School of Mathematical and Physical Sciences PHYS1220 20 August, 20021 PHYS1220 – Quantum Mechanics Lecture 1 August 20, 2002 Dr J. Quinton Office: PG

20 August, 2002 1

School of Mathematical and Physical Sciences PHYS1220School of Mathematical and Physical Sciences PHYS1220

PHYS1220 – Quantum Mechanics

Lecture 1August 20, 2002

Dr J. QuintonOffice: PG 9 ph [email protected]

20 August, 2002 2


PHYS 1220 – Quantum Mechanics

Early Quantum Theory 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

Thompson Rutherford Bohr

Correspondence Principle

Quantum Mechanics Wave functions Quantum Mechanics Schrödinger Equation Heisenberg Uncertainty

Principle Particle in a box

Infinite Potential Well Finite Potential Well

Barrier potential Electron Tunnelling Applications of Quantum

Mechanics

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At the turn of the 20th Century, it was thought that physics had just about explained all natural phenomena.

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) Newtonian Mechanics

Forces and motion of Particles, fluids, waves, sound Universal theory of gravity

Maxwell’s Theory of Electromagnetism (EM) Unified electric and magnetic phenomena Thoroughly explained electric and magnetic behaviour Predicted existence of electromagnetic waves

Thermodynamics Thermal processes Kinetic theory of gases and other materials

The Success of Classical Physics

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Wave Theory of Light - Classical Physics Light is an electromagnetic wave, produced by accelerating charges (Maxwell)Electromagnetic Spectrum

700nm 600nm 500nm 400nm

IR UV

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0 0

1c f

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

299792458 m/s (exact)

Nature of Light - Classical Physics

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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.

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)

, and 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.

Quantum Theory & Relativity

We will be discussing Quantum Mechanics from its beginnings

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Blackbody RadiationRecall Stefan-Boltzmann law (1879, 1884)

Describes energy dissipated through radiation Stefan-Boltzmann constant =5.67x10-8 W/m2.K4

The emissivity (0<e<1) is a measure of the materials’ ability to emit (and absorb) radiation

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

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

4dQe AT

dt

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32.898 x 10pT m K

Total Intensity increases with TPeak wavelength moves to shorterwavelengths with increasing Tillustrates that the apparent colourof an object depends on its temperature.Question: What is the colourprogression (with increasing T)for incandescent materials?

Wien’s Law

where p is the wavelength at the peak of the spectrum

Wien was awarded the 1911 Nobel Prize in Physics for this work.

Blackbody Emission Spectrum

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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

And so converting to degrees Celsius

32.898 x 10pT m K 3 3

9

2.898 x 10 2.898 x 105773

502 x 10p

T K

( ) ( ) 273.16 5773 273.16 5500o oT C T K C

Example - The Solar Spectrum

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I(,T)d is the radiated power/area in wavelength interval dRadiation results from oscillating charges (due to molecular vibrations) within the materialFull classical treatment led Lord Rayleigh and J. Jeans to

where kB = 1.381x10-23 J/K is the Boltzmann constant

fits data well for long wavelengthsmajor disagreement at short wavelengthsLimit as 0, I(,T) Energy density should become infinite for short wavelengthsKnown in scientific folklore as the “The Ultraviolet Catastrophe”

Theory Development

Rayleigh-Jeans Theory

Experimental Data

4

2, Bck TI T

Rayleigh-Jeans Law

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Planck’s Approach

2 52

,1Bhc k T

hcI T

e

Planck’s Law

Planck proposed an empirical formula (Dec 1900) which nicely fit the data.

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(,T) using the (a) Rayleigh-Jeans and (b) Planck’s theories for =100nm (UV) and T=300K(a) I(,T) = 2ckT/4

= (2 x 3.14159 x 2.997x108 x 1.38x10-23J/K x 300K)/(100x10-

9m)4

= 7.8x1016 W.m-3

(b) I(,T) = 2hc2/[5(ehc/kT-1)]= 1.6x10-189 W.m-3

Difference is 205 orders of magnitude!

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To produce a theory that resulted in his equation, Planck had to make a radical assumption, called Planck’s Quantum Hypothesis

Oscillating charges possess quantised (or discrete) energies, related to the oscillation frequency (cf. acoustic modes of strings and pipes)

is referred to as the quantum of energy.

Planck (and everyone else) didn’t believe this to be the ‘real’ story

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?

Tutorial Exercise: Giancoli Chapter 38, problem 7.

, 1, 2,3,...E nh f n

Planck’s Law - Implications

minE h f

-3 4

0

( , ) 0 2.898x10 m.K ? ( , ) ?pI T T I T d T

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Photoelectric EffectHertz (1887) observed that light can produce electricityAfter receiving energy from the incoming light, electrons are ejected from the surface of a metalLight strikes the photocathode (P) and ejects electrons, which get accelerated to the collector (C).

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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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:

V0 is called the stopping potential

Experiments by Lenard (1902) showed that KEmax is linearly dependent on light frequency!

constant of proportionality = h !! A ‘cut-off’ frequency, f0 exists. Below this, no

current will be produced, regardless of the incident light intensity

Photoelectric Effect

max 0KE eV

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School of Mathematical and Physical Sciences PHYS1220School of Mathematical and Physical Sciences PHYS1220Predictions of Classical Wave TheoryThe electric field of an EM wave can exert a force on electrons in the metal and eject some of themLight has two important properties

Intensity Wavelength (or frequency)

If the light intensity is increased, Electric field amplitude is greater number of electrons ejected (and measured current) increases kinetic energy (and KEmax) of ejected electrons increases

If the frequency of the light is increased, Nothing should happen. The kinetic energy of photoelectrons

should be independent of the incident light frequency

A time delay should exist before electrons are emitted The energy required to remove electrons will need to build up

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Albert Einstein proposed the following (1905)Light ‘quanta’ possess a corpuscular natureEnergy is related to frequency and wavelength bywhere h is Planck’s constant

The KE of an emitted electron is given bywhere 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 electronsEjection of the first electron should be instantaneousEinstein won the 1921 Nobel Prize in Physics principally for this work

Einstein’s Corpuscular Theory of Light

hcE hf

KE h f W

0 0W h fmax 0KE h f W

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Work functions of MaterialsThe work function of a metal is typically ~ a few eV

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)

Metal W Metal W Metal W

Li 2.38 Ca 2.80 In 3.80

Na 2.35 Sr 2.35 Ga 3.96

K 2.22 Ba 2.49 Tl 3.70

Rb 2.16 Nb 3.99 Sn 4.38

Cs 1.81 Fe 4.31 Pb 4.00

Cu 4.40 Mn 3.83 Bi 4.40

Ag 4.30 Zn 4.24 Sb 4.08

Au 4.30 Cd 4.10 W 4.50

Be 3.92 Hg 4.52

Mg 3.64 Al 4.25

Work Functions of Typical Metals (eV)

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ExampleWhat is the energy of near infrared light of wavelength 1m?

A photocell made from Tungsten has a work function of 4.50 eV. Calculate the cut-off frequency.

If light of wavelength 10nm (UV) is used to illuminate the surface, what is the maximum kinetic energy of emitted electrons?

What is the stopping potential?

34 8 1

-19 -1 6

6.626x10 . 3x10 .1.24

1.602x10 . 1x10

hc J s m sE h f eV

J eV m

19 1150

0 0 0 34

4.50 1.602x10 .1.088x10 Hz

6.626x10 .

W eV J eVW hf f

h J s

-34 8 1

max 0 -19 -1 9

6.626x10 . 3x10 .- 4.50 119.6

1.602x10 . 10x10

hc J s m sKE W eV eV

J eV

0 119.6V V

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School of Mathematical and Physical Sciences PHYS1220School of Mathematical and Physical Sciences PHYS1220Applications of the Photoelectric Effect

Photonic switches, burglar and smoke alarms 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, etcLight metersPhotosynthesisOptical sound track on movie filmThe first lasers (optically pumped)X-ray Photoelectron Spectroscopy (XPS) is used for chemical analysis by obtaining elemental fingerprints of material surfacesAnd many others

The photoelectric effect dominates interactions between light (near IR-soft X-rays) and matter

Documents

School of Mathematical and Physical Sciences PHYS1220 20 August, 20021 PHYS1220 – Quantum Mechanics Lecture 1 August 20, 2002 Dr J. Quinton Office: PG