Download OHSE 1210 - Physics

School of Mathematical and Physical Sciences PHYS1220
PHYS1220 – Quantum Mechanics
Lecture 1
August 20, 2002
Dr J. Quinton
Office: PG 9
ph 49-21-7025
[email protected]
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School of Mathematical and Physical Sciences PHYS1220
PHYS 1220 – Quantum Mechanics
Early Quantum Theory
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Physics circa 1900
The Revolution in Physics
Blackbody Radiation
Photoelectric Effect
Compton Effect
Pair Production
Wave-Particle Duality
de Broglie’s Hypothesis
Early Atomic Models
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Wave functions
Quantum Mechanics
Schrödinger Equation
Heisenberg Uncertainty
Principle
Particle in a box
 Infinite Potential Well
 Finite Potential Well
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 Thompson
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 Rutherford
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 Bohr
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Quantum Mechanics
Barrier potential
Electron Tunnelling
Applications of Quantum
Mechanics
Correspondence Principle
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The Success of Classical Physics
At the turn of the 20th Century, it was thought that physics had just
about explained all natural phenomena.
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The known fundamental gravitational, electric and magnetic forces
were quite well understood and (successful!) theories existed to
describe them.
During the preceding 3 centuries (~1600-1900)
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Newtonian Mechanics
 Forces and motion of Particles, fluids, waves, sound
 Universal theory of gravity
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Maxwell’s Theory of Electromagnetism (EM)
 Unified electric and magnetic phenomena
 Thoroughly explained electric and magnetic behaviour
 Predicted existence of electromagnetic waves
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Thermodynamics
 Thermal processes
 Kinetic theory of gases and other materials
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Wave Theory of Light - Classical Physics
Light is an electromagnetic wave, produced by accelerating
charges (Maxwell)
Electromagnetic Spectrum
UV
IR
700nm
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600nm
500nm
400nm
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Nature of Light - Classical Physics
Light propagates by mutual induction of orthogonal electric and
magnetic fields (without the need for a medium, ie aether)
We know velocity (in free space) from wave theory
c f 
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 0 0
 299792458 m/s (exact)
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The Birth of Modern Physics
~ 1900 – only a few phenomena were not fully understood, and
were not explainable using then-known principles.
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The spectrum of light emitted by hot objects
“Light electricity” (Hertz, 1887; Hallwarchs 1888)
Hydrogen emission spectrum (Balmer, 1885)
X-rays (Roentgen, 1896)
Cathode Rays, discovery of electron (J.J. Thomson 1895-97)
Radioactivity (Becquerel 1896, Marie and Pierré Curie 1898)
 a, b and g radiation
 The big question - “What is the structure of the atom?”
However, attempts to explain these led to a revolution in physics
during the early part of the 20th century, primarily due to the
emergence of two new theories.
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Quantum Theory & Relativity
We will be discussing Quantum Mechanics from its beginnings
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Blackbody Radiation
Recall Stefan-Boltzmann law (1879, 1884)
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Describes energy dissipated through radiation
Stefan-Boltzmann constant s=5.67x10-8 W/m2.K4
dQ
 e AT 4
dt
The emissivity (0<e<1) is a measure of the materials’
ability to emit (and absorb) radiation
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For very black surfaces, e is close to 1
For bright, shiny surfaces, e is closer to zero
A blackbody is the theoretical name for the ‘ideal’ case
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All radiation that falls upon it is absorbed
Emissivity e=1
A cavity is the closest real approximation
Perfect absorbers are perfect emitters
 All thermal energy is converted to radiation
 A reasonable approximation for crystalline solids, most liquids,
many gases
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Blackbody Emission Spectrum
Total Intensity increases with T
Peak wavelength moves to shorter
wavelengths with increasing T
illustrates that the apparent colour
of an object depends on its
temperature.
Question: What is the colour
progression (with increasing T)
for incandescent materials?
Wien’s Law
pT  2.898 x 103 m  K
where lp is the wavelength at the peak of the spectrum
Wien was awarded the 1911 Nobel Prize in Physics for this work.
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Example - The Solar Spectrum
Question: The solar radiation spectrum
possesses a maximum intensity at a wavelength
of ~ 502nm (visible, green!). Assuming that ‘Sol’
is a blackbody, calculate its approximate ‘surface’
temperature in degrees Celcius.
Answer: Using Wien’s law
T
2.898 x 103
p
pT  2.898 x 103 m  K
2.898 x 103

 5773K
9
502 x 10
And so converting to degrees Celsius
T ( oC )  T ( K )  273.16  5773  273.16  5500 oC
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School of Mathematical and Physical Sciences PHYS1220
Theory Development
I(l,T)dl is the radiated power/area in wavelength interval dl
Radiation results from oscillating charges (due to molecular
vibrations) within the material
Full classical treatment led Lord Rayleigh and J. Jeans to
I ,T  
2 ck BT

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Rayleigh-Jeans Theory
Rayleigh-Jeans Law
where kB = 1.381x10-23 J/K
is the Boltzmann constant
Experimental
fits data well for long wavelengths
Data
major disagreement at short wavelengths
Limit as l  0, I(l,T)  
Energy density should become infinite for short wavelengths
Known in scientific folklore as the “The Ultraviolet Catastrophe”
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Planck’s Approach
Planck proposed an empirical formula (Dec 1900) which nicely fit
the data.
2 hc 2  5
I   , T   hc  k T
Planck’s Law
e B  1
The constant, h, introduced by Planck, was measured from fitting
the equation to data (currently accepted as 6.626x10-34 J.s)
Example: Calculate the value of I(l,T) using the (a) Rayleigh-
Jeans and (b) Planck’s theories for l=100nm (UV) and T=300K
(a) I(l,T) = 2pckT/l4
= (2 x 3.14159 x 2.997x108 x 1.38x10-23J/K x 300K)/(100x10-9m)4
= 7.8x1016 W.m-3
(b) I(l,T) = 2phc2/[l5(ehc/lkT-1)]
= 1.6x10-189 W.m-3
Difference is 205 orders of magnitude!
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Planck’s Law - Implications
To produce a theory that resulted in his equation, Planck had to
make a radical assumption, called Planck’s Quantum Hypothesis
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Oscillating charges possess quantised (or discrete) energies, related to
the oscillation frequency (cf. acoustic modes of strings and pipes)
E  n h f , n  1, 2,3,...
Emin  h f is referred to as the quantum of energy.
Planck (and everyone else) didn’t believe this to be the ‘real’ story
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Merely a mathematical tool to “get the right answer”
Continued looking for a theory based on classical approaches
Won the 1918 Nobel Prize in Physics for this work
Question: Is Planck consistent with Wien and Stefan-Boltzmann?
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Tutorial Exercise: Giancoli Chapter 38, problem 7.

I ( , T )  0   pT  2.898x10-3m.K ?

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
4
I
(

,
T
)
d


T
?

0
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School of Mathematical and Physical Sciences PHYS1220
Photoelectric Effect
Hertz (1887) observed that light can
produce electricity
After receiving energy from the
incoming light, electrons are ejected
from the surface of a metal
Light strikes the photocathode (P) and
ejects electrons, which get accelerated
to the collector (C).
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The applied potential V creates an
accelerating electric field between the
collector and the Photocathode
If the metal is continually illuminated, a
steady state current is produced and
can be read at the ammeter.
The photoelectron current increases
with light intensity
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School of Mathematical and Physical Sciences PHYS1220
Photoelectric Effect
If the polarity of the voltage source is reversed
and the potential varied, the maximum KE of the
electrons may be measured.
When the current goes to zero, i.e. no electrons
make it to the collector, the maximum KE of all
emitted electrons is given by:
KEmax  eV0
V0 is called the stopping potential
Experiments by Lenard (1902) showed that
KEmax is linearly dependent on light frequency!
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constant of proportionality = h !!
A ‘cut-off’ frequency, f0 exists. Below this, no
current will be produced, regardless of the
incident light intensity
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School of Mathematical and Physical Sciences PHYS1220
Predictions of Classical Wave Theory
The electric field of an EM wave can exert a force on electrons in
the metal and eject some of them
Light has two important properties
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Intensity
Wavelength (or frequency)
If the light intensity is increased,
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Electric field amplitude is greater
number of electrons ejected (and measured current) increases
kinetic energy (and KEmax) of ejected electrons increases
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If the frequency of the light is increased,
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Nothing should happen. The kinetic energy of photoelectrons should
be independent of the incident light frequency
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A time delay should exist before electrons are emitted
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The energy required to remove electrons will need to build up
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School of Mathematical and Physical Sciences PHYS1220
Einstein’s Corpuscular Theory of Light
Albert Einstein proposed the following (1905)
Light ‘quanta’ possess a corpuscular nature
Energy is related to frequency and wavelength by
where h is Planck’s constant
The KE of an emitted electron is given by
where W is the energy required to remove that electron
from (the surface of) the material
If the light frequency is below f0 , then no electrons will
be emitted (no matter how great the intensity)
The minimum energy required to eject electrons from
the material is called the work function, W0 and is
related to the cut-off frequency (and KEmax) by
More intensity → more quanta → more electrons
Ejection of the first electron should be instantaneous
Einstein won the 1921 Nobel Prize in Physics principally
for this work
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E  hf 
hc

KE  h f  W

W0  h f0
KEmax  h f  W0


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School of Mathematical and Physical Sciences PHYS1220
Work functions of Materials
The work function of a metal is typically ~ a few eV
Work Functions of Typical Metals (eV)
Metal
W
Metal
W
Metal
W
Li
Na
K
Rb
Cs
Cu
Ag
Au
Be
Mg
2.38
2.35
2.22
2.16
1.81
4.40
4.30
4.30
3.92
3.64
Ca
Sr
Ba
Nb
Fe
Mn
Zn
Cd
Hg
Al
2.80
2.35
2.49
3.99
4.31
3.83
4.24
4.10
4.52
4.25
In
Ga
Tl
Sn
Pb
Bi
Sb
W
3.80
3.96
3.70
4.38
4.00
4.40
4.08
4.50
Source: V. S. Fomenko, Handbook of Thermal Properties, G. V. Samsanov, ed., Plenum Press Data Division, New
York, 1966. (Values given are the author’s distillation of many different experimental determinations)
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School of Mathematical and Physical Sciences PHYS1220
Example
What is the energy of near infrared light of wavelength 1mm?
6.626x1034 J .s  3x108 m.s 1
Eh f 

 1.24eV
 1.602x10-19 J .eV -1 1x106 m
hc
A photocell made from Tungsten has a work function of 4.50 eV.
Calculate the cut-off frequency.
W0 4.50eV 1.602x1019 J .eV 1
W0  hf 0  f 0 

 1.088x1015 Hz
34
h
6.626x10 J .s
If light of wavelength 10nm (UV) is used to illuminate the
surface, what is the maximum kinetic energy of emitted
electrons?
KEmax
6.626x10-34 J .s  3x108 m.s 1

 W0 
- 4.50eV  119.6eV
-19
-1
9

1.602x10 J .eV 10x10
hc
What is the stopping potential?
V0  119.6 V
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School of Mathematical and Physical Sciences PHYS1220
Applications of the Photoelectric Effect
Photonic switches, burglar and smoke alarms
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The phototube acts much like a switch in an
electric circuit.
Photodiodes and light dependent resistors
(LDRs) and are modern equivalent to phototube
IR detectors, such as remote controls, etc
Light meters
Photosynthesis
Optical sound track on movie film
The first lasers (optically pumped)
X-ray Photoelectron Spectroscopy (XPS) is
used for chemical analysis by obtaining
elemental fingerprints of material surfaces
And many others
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The photoelectric effect dominates interactions
between light (near IR-soft X-rays) and matter
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