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Announcements
Homework assignments updated
Two are due on Fridays to allow for
time after the exam.
Chapter 40 (“Quantum Mechanics”) has
two homework assignments.
I am away at KEK (Japan) Oct 13-25.
Professor Lam will replace me and
continue the iclicker reading quizzes.
Copyright © 2012 Pearson Education Inc.
Today: Heisenberg’s Uncertainty Principle
Some “left-overs” on
Compton scattering/pair
production.
We will then read the
original 1927 paper in
German by Heisenberg.
Not joking: Will have
some extra (4) clicker
questions towards the
end of class to check
your understanding of
the Heisenberg
uncertainty principle.
Copyright © 2012 Pearson Education Inc.
Left overs: Compton scattering in “real life”
Copyright © 2012 Pearson Education Inc.
Compton scattering example
An photon with a wavelength of 0.100nm collides with an
electron initially at rest. The x-ray’s final wavelength is
0.110 nm. What is the final kinetic energy of the electron ?
Energy is conserved
hc
E=
KEelectron =
(6.63 ´ 10
l
hc
l
-34
-
;E =
'
hc
l
'
hc
l'
E-E =
'
hc
l
-
hc
l
'
= KEelectron
=
1
1
Js)(3 ´ 10 m / s)(
)
-9
-9
0.10 ´ 10 m 0.11 ´ 10 m
Copyright © 2012 Pearson Education Inc.
8
Compton scattering example (result in Joules and keV)
A photon with a wavelength of 0.100nm collides with an
electron initially at rest. The x-ray’s final wavelength is
0.110 nm. What is the final kinetic energy of the electron ?
KEelectron =
(6.63 ´ 10
hc
l
-34
-
hc
l
'
=
1
1
Js)(3 ´ 10 m / s)(
)
-9
-9
0.10 ´ 10 m 0.11 ´ 10 m
8
KEelectron = 1.81´10-16 J = 1.13keV
Question:: How do you convert to eV ?
Ans: Remember 1 e =1.6 x 10-19 C
Copyright © 2012 Pearson Education Inc.
Compton scattering conceptual question
Question: Do you expect to notice a wavelength shift in
Compton scattering from visible light ?
Ans: No, the quantity h/(mec)= 2.42 x 10-12m =
0.00242nm. Compare to visible light wavelengths
380-750 nm.
The effect is much more dramatic with X-rays.
Copyright © 2012 Pearson Education Inc.
Pair production by a single photon
Question : Why are
the two tracks
curving in opposite
directions ?
Ans: Particle and
antiparticle have
opposite charges ?
Copyright © 2012 Pearson Education Inc.
Bohr’s principle of complementarity
1928: The wave descriptions and the particle descriptions
of nature are “complementary”. We need both to describe
nature, but we will never need to use both at the same time
to describe a single part of an occurrence.
Copyright © 2012 Pearson Education Inc.
Scene from the foundations of quantum mechanics.
1928-1935: In this period, Niels Bohr and Albert Einstein
had many discussions about quantum mechanics.
Einstein:
“God does
not play dice
with the
universe”
Bohr: Einstein
stop it ! Stop
telling God
what to do !
Copyright © 2012 Pearson Education Inc.
Let’s think about formation of Optical Images
The point:
Processes that seem to be continuous may,
in fact, consist of many microscopic “bits”.
(Just like water flow.)
For large light intensities, image formation
by an optical system can be described by
classical optics.
For very low light intensities, one can see the
statistical and random nature of image formation.
Use a sensitive camera that can detect single photons.
Copyright © 2012 Pearson Education Inc.
Exposure time
A. Rose, J. Opt. Sci. Am.
43, 715 (1953)
Diffraction and the particle nature of light
• A diffraction pattern is the result of many photons hitting
the screen. The pattern appears even if only one photon is
present at a time in the experiment.
Copyright © 2012 Pearson Education Inc.
Diffraction and uncertainty
• A diffraction pattern is the result of many photons hitting
the screen. The pattern appears even if only one photon is
present at a time in the experiment.
Question: Diffraction is the result of the wave
nature of light ? How is this possible ?
Ans: Light is both a wave and a particle. Different measurements
reveal different aspects. “Complementarity”.
Copyright © 2012 Pearson Education Inc.
Diffraction and the road to the uncertainty principle
• When a photon passes through a narrow slit, its momentum
becomes uncertain and the photon can deflect to either side
A diffraction pattern is the result of many photons hitting
the screen. The pattern appears even if only one photon is
present at a time in the experiment.
Copyright © 2012 Pearson Education Inc.
Diffraction and uncertainty
• Question: What is the location of the first minimum in
single slit diffraction ?
ml
sin q =
a
Although photons are point
particles, we cannot predict
their paths exactly.
All we can do is predict
probabilitiesThere are
uncertainties in their positions
and momenta
Copyright © 2012 Pearson Education Inc.
In the small angle
approximation
Uncertainty in momentum and position (rough argument)
• What is the uncertainty in the momentum of each photon ?
~85% of the photons lie in the
angular range between [+θ1 , -θ1]
py = px tanq1 » pxq1
Using the small
angle approximation
py = pxq1 = px
l
a
The average value of py is zero.
However, the value of py is uncertain
l
h
Dpy ³ px = px
Þ Dpy a ³ h
a
px a
Copyright © 2012 Pearson Education Inc.
Uncertainty in momentum and position (rough argument)
• How are uncertainties in the momentum and position of
each photon related ?
l
h
Dpy ³ px = px
Þ
a
px a
Dpy a ³ h
Here a is related to the uncertainty in the yposition of the photon while Δpy is the ymomentum uncertainty.
Question: What happens to the width of the central
maximum in diffraction when the slit width a decreases ?
Ans: The peak becomes broader. What does this mean
for uncertainties ?
Copyright © 2012 Pearson Education Inc.
Heisenberg Uncertainty Principle (exact result)
• How are uncertainties in the momentum and position of
each photon related ?
DxDpx ³ h / 2
Here Δx and Δpx are the standard deviation or
(rms) uncertainties in x and px
The quantity on the right is “h-bar” over 2.
h
-34
h=
= 1.05 ´10 J · s
2p
Warning:
“h-bar” is
not the
same as h
DxDpx ³ h / 2;DyDpy ³ h / 2;DzDpz ³ h / 2
Copyright © 2012 Pearson Education Inc.
Heisenberg Uncertainty Principle (another version)
• How are uncertainties in the energy and time localization
related ?
DEDt ³ h / 2
Here ΔE and Δt are the standard deviation or
(rms) uncertainties in E and t.
The quantity on the right is “h-bar” over 2.
h
-34
h=
= 1.05 ´10 J · s
2p
In Chapter 39, we will see that the Heisenberg
uncertainty principle also applies to matter particles
Copyright © 2012 Pearson Education Inc.
The Heisenberg Uncertainty Principle
1932
Nobel
Prize in
Physics
• You cannot simultaneously know
the position and momentum of a
photon with arbitrarily great
precision. The better you know
the value of one quantity, the less
well you know the value of the
other.
• In addition, the better you know
the energy of a photon, the less
well you know when you will
observe it.
Copyright © 2012 Pearson Education Inc.
The Heisenberg Uncertainty Principle
Can also be understood in terms of
“wave packets”.
Combine two waves with slightly different
frequencies (slightly different momenta, slightly
different energies)gives a wave packet with an
energy spread but localized in time and space.
Copyright © 2012 Pearson Education Inc.
QM: Clicker question I
A photon has a momentum uncertainty of 2.00 x 10–28 kg •
m/s. If you decrease the momentum uncertainty to 1.00 x 10–
28 kg • m/s, how does this change the position uncertainty of
the photon?
A. the position uncertainty becomes 1/4 as large
B. the position uncertainty becomes 1/2 as large
C. the position uncertainty is unchanged
D. the position uncertainty becomes twice as large
E. the position uncertainty becomes 4 times larger
Copyright © 2012 Pearson Education Inc.
QM Clicker question I
A photon has a momentum uncertainty of 2.00 x 10–28 kg • m/s.
If you decrease the momentum uncertainty to 1.00 x 10–28 kg •
m/s, how does this change the position uncertainty of the
photon?
A. the position uncertainty becomes 1/4 as large
B. the position uncertainty becomes 1/2 as large
C. the position uncertainty is unchanged
D. the position uncertainty becomes twice as
large
E. the position uncertainty becomes 4 times
larger
Copyright © 2012 Pearson Education Inc.
QM: Clicker question II
Through which of the following angles is a photon of wavelength λ
most likely to be deflected when passing through a slit of width a ?
A. θ=λ/a
B. θ=3λ/2a
C. θ=2λ/a
D. θ=3λ/a
E. Not enough information to decide.
Copyright © 2012 Pearson Education Inc.
QM: Clicker question II
Through which of the following angles is a photon of wavelength λ
most likely to be deflected when passing through a slit of width a ?
A. θ=λ/a (zero in diffraction)
B. θ=3λ/2a (non-zero intensity in the diffraction pattern
!)
C. θ=2λ/a (zero in diffraction)
D. θ=3λ/a (zero in diffraction)
E. Not enough information to decide.
Copyright © 2012 Pearson Education Inc.
QM: Clicker question 3
A photon has a position uncertainty of 2.00 mm. If you
decrease the position uncertainty to 1.00 mm, how does this
change the momentum uncertainty of the photon?
A. the momentum uncertainty becomes 1/4 as large
B. the momentum uncertainty becomes 1/2 as large
C. the momentum uncertainty is unchanged
D. the momentum uncertainty becomes twice as large
E. the momentum uncertainty becomes 4 times larger
Copyright © 2012 Pearson Education Inc.
QM Clicker question 3
A photon has a position uncertainty of 2.00 mm. If you
decrease the position uncertainty to 1.00 mm, how does this
change the momentum uncertainty of the photon?
A. the momentum uncertainty becomes 1/4 as large
B. the momentum uncertainty becomes 1/2 as large
C. the momentum uncertainty is unchanged
D. the momentum uncertainty becomes twice as large
E. the momentum uncertainty becomes 4 times larger
Copyright © 2012 Pearson Education Inc.
QM Clicker question 4
A beam of photons passes through a narrow slit. The
photons land on a distant screen, forming a diffraction
pattern.
In order for a particular photon to land at the center of the
diffraction pattern, it must pass
A. through the center of the slit.
B. through the upper half of the slit.
C. through the lower half of the slit.
D. impossible to decide
Copyright © 2012 Pearson Education Inc.
QM Clicker question 4
A beam of photons passes through a narrow slit. The
photons land on a distant screen, forming a diffraction
pattern.
In order for a particular photon to land at the center of the
diffraction pattern, it must pass
A. through the center of the slit.
B. through the upper half of the slit.
C. through the lower half of the slit.
D. impossible to decide
Copyright © 2012 Pearson Education Inc.