Physical Chemistry Lecture 3

1
Physical Chemistry Lecture 3

• Hamiltonian Mechanics
• Classical Mechanics Breakdown
• Particle Wave Duality
• Heisenberg Indeterminacy Principle
• Quantum Mechanics Fundamentals

Fred J. Grieman
2
Hamiltonian Mechanics

• q generalized coordinate p generalized
  momentum t time
• H E T(p,q) V(q,t)
• Equations of motion
• (?H/?pi)qi,pj dqi/dt
  (?H/?qi)pi,qj -dpi/dt
• Harmonic oscillator example
• Newtonion Result µ d2r/dt2 -kr
• Hamiltonian r is the coordinate and pr is the
  momentum
• H T V ½ µ(dr/dt)2 ½ kr2 pr2/2µ
  k/2 r2 write in terms of
• 
  
  momentum !!!
• (?H/?r)pr kr -dpr/dt
• -kr d(µ dr/dt)/dt
• -kr µ d2r/dt2 Same as
  Newtonian!!!!

3

• Classical Mechanics Breakdown
• Blackbody Radiation
• Matter Energy in quanta, 1 quantum h?
• h is Plancks constant (energytime
  action)
• (handout)
• 2) Photoelectric Effect
• Light particle behavior
• a photon is a quantum of light
• e h?, Einstein Radiation Law (ERL)
• 3) Atoms
• Rutherford Planetary Atom
• Bohr Atom quantized energy and angular
  momentum (handout)
• Atomic Spectra lines explained by ERL with
  quantized levels
• 4) de Broglie Matter Waves
• derivation e mc2 ? e h? hc/?

Photons absorbed or emitted
?E Ef Ei elight h?
hc/?
E
Matter energy levels
4
Matter Waves/Implications via Thought Experiments

• Bullets

• Conditions
• Slits screen immovable
• Slits just big enough for bullets
• Gun fired slowly fire 1, detect 1

P12 ?
Bullet detector
P1
Slit 1

• Observables
• Lumps particles remain whole
• Only 1 lump at a time
• Always same size identical

Gun
x
Slit 2
P2

• Results
• Slit 1 only P1 Distribution
• Slit 2 only P2 Distribution

3) Both Slits? P12 P1 P2 probabilities
add, as in mathematics
Waves

• Conditions
• Slits screen immovable
• No Reflection
• Intensity (height)2 ? Energy

I12 ?
Intensity detector
I1
Wave generator
Slit 1

• Observables
• Intensity can have any value
• Continuous, not lumpy

x
I2
Slit 2

• Results
• Slit 1 only I1 Distribution
• Slit 2 only I2 Distribution

3) Both Slits? I12 Interference Pattern
Mathematics we know
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Mathematics for Interference

• Height h(t) Slit 1 h1ei?t Slit 2 h2ei?t
  h1 h2 can be complex
• One slit open I1 ? h12 h1h1 I2 ?
  h22 h2h2 complex conjugate
• Both slits open (h1 h2) ei?t I12 ? h1
  h22
• I12 ? h12 h22 2
  h1 h2cosd d phase difference
• I12 I1 I2
  2(I1I2)1/2cos d
• for maxima d 2np minima d
  (2n-1)p
• From mathematical analysis of pattern you
  can determine ?

Electrons

• Conditions
• Slits screen immovable
• Slits just big enough for electrons
• Gun fired slowly fire 1, detect 1

P12 ?
Faraday Cup
Electron Gun
P1
Slit 1

• Observables
• Lumps Just like bullets!!!
• Always go through 1 or 2
• Always same size identical

x
Slit 2
P2

• Results
• Slit 1 only P1 Distribution
• Slit 2 only P2 Distribution

3) Both Slits? P12 Interference Pattern
like waves!!!
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Result Particles Behave Like Waves!

• Thats okay we have the mathematics
• P1 ?12 ?1?1 P2 ?22 ?2?2
• P12 ?1 ?22
• Test strange behavior

Electrons

• Conditions
• Scattered Light indicates which slit
• See scattered light for each electron
• Position of electron determined

P12 ?
Electron Gun
Light Source
P1
Slit 1

• Observables
• Lumps like before
• Always go through 1 or 2

Slit 2
P2

• Results
• Slit 1 only P1 Distribution
• Slit 2 only P2 Distribution

3) Both Slits? P12 Interference Pattern
Gone Why?!!! Did light destroy interference
pattern?
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Electrons
P12
Investigate Light Effect
Electron Gun
Light Source
P1
Slit 1
Slit 2
P2
1) Dim light Stays the same 1 e-, one flash,
same pattern Dim more e- detected at screen,
but no flash - How can this be? Light
behaving as particles (photoelectric effect)
Pattern? e-s seen, no interference e-s not
seen, interference pattern!! Turn the
brightness back up 2) Change ? de Broglie p
h/? Increase ? decreases momentum

Treat light like matter waves At first no
change, still no interference pattern
Increase ? more, interference pattern back -
Yay!!! But!!!! Now wavelength gt distance
between slits We lost position information
when we get interference pattern Conclusion
Impossible to define position have interference
pattern
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Heisenberg Indeterminacy Principle

• Impossible to build apparatus that will determine
  e- position
• and maintain interference pattern.
• What does this mean?
• a) Slit is just size of e-, so defines e-
  position exactly
• b) From interference pattern can determine ?
  which gives p
• c) Thus, we cannot know exact position and
  exact momentum simultaneously
• ?x ?px ? 0
• Complimentary variables x, px no longer
  treated classically
• Heisenberg via Born (Jordon overhears on train)
• x px do not commute (xpx pxx) ? 0
• 
  ?
• 
  i h/2p ih

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• Q. M. Fundamentals
• xpx-pxx x,px ih
• Particles decribed by ? (Wavefunction)
• ?? interpreted as probability density
• dP ??dt
• Bullets?? Are they different?
• No!
• Too fine of interference pattern to detect!