Quantum Theory

1
Quantum Theory
Micro-world Macro-world Lecture 14
2
Light propagates like a wave, interacts as a
particle
h l
p
Ehf
3
Louis deBroglie
Wave-particle duality is a universal phenomenon
If light behaves as particles, maybe other
particles (such as electrons) behave as waves
h l
h p
p
Photons
? l
h p
particles
l

• deBroglie wavelength

4
Ordinary-sized objects have tiny wavelengths
30m/s
0.2kg
6.6x10-34Js 0.2kg x 30 m/s
h p
h mv


l
6.6x10-34Js 6.0 kg m/s
1.1x10-34m


Incredibly small
5
the wavelength of an electronis not so small
9x10-31 kg
6x106 m/s
-
h p
h mv
6.6x10-34Js 9x10-31kg x 6x106 m/s


l
6.6x10-34Js 5.4x10-24 kg m/s
1.2x10-10m


About the size of an atom
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Send low-momentum electrons thru narrow slits
See a diffraction pattern characteristic
of wavelength lh/p as predicted by de Broglie
7
or through a small hole
Diffraction rings
8
Matter waves(electrons through a crystal)
Diffraction rings
9
Electron waves through a narrow slit acquire some
py
y
x
py
Dy
py
10
Waves thru a narrower slit
y
x
py
Dy
wider
py
When the slit becomes narrower, the spread in
vertical momentum increases
11
y
x
q
Dy/2
l/2
12
Heisenberg Uncertainty Principle
Dy Dpy gt h
Uncertainty in momentum in that direction
Uncertainty in location

If you make one of these smaller, the other has
to become bigger
13
Heisenberg tries to measure the location of an
atom
For better precision, use a shorter wavelength
But then the momentum change is higher
Dx Dpx gt h
14
Localize a baseball
h Dx
Dpx gt
Dx Dpx gt h
0.2kg
Suppose Dx 1x10-10m
About the size of a single atom
6.6x10-34Js 1x10-10m
6.6x10-24kgm/s
Dpx gt
A very tiny uncertainty
Dpx m
6.6x10-44Js 0.2kg
Dvx gt
3.3x10-23 m/s

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Localize an electron
-
h Dx
Dx Dpx gt h
Dpx gt
me9x10-31kg
Suppose Dx 1x10-10m
About the size of a single atom
6.6x10-34Js 1x10-10m
6.6x10-24kgm/s
Dpx gt
Huge, about 2 of c
Dpx me
6.6x10-24Js 9x10-31kg
Dvx gt
7x106 m/s

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uncertainty is inherentin the quantum world
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The Bohr-Rutherford Atom
Nils Bohr
Ernest Rutherford

• Physics 100
• Chapt 23

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1895 J.J. Thomson discovered electron
Vacuum flask
cathode

- - -
cathode rays

-

anode
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Cathode rays have negative chargeand very small
mass
S
cathode

- - -

-

anode
N
m0.0005MHydrogen
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Plum pudding?
Positively charged porridge
Negatively charged raisins (plums)




-
-
-





-
-
-




-
-


-

10-10m
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Planetary-like?
Positively charged dense central nucleus
Negatively charged orbiting electrons
-
-
-

-
-
10-10m
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Rutherford Experiment
Vacuum flask
a-rays
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Whats in the box?
or is all the mass concentrated in a
dense ball-bearing?
Is all the mass spread throughout as in a box of
marshmallows?
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Figure it out without opening (or shaking)
Shoot bullets randomly through the box. If it
is filled with marshmallows, all the bullets
will go straight through without (much) deflection
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Figure it out without opening (or shaking)
If it contains a ball-bearing most the bullets
will go straight through without deflection---but
not all
Occasionally, a bullet will collide nearly
head-on to the ball-bearing and be deflected by a
large angle
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Rutherford used a-ray bullets to distinguish
between the plum-pudding planetary models
Plum-pudding




a
-
-
-




a

-
-
-


a


-
-
a


-

no way for a-rays to scatter at wide angles
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distinguishing between the plum-pudding
planetary models
-
a
a
-
-

a
a
-
-
Occasionally, an a-rays will be pointed head-on
to a nucleus will scatter at a wide angle
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Rutherford saw 1/10,000a-rays scatter at wide
angles
-
a
-
-
a

a
a
-
-
from this he inferred a nuclear size of about
10-14m
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Rutherford atom
10-10m
Not to scale!!!
If it were to scale, the nucleus would be too
small to see
10-14m

Even though it has more than 99.9 of the atoms
mass
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Relative scales
Aloha stadium
Golf ball
x10-4
Nucleus 99.97 of the mass
Atom
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Classical theory had trouble with Rutherfords
atom
Orbiting electrons are accelerating
Accelerating electrons should radiate light
According to Maxwells theory, a Rutherford atom
would only survive for only about 10-12 secs
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Other peculiar discoveries
sola
Solar light spectrum
Fraunhofer discovered that some wavelengths
are missing from the suns black-body spectrum
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Other discoveries
Low pressure gasses, when heated, do not radiate
black-body-like spectra instead they radiate
only a few specific colors
34
bright colors from hydrogen match the missing
colors in sunlight
Hydrogen spectrum
Solar spectrum
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Bohrs idea
Allowed orbits
36
Hydrogen energy levels
4
3
1

37
Hydrogen energy levels
2
1

38
What makes Bohrs allowed energy levels allowed?
Recall what happens when we force waves into
confined spaces
39
Confined waves
Only waves with wavelengths that just fit in
survive (all others cancel themselves out)
40
Electrons in atoms are confined matter waves ala
deBroglie
However, if the circumference is exactly an
integer number of wavelengths, successive turns
will interfere constructively
This wave, as it goes around, will interfere with
itself destructively and cancel itself out
Bohrs allowed energy states correspond to
those with orbits that are integer numbers of
wavelengths
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Bohr orbits
42
Bohr orbits
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Quantum Mechanics
Erwin Schrodinger
Schrodingers equation
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Matter waves are probability waves
Probability to detect the electron at some place
is ? y2 at that spot
y
Electrons will never be detected here
Electrons are most likely to be detected here
or here
45
Electron clouds in Hydrogen
46
Nobel prizes for Quantum Theory

• Max Planck Ehf
• 1921 Albert Einstein photons
• 1922 Niels Bohr atomic orbits
• 1929 Louis de Broglie matter waves
• 1932 Werner Heisenberg uncertainty principle