Transcript Lecture 1 - UW Canvas

LECTURE 6
QUANTUM PHYSICS II
Instructor: Shih-Chieh Hsu
Photoelectric effect
2

In photoelectric effect, electrons are emitted from
matter after the absorption of energy from
electromagnetic radiation.
Kmax = hf - f
The slope is h.
Photons
3


Einstein explained the photoelectric effect
experimental result using light quanta, photons.
The energy E of each photon is given by
E = hf =

hc
l
Einstein equation
where f and  are the frequency and the wavelength of the
light, respectively, h is called Planck’s constant and
measured to be:
h = 6.626  10-34 J·s = 4.136  10-15 eV·s.
Momentum of a photon
4

Energy, Momentum, Mass equation
( )
E = p c + mc
2


2 2
2
2
Photon rest mass is zero. The momentum of photon
and energy is related by
E = pc
Therefore the momentum of a photon is
E hf h
p= =
=
c
c l
Compton scattering
5



In Compton scattering only
some of the energy of the
photon is transferred to an
electron upon their collision.
The electron would recoil and
thus absorb energy.
The scattered photon would
have less energy, and
therefore a lower frequency
and longer wavelength, than
the incident photon.
Matter Waves
6
Tipler chapter 34-5 to 34-10






Electrons and Matter waves
The Interpretation of the Wave function
Wave-Particle Duality
A Particle in a box
Energy quantization in Other system
Development of Quantum Mechanics
7

In 1862, Kirchhoff coined black body
radiation or known as cavity radiation


The experiments raised the question of the failure
of classical EM theories
In 1887, Heinrich Hertz discovers photoelectric
effect

Another experiment raised the concern of the wave and
particle nature of the light.
Theory of quantization of light
8

In 1900, Max Planck resolves the blackbody
radiation issues by introducing “quantum” concept
of the discrete energy element



The energy element is discrete and the energy is
proportional to the frequency
The invention of Planck constant h
In 1905, Einstein explained photoelectric effect by
using Max Planck’s light quantization concept

Photon is introduced by Gilbert N. Lewis in 1926
More Quantization System
9


In 1909, Robert Millikan conducted
oildrop experiment and showed
that electric charge is quantized.
In 1911, Ernest Rutherford’s Gold Foil
Experiment disproved the plum
pudding model of the atom.
Old Quantum Theory
10


In 1913, Niels Bohr explains the spectra
line of the hydrogen atom – using
quantization ideas.
1918-1923 expansion of quantum
mechanics research works! e.g.
Stern-Gerlach demonstrated spin
property of electrons in 1920
Stern
Gerlach
New Quantum Theory
11


In 1924, Louise de Broglie proposes
matter wave theory.
In 1925, Matrix Mechanics is invented

Heisenberg Uncertainty
was proposed in 1927
Werner Heisenberg
Max Born
Pascual Jordan
Completion of Quantum Mechanics
12


In 1925, Erwin Schrödinger invented wave
mechanics and non-relativistic Schrödinger
equation as generalization of de Broglie’s
theory
1927, Paul Dirac began the process of
unifying quantum mechanics with special
relativity by proposing the Dirac equation
for the electron.
Black-Body Radiation
13


"Blackbody radiation" or "cavity radiation" refers
to an object or system which absorbs all radiation
incident upon it.
It re-radiates energy which is characteristic of this
radiating system only, not dependent upon the type
of radiation which is incident upon it.
Black-Body Radiation Spectrum
14

The spectrum of blackbody radiation shows that some
wavelengths get more energy than others.
Otto Lummer and Ferdinand Kurlbaum (1898)
Classical Theory Prediction
15

Classical theory predicts Energy density is higher at
higher frequency.
Quantum Theory comes the rescue
16

Planck solves the problem by introducing quantum of
energy element as function of frequency
Discovery of electron
17




J. J. Thomson discovered electrons using
a cathode-ray tube in 1897.
By applying electric and magnetic fields to the ray
and observing that it deflects, he concluded that the
ray is negatively charged.
By measuring the amount of deflection, he measured
the charge-to-mass ratio.
The particles in the ray always had the same ratio,
so he concluded that they are fundamental particles.
Thomson’s plum-pudding model
18

J. J. Thomson proposed the “plum-pudding model,”
the first atomic model.
consist of a “cloud” of positive charge roughly 1010 m in diameter.
 Atoms
First nuclear physics experiment
19

Around 1911 Ernest Rutherford, with his students, Geiger and
Marsden, showed that the positive charge and most of the
mass of an atom is concentrated in a very small region
(~1 fm), now called the nucleus.
Rutherford scattering
20
Rutherford scattering
21

For  particles to scatter at ~180 the Coulomb force must be
extremely strong.
Bohr’s classical model of H
22

Niels Bohr proposed a model of
the hydrogen atom that
successfully predicted the
observed spectra.
 The
electron of the hydrogen atom
moves in a circular orbit around
the positive nucleus according to
Coulomb’s law and classical
mechanics like the planets orbiting
around Sun.
Flaw in the classical model
23


Classical EM theory says that an electron in a
circular orbit is accelerating, so it would radiate an
EM wave and loses its energy.
This atom would quickly collapse as the electron
spirals into the nucleus and radiates away the
energy.
Spectral Lines
24
http://chemistry.tutorvista.com/inorganic-chemistry/spectral-lines.html
Hydrogen Spectrum
25
Bohr’s semi-classical model (1913)
26
Energy Quantization
In Aton
En energy level
|ΔEn| = hf=hc/λ
27
Old Quantum Theory
Three Failures of Classical Physics
28

Black Body Radiation.


Photoelectric Effect.
The Hydrogen Atom
29
New Quantum Theory
de Broglie hypothesis
30

In 1924 Louis de Broglie hypothesized:
 Since
light exhibits particle-like properties and act as a
photon, particles could exhibit wave-like properties and
have a definite wavelength.

The wavelength and frequency of matter:
h
l=
p
 For
E
f =
h
macroscopic objects, de Broglie wavelength is too
small to be observed.
Example 1
31

One of the smallest composite microscopic particles
we could imagine using in an experiment would be a
particle of smoke or soot. These are about 1 m in
diameter, barely at the resolution limit of most
microscopes. A particle of this size with the density of
carbon has a mass of about 10-18 kg. What is the de
Broglie wavelength for such a particle, if it is moving
slowly at 1 mm/s?
h
6.626 ´10-34 Js
h
h
-13
=
=
6.626
´10
m
l = = = -18
-18
-3
-3
p mv 10 kg ´10 m / s 10 kg ´10 m / s
h = 6.626  10-34 J·s = 4.136  10-15 eV·s
Diffraction of matter
32




In 1927, C. J. Davisson and L. H. Germer first observed the
diffraction of electron waves using electrons scattered from a
particular nickel crystal.
G. P. Thomson (son of J. J. Thomson) showed electron diffraction
when the electrons pass through a thin metal foils.
Diffraction has been seen for neutrons, hydrogen atoms, and
alpha particles.
In all cases, the measured  matched de Broglie’s prediction.
X-ray diffraction
electron diffraction
neutron diffraction
Interference and diffraction of matter
33


If the wavelengths are made long enough (by using very slow
moving particles), interference patters of particles can be
observed.
These figures show the build up of the electron two-slit
interference pattern as the electrons arrive at the detector.
Electron microscope
34


When viewing details of objects with visible light, the
details can be resolved only if they are larger than the
wavelength of the light.
In electron microscopes, beams of electrons, with small
wavelength is used to “see” small objects.
Pollen
grains
Clicker Question 18-1

The electron microscope is a welcome addition to the
field of microscopy because electrons have a
__________ wavelength than light, thereby increasing
the __________ of the microscope.




longer; resolving power
longer; breadth of field
shorter; resolving power
longer; intensity
Classical waves vs. particles
36

A classical wave behaves like a sound wave.
• It exhibits diffraction and interference.
• Its energy is spread out continuously in space and time.

A classical particle behaves like a piece of shot.
It
can be localized and scattered.
It exchanges energy suddenly at a point in space.
It obeys the laws of conservation of energy and
momentum in collisions.
It does not exhibit interference or diffraction.
Wave-particle duality
37

Light, normally thought of as a wave, exhibits particle
properties when it interacts with matter.
• photoelectric effect

Electrons, normally thought of as particles, exhibit the wave
properties when they pass near the edges of obstacles.
• interference and diffraction


All carriers of p and E exhibit both wave and particle
characteristics.
In classical physics, the concepts of waves and particles are
mutually exclusive.
Wave-particle duality
38

The classical concepts of waves and particles do not
adequately describe the complete behavior of any
phenomenon.
Everything propagates like a wave and exchanges energy
like a particle.

Often the concepts of the classical particle and the classical wave
give the same results.
If  is very small,
diffraction and interference are not observable.
 If there are a lot of particles,
they can be treated as a wave.

Clicker Question 19-1

If the wavelength of an electron is equal to the wavelength of
a proton, then.
1.
2.
3.
4.
5.
the speed of the proton is greater than the speed of the electron
the speeds of the proton and the electron are equal
the speed of the proton is less than the speed of the electron
the energy of the proton is greater than the energy of the electron,
both (1) and (4) are correct.
h
h
l= =
p
mv
mv
E=
2
2
p
=
2m
2
h2
= 2
2ml
Uncertainty principle
40



If we use light with  to measure the position of an object, x, its
uncertainty, x, cannot be less than ~ because of diffraction.
If we use photons with p = h/ to measure the momentum of
an object, p, p of the object cannot be less than ~h/ since
the photon changes the momentum of the object upon
scattering.
The Heisenberg uncertainty principle states that:
It is impossible to simultaneously measure both the
position and the momentum of a particle with unlimited
precision.
DxDpx ³
2
, where
º
h
2p
Quantum Mechanics (1923)
41

In quantum mechanics, a particle is described by a wave
function  that obeys a wave equation called the Schrödinger
equation.
¶
Ñ Y ( r ,t ) + U ( r ) Y ( r ,t ) = i
Y ( r ,t )
2m
¶t
2
2
You absolutely do not need to memorize the formula.

The solution of the equation by itself has no physical meaning.
However, the probability to find a particle in a certain spacetime is:
Time-Independent Schrodinger Equation
42

Solution of the Schrödinger equation.
¶
Ñ Y ( r ,t ) + U ( r ) Y ( r ,t ) = i
Y ( r ,t )
2m
¶t
2
2
Wave function
43


The Schrödinger equation describes a single
particle.
The probability density P(x), the probability per unit
volume (or length in 1-D), of finding the particle as
a function of position is given by
()
()
P x =y2 x

The probability is probability times unit volume, i.e.
P(x) Δx
Normalization condition
44

If we have a particle, the probability of finding the
particle somewhere must be 1. Therefore the wave
function must satisfy the normalization condition.
¥
¥
ò P ( x ) dx = ò y ( x ) dx = 1
2
-¥

-¥
For  to satisfy the normalization condition, it must
approach zero as |x| approaches infinity.
45
Backup
Compton scattering: conservation of p and E
46

From conservation of momentum:
pi = ps + pe ®
(
pe = pi - ps
)(
pe × pe = pi - ps × pi - ps
)
= pi × pi + ps × ps - 2 pi × ps
= pi2 + ps2 - 2 pi ps cos q

From conservation of energy:
Ei + Ee, i = Es + Ee, f
pic + me c = ps c +
2
(
p c + me c
2 2
e
2
)
2
Compton equation
47

Combining the momentum and energy conservation
equations, we get
pe2 = pi2 + ps2 - 2 pi ps cos q
pic + me c = ps c +
2
(
p c + me c
2 2
e
2
)
2
1
1
1
- =
(1 - cos q )
ps pi me c
h
ls - li =
(1 - cos q ) º lC (1 - cos q ) = 2.426 pm(1 - cosq )
me c
Expectation Value
48
Example
49
A particle in a one-dimensional box of length L is in the
ground state. Find the probability of finding the particle (a) in
the region that has a length Δx = 0.01L and is centered at x =
L and (b) in the region 0 < x < L.
(a) 0
(b) 1
Example
50
A particle in a one-dimensional box of length L is in the
ground state. Find the probability of finding the particle (a) in
the region that has a length Δx = 0.01L and is centered at x =
L/2 and (b) in the region 0 < x < L/4.
Example 3
51

The photons in a monochromatic beam are scattered
by electrons. The wavelength of the photons that are
scattered at an angle of 135° with the direction of
the incident photon beam is 2.3 percent more than
the wavelength of the incident photons.
a)
b)
What is the wavelength of the incident photons?
What is the kinetic energy of the electron?
Harmonic oscillator potential well
52


Consider a particle with mass, m, on a spring with force
constant, k.
Potential energy function for a harmonic oscillator is parabolic.
()
U x = 12 kx 2 = 12 mw 02 x 2
where w 0 = k m is the natural

frequency of the oscillator.
Classically, the object oscillates between
±A, and its total energy, E, can have
any nonnegative value, including zero.
1
Eclassic = mw 02 A2
2
Parabolic well
Harmonic oscillator: allowed energies
53

Normalizable n(x) occur only for discrete values of
the energy En given by
(
)
En = n + 12 hf0
n = 0, 1, 2,
Note that the ground
state energy is not 0.
Equally spaced levels: hf0
Example 3
54

An electron in a harmonic oscillator is initially in the
n = 4 state. It drops to n = 2 state and emits a
photon with wavelength 500 nm. What is the
ground state energy of this harmonic oscillator?
(
)
En = n + 12 hf 0
E4 - E2
= 2hf 0
1240ev ·nm
=
= 4.96eV
500nm
hc = 1240 eV·nm = 1.988
 10-25 J·m
(
)
E0 = 0 + 12 hf 0 = 4.96eV / 4 =1.24eV