Transcript Micro_lect14

Quantum Theory
Micro-world Macro-world
Lecture 14
Light propagates like a wave,
& interacts as a particle
p=
E=hf
h
l
Louis deBroglie
Wave-particle duality is
a universal phenomenon
If light behaves as particles,
maybe other “particles” (such
as electrons) behave as waves
Photons:
particles:
h
p =
l
h
 l =
p
h
 “deBroglie
l =
p
wavelength”
Ordinary-sized objects have tiny
wavelengths
30m/s
-34Js
6.6x10
h = h =
l =
p
mv 0.2kg x 30 m/s
=
6.6x10-34Js = 1.1x10-34m
6.0 kg m/s
Incredibly small
the wavelength of an electron
is not so small
9x10-31 kg
6x106 m/s
h = h =
l =
p
mv
=
6.6x10-34Js
9x10-31kg x 6x106 m/s
6.6x10-34Js
5.4x10-24 kg m/s
= 1.2x10-10m
About the size of an atom
Send low-momentum electrons thru
narrow slits
See a diffraction pattern characteristic of
wavelength l=h/p as predicted by de Broglie
or through a small hole
“Diffraction”
rings
Matter waves
(electrons through a crystal)
“Diffraction”
rings
Electron waves through a narrow
slit acquire some py
y
x
Dy
py
py
Waves thru a narrower slit
y
x
py
Dy
wider
py
When the slit becomes narrower, the
spread in vertical momentum increases
y
x
l
Dy/2
h
h
p
2
sin  


Dy
Dy p  Dy
2
p sin   Dy  h
p y Dy  h
Heisenberg Uncertainty Principle
Dy Dpy > h
Uncertainty
in location
Uncertainty in
momentum in
that direction
If you make one of these smaller, the
other has to become bigger
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 > h
Localize a baseball
Dx Dpx > h
h
Dpx >
Dx
Suppose Dx= 1x10-10m
Dpx >
6.6x10-34Js
1x10-10m
About the size of
a single atom
= 6.6x10-24kgm/s
A very tiny uncertainty
Dvx >
6.6x10-44Js
Dpx
-23 m/s
=
3.3x10
m =
0.2kg
Localize an electron
-
me=9x10-31kg
h
Dpx >
Dx
Dx Dpx > h
About the size of
a single atom
Suppose Dx= 1x10-10m
Dpx >
6.6x10-34Js
1x10-10m
= 6.6x10-24kgm/s
Huge, about 2% of c
Dvx >
6.6x10-24Js
Dpx
me = 9x10-31kg
= 7x106 m/s
uncertainty is inherent
in the quantum world
The Bohr-Rutherford Atom
Nils
Bohr
Ernest
Rutherford
Physics 100
Chapt 23
1895 J.J. Thomson discovered
electron
Vacuum flask
-
+
++
++
---
cathode
anode
“cathode rays”
Cathode rays have negative charge
and very small mass
S
-
+
++
++
---
cathode
anode
N
m=0.0005MHydrogen
Plum pudding?
Positively charged
porridge
Negatively charged
raisins (plums)
++
+
- -
+
+
+
+
- - + -+
+
+
+
-+
-+
- +
+
10-10m
Planetary-like?
Positively charged
dense central nucleus
-
-
+
-
-
10-10m
Negatively charged
orbiting electrons
Rutherford Experiment
Vacuum
flask
a-rays
What’s in the box?
Is all the mass spread
throughout as in a box
of marshmallows”?
or is all the mass
concentrated in a dense
“ball-bearing”?
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
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
Rutherford used a-ray “bullets” to
distinguish between the plumpudding & planetary models
Plum-pudding:
a
++
a
a
- -
+
+
+
+
- - + -+
+
+
+
-+
-+
- +
+
a
+
no way for a-rays to scatter at wide angles
distinguishing between the plumpudding & planetary models
a
a
a
a
-
+
-
-
Occasionally, an a-rays will be pointed head-on
to a nucleus & will scatter at a wide angle
Rutherford saw ~1/10,000
a-rays scatter at wide angles
a
a
a
a
-
+
-
-
from this he inferred a nuclear
size of about 10-14m
Rutherford atom
10-10m
Not to scale!!!
10-14m
+
If it were to scale,
the nucleus would
be too small to see
Even though it has
more than 99.9%
of the atom’s mass
Relative scales
Aloha stadium
Golf ball
x10-4
Atom
Nucleus
99.97% of the mass
Classical theory had trouble with
Rutherford’s atom
Orbiting electrons are accelerating
Accelerating electrons should radiate light
According to Maxwell’s
theory, a Rutherford
atom would only survive
for only about 10-12 secs
sola
Other peculiar discoveries:
Solar light spectrum:
Fraunhofer discovered that some wavelengths are
missing from the sun’s black-body spectrum
Other discoveries…
Low pressure gasses, when heated,
do not radiate black-body-like spectra;
instead they radiate only a few specific colors
bright colors from hydrogen match
the missing colors in sunlight
Hydrogen spectrum
Solar spectrum
Bohr’s idea
“Allowed”
orbits
Hydrogen energy levels
4
3
1
+
Hydrogen energy levels
2
1
+
What makes Bohr’s allowed energy
levels allowed?
Recall what happens when we force
waves into confined spaces:
Confined waves
Only waves with wavelengths that just fit
(all others cancel themselves out)
in survive
Electrons in atoms are confined
matter waves ala deBroglie
However, if the circumference is exactly an
integer
number
of wavelengths,
This
wave,
as it goes
around, will successive
interfere
turns
will interfereand
constructively
with itself
destructively
cancel itself out
Bohr’s allowed energy states correspond to those
with orbits that are integer numbers of wavelengths
Bohr orbits
Bohr orbits
Quantum Mechanics
Erwin Schrodinger
h

2
 d


V
(
x
)


E

2
2m dx
2
2
Schrodinger’s equation
Matter waves are “probability”
waves
Probability to detect the
electron at some place
is  2 at that spot

Electrons will never be
detected here
or here
Electrons are most likely
to be detected here
Electron “clouds” in Hydrogen
Nobel prizes for Quantum
Theory
1918 Max Planck
1921 Albert Einstein
1922 Niels Bohr
1929 Louis de Broglie
1932 Werner Heisenberg
…
E=hf
photons
atomic orbits
matter waves
uncertainty principle