constant h again.
• Energy n h f, momentum n h f/c
• Energy density E e0E2/2 B2 / (2 m0 )
• Energy density times volume of wave Energy n
  h f
• Amplitude and frequency of light waves are linked

9
Black Body Radiation

• Temperature is just average energy in each
  microscopic degree of motion (translation,
  rotation, vibration of molecules)

• (1/2)kT (1/2)RT/N0 thermal energy in each
  degree of motion.
• k Boltzmans constant, R Ideal gas law
  constant, R N0 k
• N0 Avogadros number
• Every object radiates light at its intrinsic
  frequencies of vibration etc.
• A Black Body absorbs all light incident, but must
  re-radiate light, whose intensity and spectrum
  depends only upon the temperature.

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The Black Body Radiation Problem

• Classical Mechanics, and Classical EM (what you
  have learned in this class) gave prediction for
  black body radiation that
• Disagreed with experiments
• Was logically inconsistent (Infinite total
  energy).

Solar spectrum
11
Cosmic Background Radiationhttp//pdg.lbl.gov/200
2/contents_sports.htmlastroetc

• Quantum Theory curve with one free parameter
  (T2.73 K)
• Excellent fit to data
• Factor of 500 range in frequency
• Factor of 4000 in intensity

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Why Quantum Mechanics?1. Black Body Radiation

• Link between frequency and amplitude?area of wave
  solved the Black body problem (Planck)
• Peak frequency proportional to Temperature
• fpeak 5.881010 Hz (T/Kelvin)
• fpeak (2.82) kT/ h
• Total power radiated per unit area

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Black Body Examples

• Surface of Sun Black body, T6000K
• Blue photons, h f 3eV 2.82 k (12000 Kelvin)
• Incandescent filament Black body, T lt
  6000 K
• Glowing embers in fire Black body, T ?
  2000 K
• Human body Black body, T 310 K (IR night
  vision)
• Universe (minus stars, galaxies) Black body T3
  K
• Doppler shifted from T 12000 K when electrons
  and ions recombined in early universe to make
  neutral atoms.
• l 1.7 mm

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Cosmic Microwave BackgroundNASA/WMAP Science
Team
Red to Blue, T 2.725?0.001 K
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Solar Intensity Distribution

• Shaded grey Above atmosphere
• White At surface (with absorption bands of O2
  and H20

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2. PhotoElectric Effect (Einstein)

• Light shining on a metal will liberate electrons,
  but the photon energy hf must be greater than a
  threshold energy (equal to binding energy of
  electron in metal.
• The threshold effect is independent of light
  intensity (energy density of light).
• Na requires 2.5eV Green

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3. Compton Effect

• Classical picture of light scattering on a free
  electron
• Incident EM wave shakes the electron (transverse)
  , the oscillating electron radiates in all
  directions (except exactly 90o).
• The electron recoils straight ahead.
• Quantum effect Photons and electrons scatter
  like two billiard balls.
• Recent ODU-Jlab experiment exploited this effect
  with 6 GeV photons on protons to look measure
  Compton Effect of photons on quarks.

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DiffractionX-rays, neutrons from Crystal

• Pathlength difference 2dsinq.
• Constructive Interference
• 2dsinq nl
• h/p
• Photon Epc
• Slow Neutron
• E p2/(2M)

See pictures in text
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Diffraction by matter waves
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Diffraction Uncertainty Principle

• Sinq l/W
• Uncertainty
• Dy ? W/2
• Dpy ? psinq (h/l) (l/W)
• Dpy Dy h/2
• Heisenberg
• Dpy Dy ? h/(2p)
• Uncertainty principle is a consequence of wave
  nature of matter.

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Example 30-5

• Neutrons of velocity 1450 m/s diffract from a
  crystal with inter-planar spacing 0.282 nm.
• Find the deBroglie wavelength lh/p of neutrons
• Find the angle of the first diffraction maximum.

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Cold Neutron Sourcev1450 m/s

• Neutron Energy (1/2) mv2
• K (0.5)(1.6710-27 kg)(1450m/s)21.76 10-21
  J 0.011eV
• Thermal energy (3/2)kT. At T 300K, (3/2)kT
  0.039eV
• Neutrons from fission reaction have typical
  energy 4 MeV
• Elastic collisions with protons in water
  (hydrogen nuclei).
• Each collision reduces energy of neutron by 50,
  on average
• N collisions to reduce mean neutron energy to
  0.04 eV
• 2-N (0.04eV) / (4106 eV) 10-8
• N 26.6 collisions

23
Electron Microscope

• Resolution of microscope is limited by wavelength
  of probe.
• It is easier to produce, collimate, and focus
  high energy electrons than x-rays, g-rays.
• Figure 10-12 Electron microscope images of
• Sea Urchin sex
• Sexually transmitted disease.

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Schroedinger Equation Tunneling

• Energy in matter wave

Wavelength l h / p varies with position.
h2/(2m l2) V(x) ? h f Classically forbidden
region EltV. Wave does not oscillate, wave is
damped Wave Tunnels into forbidden region.
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Optical Tunneling

• Totally internally reflected wave actually
  tunnels out several wavelengths into air.

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Tunneling Electron Microscope
8,000
Single atomic tip is fabricated with a piece of
wire and ordinary pliers. With a voltage (100V)
between tip and surface, electrons tunnel across
gap, producing a current. Voltage is kept
constant, piezo electric crystal raises and
lowers tip to keep current constant as tip is
scanned across surface. Generate a map of the
height of the tip as a function of position
topographic map of electron density at sub-atomic
resolution.
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Standing waves on Guitar String ? electron waves
in atom
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Standing Electron waves in an atomic corral
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Atomic Spectra Quantum Color

• Atomic Spectra Quantum Color
• Every Atom vibrates at only special frequencies
• Electron wave standing wave in a box.
• Box size L
• Wavelength 2L, 2L/2, 2L/3, 2L/n
• Frequency fn nv/(2L),
• v velocity of electron in atom
• v p/m h/(lm) n h / (2Lm).
• Frequency fn n2 h/ m(2L)2
• Energy En h fn nh / (2L)2 / m
• Size L of box dictates energy levels En

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Photons in a laser

• How many photons per second arrive on the screen
  from a 1 mW red laser?
• l 780 nm, hf 1.5 eV
• (Dn/Dt)hf Power 1.e-3 J/s
• 1eV 1.6e-19 CoulombVolt 1.6e-19 Joule
• hf (1.5 eV)(1.6e-19 J/eV) 2.4e-19 J/photon
• Dn/Dt P/(hf) (1.0e-3 J/s) / (2.4e-19
  J/photon)
• Dn/Dt 4.17e15 photons/second

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Walker Problem 25, page 1007
Zinc and cadmium have photoelectric work
functions given by WZn 4.33 eV and WCd 4.22
eV, respectively. (a) If both metals are
illuminated by UV radiation of the same
wavelength, which one gives off photoelectrons
with the greater maximum kinetic energy?
Explain. (b) Calculate the maximum kinetic
energy of photoelectrons from each surface if l
275 nm.
f c/l hf hc/l hc 1237eVnm hf
(1237eVnm)/(275 nm) hf 4.5 eV Kmax(Zn)
4.5eV 4.33 eV 0.17eV
Kmax hf W0 Kmax(Zn) hf 4.33 eV Kmax
(Cd) hf 4.22 eV Kmax (Cd) gt Kmax (Zn)
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Stability of Atoms

• Rutherford, and his graduate students Geiger
  Marsden discovered that an atom of atomic number
  Z has a tiny nucleus of charge Ze at the center,
  surrounded by a cloud of Z electrons.
• According to Classical Mechanics, the orbiting
  electrons should continuously emit radiation.
• Bohr suggested the atom is stabilized, because
  the electron wave must form a stable wave, with
  an integer number of wavelengths in one orbit
• Schroedinger Equation later made this more exact

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Bohr Model of Hydrogen Atom

• Consider an electron of momentum p mv in an
  orbit of radius r,
• Coulomb attraction to proton (nucleus) F k
  e2/r2
• Centripetal acceleration v2/r
• Fma k e2/r2 mv2/r p2/(mr)
• Equivalent to Virial Theorem E KV p2/(2m) -
  ke2/r
• K - V/2 This constraint defines the size of
  the box
• p2 m k e2 / r p2r m k e2
• Circumference 2p r nl nh/p r nh/(2p p)
• Integer (n1,2..) number of oscillations of wave
  in one circumference of orbit.
• pr nh/(2p) angular momentum of orbit
  (quantized).
• m k e2 p2r p nh/(2p)
• p 2p m k e2/ (nh), n 1, 2, Momentum is
  Quantized
• E KV p2/(2m) - k e2 /r - (2p m k e2/ (nh)
  2/(2m)
• E (-13.6eV)/n2 Atomic Energy is Quantized, n
  1,2,3
• 1eV e(1Volt) (1.6e-19 C)(1 V) 1.6e-19 J
• Exact Wave functions and probability densities at
  e.g.
• http//www.phys.unm.edu/finley/P262/Hydrogen/Wave
  Fcns.html

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Bohr Model of Hydrogen atom
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Hydrogen Energy Levels Transitions
36
Hydrogen Emission Absorption spectra.

• Sun and stars are made of Hydrogen and Helium,
• The galaxies are receding from us (redshift)
• Balmer Series

Complete Solar Spectrum
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Walker, Chapter 31, Problem 11

• What is the frequency of the orbit of an electron
  in the nth Bohr orbit of Hydrogen?
• Quantum Solution
• Quantum wave oscillates with frequency
• fn -En/h E0/(n2h)
• In the nth orbit, it takes n oscillations of the
  wave for the electron to complete one orbit
• Frequency of orbit fn /n E0/(n3h)

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Walker, Chapter 31, Problem 11

• What is the frequency of the orbit of an electron
  in the nth Bohr orbit of Hydrogen?
• Classical Solution

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Quiz 8 April 19, 2004 Name

• What is the kinetic energy of an electron if it
  is confined to a box the size of the atomic
  nucleus (l 1fm 1.e-15 m)
• Use the following values
• K p2/(2m)
• mc2 0.5e6 eV m (0.5e6 eV)/c2
• hc 2p (200.e-9 eV m)
• p h/l

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Nuclear Energy

• What is the kinetic energy of a proton if it is
  confined to a box the size of the Pb nucleus
  (radius 1 fm (208)1/3 6fm 610-15 m)
• Use the following values
• K p2/(2m)
• Mc2 1GeV109 eV m (109 eV)/c2
• hc 2p (200.e-9 eV m)

l/2 2r l24 10-15 m
Nuclear energies are a million times greater than
molecular energies
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Optical Molasses

• Rb atom has a series of excited states of
  energies En.
• Each state can decay to a lower energy state Em
  (mltn) by emitting a photon of energy En - Em.
• If the mean lifetime for decay is tn, then by the
  uncertainty principle, the state En has a width
  Gn h/tn.
• The width Gn is just like the resistive damping R
  in a RLC oscillator.
• Cool a cloud of atoms by exploiting the doppler
  shift with a detuned laser beam.

42
Laser Manipulation of atoms.

• Each photon absorbed delivers momentum hf/c
  Each decay gives a random kick average momentum
  0.
• If atom is traveling towards laser, frequency is
  doppler shifted to f0/(1-v/c).
• Atoms traveling towards laser get much stronger
  damping than atoms traveling away.

Laser frequency
Doppler shifted laser, atom traveling towards
laser
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Magneto-Optic Trap

• 6 counter propagating circularly polarized laser
  beams cool beams to micro-Kelvin temperatures
• Anti-Helmholtz coils confine atoms to central
  region

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Periodic Chart of elements

• Waves in the atom have both radial and angular
  variables.
• For complex atoms with Z electrons (Z1,2,3, H,
  He, Li,) there is no simple formula for the
  energy levels, But energy levels are still
  quantized.
• Last electron sees effect of nucleus of charge Ze
  minus (Z-1) electrons ? Hydrogen-like.

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Hydrogen-like Wave Functions

• Principle Quantum number n 1, 2
• Orbital Quantum number l 0, 1, n-1
• Magnetic quantum number m -l, -l1, , l-1, l
• Orbital angular momentum, projected only any axis
  mh/(2p)
• Spin Quantum number ms?1/2
• The electron has intrisic spin, of angular
  momentum h/(4p)
• Energy Levels
• En (-13.6 eV)/n2 increase with l,
• Pauli Exclusion Principle each state, described
  by (n,l,m,ms) can be occupied by at most one
  electron

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Approximate Energy Levels in atoms of Zgt1
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Periodic Chart
Closed Shell
l1
l0
                                               
                             
n1 n2, n3, n4,
l2
l3
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Molecular Forces Van der Waals

• Electric field from point charge q E kq/r2
• Electric field from electron (-e) and rest of
  atom (e) separated a distance a E k(ae)/r3
• Charges in 2nd atom separate a distance
  proportional to strength of electric field from
  first atom (remember balloons stuck to wall)
  a?E ?1/r3
• Energy of second atom, in dipole field of first
  atomUEa ?1/r6

-e
e
r
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Molecular Forces Covalent Binding

• As two atoms approach, an electron from each atom
  can tunnel through the energy barrier from one
  atom to the next.
• This increases the effective size of the box
• The wavelength of each electron increases
• The momentum of each electron decreases
• The kinetic energy of each electron decreases
• The Energy of the two atom system decreases
• A molecule is born!!