In the last episode of the show…
Text Example 29.1
• Photons (massless) are both particles & waves;
• Particles of matter are both particles & waves;
E = h f & p = h/!
v = !(2K/m) = !(2•1.76#10-17/9.11#10-31) = 6.22#106ms-1
p = mv = 6.22#106 • 9.11#10-31 = 5.66 # 10-24 kg ms-1
! = h/p = 6.63#10-34/5.66#10-24 = 1.17 # 10-10 m " 1.2Å
same wavelength range as X-rays ••• can do "wave-like"
e- diffraction experiments with a crystal
& for particles (de Broglie);
p = mv = !(2mK) = h/!
Hecht §29.1
Environmental: Radiation Module
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Follow-up Example
Hecht §29.1
Environmental: Radiation Module
• Solution:
p = mv = 5.7 # 10-2kg • 50ms-1 = 28.5 kg ms-1
! = h/p = 6.63 # 10-34/28.5 = 2.33 # 10-34 m
which is so small (atom spacing ~10-10m) that one could
never measure its wave properties. Definitely "particlelike"
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Schrödinger's Wave Equation
• Particle-like properties and wave-like properties
are complementary, 2 sides of a coin i.e. not
apparent at the same time.
• Mathematical example;
Remember; p = h/! & E = h f = h/T (T = wave period)
•
• • ET = p! = h
E & p are particles-like properties. T & ! are
wave-like properties. If E & p are large then T & !
are too small to detect and vice-versa
Hecht §29.2
Environmental: Radiation Module
• Use simplest example of a wavefunction ";
than de Broglie's equations; E = h f & p = h/!
e.g.
The function which describes a de Broglie wave is called
"wavefunction" "(x) (or " for simplicity)
• Let's try taking "double derivative":
but
Start with classical mechanics equations;
• K= 1/2 mv2 = p2/2m
• E = Ekin + Epot = p2/2m + U
# both sides of equation by
But p2 = h /!2 so Ekin = h2/2m!2
2
("kinetic energy operator") &
• To find Ekin we need a method to extract !2 from "(x)
Environmental: Radiation Module
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Schrödinger's Wave Equation 2
Schrödinger developed a more general equation
Hecht §29.3
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Complementarity Principle (Bohr)
• Calculate de Broglie ! for a tennis ball (57 g) with
speed 50.0ms-1.
Environmental: Radiation Module
• Calculate de Broglie ! for e- accelerated through
110V. Compare with typical X-ray wavelength.
Assume non-relativistic speed i.e. K= 1/2 mv2
• Solution: 1st find electron kinetic energy;
K = Vq = 110 • 1.60 # 10-19 = 1.76 # 10-17 J = 1/2 mv2
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Hecht §29.3
Environmental: Radiation Module
so…
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Schrödinger's Wave Equation!
What is a wavefunction?
• We used a simple sine wave to show this, but it's
true for all quantum wavefunctions.
• To find a wavefunction "(x), substitute in a function to
represent potential energy U and solve. "differential equation"
DON'T PANIC…
P.S. This treatment is only meant to be qualitative. You
don't need to derive or solve this equation in the exam.
Hecht §29.3
Environmental: Radiation Module
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Wavefunction & probability
• Red arrows % mark positions where the particle is least
likely to be found
• "Found" means, absorbed, detected, scattered etc. in
other words, any experiment which reveals the particlelike property "position".
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HUH?!!
8
• At first pattern of photons appears random, but after
a long time, all the individual photons build up to
reveal an interference pattern (wave property)
• Which slit did each photon go through? BOTH at the
same time! Can only get interference if wave passes
through both slits. HUH?!!
• Also works with beams of e-, individual atoms and
(theoretically!) human beings
Hecht Fig 29.3
Environmental: Radiation Module
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Please stop freaking me out
• While light (or e- etc.) travels through extended space,
source & slits & screen, its wavefunction is wave-like,
passing through both slits, undergoing interference.
• BEFORE any photon is seen, @ screen, there is an
undetectible interference pattern of bands of large "2 and
small "2. Where "2 is large, from time to time, a photon
is most likely to be scattered and made visible as a small
flash. Over time, all the flashes add up to a pattern of
bright interference fringes.
• You can't measure ",only individual photon positions.
Environmental: Radiation Module
Environmental: Radiation Module
• Reduce intensity until only one photon (particle) at
a time passes through slits then appears on the
screen. (Permanently record photon positions).
• Black arrows $ mark positions where
the particle is most likely to be found
Hecht §29.3
Hecht §29.3
Young's double slit:
• Square it to find probability density "2
Environmental: Radiation Module
• "2 is called "probability density"
In 2 places at once..
• Consider a simple wavefunction " for a
single particle
Hecht §29.3
• A "ripple" is a wave on the surface of water
• Sound is a pressure wave in air
• Light is wave in the electromagnetic field
" So what kind of wave is a quantum wavefunction?
" Max Born 1926; "(x) is a wave of probability.
The probability of finding the particle at a
particular position is proportional to the square of
the wavefunction "2(x) at that position.
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• Exactly when & where photon (or e- etc) appears is
random, but its probability is proportional to "2.
On small scale a particle "trajectory" has little meaning.
• @ the moment photon is detected (converted to a flash)
its wave-like, spread-out wavefunction "collapses" to be
replaced by a concentrated particle-like wavefunction.
• "Collapse of the wavefunction" is not yet understood but
is believed to be instantaneous.
• The act of measuring particle position causes the
wavefunction to collapse into a definite position.
• Is the moon really there if no one's there to see it?
Environmental: Radiation Module
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