Introduction Chapter 7: Idea 6 You can’t predict or know...

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Chapter 7: Idea 6

Quantum Theory d th

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and the End of Causality

You can’t predict or know everything.

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Introduction

• Universe Shaking Idea #6

• Quantum Theory– The modern theory of Atoms

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– It is a somewhat counterintuitive theory…

• Like the Theory of Relativity– Our everyday experiences do not apply– And it was the solution to a problem (as usual)

• Also known as “Quantum Mechanics”

• The most successful scientific theory ever!– Never failed! Undefeated!

Quantum Theory

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• Explains a wide range of phenomena– Quarks, Nuclei, Atoms, Molecules

• Very small scales– 21-centimeter line, White Dwarfs, Neutron Stars

• Very large scales

• Classical Physics is the “old” stuff– Approximately before 1900

• Classical Physics includes

Classical Physics

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y– Idea 1: Copernican Astronomy– Idea 2: Newtonian Mechanics– Idea 3: Energy and Conservation Laws– Idea 4: Thermodynamics– Idea 5: Maxwell’s theory of E&M

Applies to “large” objects at everyday speeds

• Classical Physics fails at high speeds– Speeds near the Speed of Light

Classical Physics

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– We need Special Relativity

• Classical Physics fails at very small sizes– Sizes near the size of Atoms– We need Quantum Mechanics

• The “old” theories are not “wrong”.– They are just limited– They are valid over a limited range

h ill f l d i h if

Classical Physics

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• They are still useful and “right” if– they are used where they are appropriate

• We cannot extend them beyond their– “range of validity”

• The range of values of– Speed and Size (and Temperature too)

• For which a particular theory is valid.

Range of Validity

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p y– Where it works!

• Classical Physics– Range of Validity: Macroscopic (large size) and

Slow speeds

Speed (m/s)

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Range of Validity

c

10-3

103

1

Dimension (m)

1010105

110-5

10-10

10-15

EarthHuman

AtomsNuclei

Sound

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2

• Notice that there are “overlaps”– Where more than one theory is valid

• Special Relativity works at all speeds

Range of Validity

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– Newton works only for the slow speeds

• Quantum Theory works at all sizes– Newton works only for macroscopic sizes

• Many thought Physics was complete……that Physics had explained almost everything

• Students were actually advised

Back to 1900 AD

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– Study other subjects, we’re almost done.

• There were only two “minor” problems– The Light/E&M Force Problem– The Ultraviolet Catastrophe

• As we saw in the last chapter– The solution to the Light/E&M Force

Problem was the Theory of Relativity

Back to 1900 AD

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Problem was the Theory of Relativity

• As we will soon see– The solution to the The Ultraviolet Catastrophe

is the Quantum Theory

But first…some definitions

• There are three parameters– which describe any wave

• including an EM wave

– Wavelength (λ)Frequency (f )

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– Frequency (f )– Amplitude (A)

• The Frequency and Wavelength are relatedc = f λ

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The Ultraviolet Catastrophe

• Metal objects “glow” when heated– Iron: Red → Orange → White hot

• They also emit other EM radiationTh

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– That we cannot see• It’s out of the range our eyes are sensitive to

• Infrared – used in night-vision scopes

• Ultraviolet – “black light” stamp at a club

• To simplify the problem– Scientists analyzed the radiation emitted – By a “perfect” absorber or emitter

• Perfect Absorber

Blackbody Radiation

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Perfect Absorber– Reflects no Light or other EM radiation– A truly black surface

• Perfect Emitter– A perfect Absorber is also a perfect Emitter

• Blackbody Radiation– the EM radiation emitted by a perfect Emitter

• Experimentally a Blackbody

Blackbody Radiation

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p y y– A “perfect” Emitter

• Can be created to a high degree of accuracy– A small hole in a furnace works well…

• The results of the experiment…

• The Blackbody Radiation Spectra

Blackbody Radiation

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• A graph that tells us– How much of each kind of Electromagnetic

Radiation is emitted by a Blackbody

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IRUV

How much

RadiationExperimental results

λmax What kind of Radiation19

• Emits Radiation at ALL wavelengths– But in varying amounts

• Curve has a distinctive shape

Blackbody Radiation

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Cu ve as a d s c ve s ape– Depends only on the object’s TEMPERATURE– Does not depend on its chemical composition

• So far so good, until…

• The theorists tried to reproduce the experimental results– using Maxwell’s E&M Theory

Th l di

Blackbody Radiation

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• The results were a disaster– Predicted an infinite amount of UV radiation– An infinite amount of Energy

• This result was obviously incorrect!

IRUV

How much

Radiation

λmax What kind of Radiation

Theoretical result

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• Theoretical result OK for long wavelengths

• But failed completely for short wavelengths

di i ll

Ultraviolet Catastrophe

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• Its predictions were all wrong!– Wrong amount of Radiation– Wrong shape for the Graph

• A Catastrophe! (for the theorists anyway)

Ultraviolet Catastrophe

• The theory was based partly upon Thermodynamic theory– The Conservation of Energy (COE)

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The Conservation of Energy (COE)

• But the results violated the COE!!!– Ouch! That’s why I’m an experimentalist!

• How did they do the theoretical calculation?

• They added up the contribution– each atom in the emitter makes to the EMR

Blackbody Radiation

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• Each atom generates some EMR

• What is the recipe for EMR?– Wiggle an electron!

• The electrons in each atom oscillate– At a certain Frequency

• And the atom emits Radiation

Blackbody Radiation

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– At the same Frequency

• The total amount of Radiation– Is the sum of all the Radiation– At all Frequencies

• They also assumed that the Atom was– A continuous system

• They thought that

Blackbody Radiation

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– the Atom could have any Energy– the Electron could oscillate at any Frequency

• In other words, – the Atom could change Energy continuously

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• Remember the “number line”?

• Think of it as the “Energy Line”

A Simple Analogy

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• Energy is “continuous”– There are no gaps in the “Energy Line”– Like the Real Number System…

Energy

Max Planck (1858-1947)

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• Born in Kiel, Germany

• A very Academic Family– Father, G-Father, G-G-Father were all professors

Max Planck

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Max Planck

• Chose Physics @ age 16– despite being told Physics was a dead subject!

• Graduated @ age 21 with a PhD

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Graduated @ age 21 with a PhD– and assumed an Academic career

• Married and had 4 children (2 boys, 2 girls)– A very tragic family!

Max Planck

• Lost 1st son in WWI

• Lost 2nd son in WWII– Executed for plotting to assassinate Hitler

• A false accusation set him up

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• A false accusation set him up

• Both daughters died in child-birth

• Lost his house to Allied bombing in WWII

• In 1899, though, Max Planck– had a bold idea

• Suppose the Energy of the Atoms

Blackbody Radiation

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– Can only change by certain amounts

• Energy comes in bundles– And Atoms can gain or lose Energy– Only one bundle at a time

• Then the Energy of the Atoms– Must be some number of bundles– Energy is discrete, like the Integers

A Simple Analogy

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• He even specified the size of a bundle: hf

0hf 1hf 2hf 3hf 4hf 5hfEnergy

• This is called the “Quantum Hypothesis”– One bundle is called a quantum of energy

• Energy is quantized!

The Quantum Hypothesis

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– Energy can only have certain values– It must be some integer number of bundles

• Energy comes in bundles!– A bundle is the same thing as a quantum!

E n hf=

One BundleThe Quantum Hypothesis

f

n = 0 1 2, , ,...Number of BundlesEnergy

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• For exampleThis is allowed: E = 3 hf 3 bundles

So is this: E = 177 hf 177 bundlesThis is not: E = ½ hf


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• Energy comes in whole bundles only!– Energy = some number of Quanta

E n hf=

• One quantum of Energy is “hf ”– “f ” is the Frequency of the oscillation ( f = c/λ)– “h” is a new constant called Planck’s Constant


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• It is an extremely small number– A decimal point, 33 zeros, then the “66”

h = × −6 6 10 34. Js

• Vibrating Atoms can change energy– Only in amounts of “hf ” – Energy changes value one quantum at a time

hi i li i h l


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• This is not a limit on the total Energy– Only a rule about how Energy changes

• Energy comes bundles – And the bundles have a certain size: “hf ”

• The size of one bundle is very small– Since Planck’s constant is very small

• Not noticeable macroscopically


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– We don’t notice in our everyday lives

• Very noticeable microscopically– Important when dealing with small systems– Like Atoms…

• When Planck used his new idea– His results agreed exactly with experiment– It eliminated the UV Catastrophe!– He won the Nobel Prize in Physics in 1918


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• But the question remained– Where these bundles “real”?– Or just a way to get the “right”answer?– There was no proof!!

• Once again, Einstein supplied the answer

• But first, some background information…


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But first, some background information…

• Need to understand the Photoelectric Effect– Discovered by Heinrich Hertz in 1887

Heinrich Hertz (1857-1894)

• Born in Hamburg, Germany– Another stellar German Physicist

• Was a Physics hobbyist as a child

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• Spoke fluent Arabic and Sanskrit

• PhD from U. of Berlin in 1880– Top of his class, unlike Einstein

Heinrich Hertz

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Heinrich Hertz

• Professor of Physics at a local tech college

• Discovered the Photoelectric Effect in 1887– Accidentally in an unrelated lab experiment

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• Generated the first radio waves in his lab (1888)– Received them with a circuit a few feet away

• Not music, just static– Thought it was interesting…

• But of no practical value!!!! Oops!

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Heinrich Hertz

• Up to an Italian graduate student– to see their potential use– Guglielmo Marconi

H t di d f bl d i i

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• Hertz died of blood poisoning– On New Years Day 1894 @ age 37

• Was honored by having the unit of frequency named after him (the Hertz)

• When Light strikes the surface– Of certain metals

El t j t d

The Photoelectric Effect (PEE)

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• Electrons are ejected– So an Electric Current is generated

The Photoelectric Effect

Light Electron

Surface of Metal

Light strikes surface Electron is ejected

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• This can generate an Electric Current– Across a vacuum – With no connecting wires

• Was considered as an alternate means


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• Was considered as an alternate means– Of delivering Electricity without wires– Was not successful: air ionizes

• Electrons collide with air molecules destroying the current


Light

Electron

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Electron

No Light ⇒ No CurrentLight ⇒ Current

+-

Battery

• When the PEE was discovered– Maxwell’s Classical E&M theory– was the prevailing theory of Light

h i d l i h


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• So they tried to explain the PEE– In terms of Maxwell’s Theory of Light

• Light is a traveling wave – of oscillating Electric and Magnetic Fields

• In Maxwell’s Wave Theory of Light

• To make the Light Brighter– A Larger “Intensity”

Classical Explanation

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• You need to make the Wave bigger– Give it a larger Amplitude

Brighter ⇒ More Intense⇒ Larger Amplitude


A Light Wave c

A Brighter Light Wave

c

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• Maxwell’s classical explanation

• When the Light wave strikes the surface– It forces the Electron to oscillate


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– It forces the Electron to oscillate– The Electron “rides the wave”

• This gives the Electron Kinetic Energy

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Light Wave

c

The electron “rides the wave”

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• If the Light has a large enough Intensity– If the Light is Bright enough

• Then the Wave amplitude is large enough


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• Then the Wave amplitude is large enough– The Electron has enough KE to escape the metal

• The Electron is knocked out of the Atom

• The Classical explanation of the PEE– Based on Maxwell’s successful theory

F il !


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• Fails!

• When we apply Maxwell’s Theory– it makes predictions that are in complete

contradiction with the experiment!

The Experimental Results

1. Only certain colors of Light worked– Color ⇒ Frequency

• Certain colors of Light never worked– no matter how Bright

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no matter how Bright– Yellow light, for example

• Other Colors always worked– no matter how Dim– Blue light, for example

2. More Light gave more Electrons

• Pick a “good” Frequency

Shi Li ht j t El t


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• Shine Light ⇒ eject Electrons – Ejected with some Kinetic Energy

• Make the Light brighter– get more Electrons with the same Kinetic Energy

3. The Electrons were ejected immediately

• There was no measurable Time delay


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• When you turn the Light on…

• … Electrons immediately start flowing.

• Results exactly and totally contradicted– Maxwell’s Classical E&M explanation

• The classical explanation...


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p– based on Maxwell’s Wave Theory of Light

• … predicted exactly the opposite– of what the experiment showed!

• All colors work– if Light Bright enough

• Certain colors work– no matter how Dim– other colors never work– no matter how Bright

Classical Predictions Experimental Results

• Brighter Light– more Kinetic Energy– same number of Electrons

• Time delay

• Brighter Light– more Electrons ejected– with same KE

• No measurable delay62

• Borrowed Planck’s Quantum idea… – about Energy and “bundles”

• … and expanded it:

Einstein to the Rescue!

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– Light Energy is transported in bundles– called Photons

• The Wave is just the means of transport– Of the Photons

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Proof of the Concept?

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• Each Photon is one bundle of Energy– one of Planck’s quanta

• The energy of the Photon is


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E h f=

Depends on Frequency! (color)

• When the Photon strikes the surface– an Atom absorbs the whole Photon

• It absorbs all the Energy at the same time


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• If the Photon has enough Energy– an Electron is ejected immediately

• Enough Energy ⇒ large enough Frequency


KE hf W= −Einstein’s prediction:

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Kinetic Energy

Energy of Photon

Work Function

• Einstein predicts a “Threshold” Frequency– a minimum Frequency fth

• If the Frequency is below the threshold


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– no Electrons are ejected: f < fth

• If the Frequency is above the threshold– an Electron is ejected: f > fth

• He also predicts more Electrons…

• One Photon ⇒ one collision⇒ one ejected Electron


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• Brighter Light means more Photons– with the same Energy

• More Photons ⇒ more collisions– more ejected Electrons with same KE

• Einstein also provides – an experimental method for determining

Planck’s constant• A very small number that is difficult to measure

Ph i i t R b t Millik


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• Physicist Robert Millikan– Awarded Nobel Prize in 1923 for measuring h

• Work Functions– The minimum amount of energy needed to eject

an electron from the atom

KineticEnergy

Metal A

Metal B

fth for “A”

Sl h

Same Slope!

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Frequency(Color)

Threshold Frequency for “B”

Slope = h

WorkFunctions

• Einstein’s Quantum explanation– was right on the money!

• It fit the experimental results perfectly– Threshold frequency More Electrons No Time


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Threshold frequency, More Electrons, No Time delay!

• Einstein won the Nobel Prize for this– “and other achievements”– No mention of Relativity!

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• An important point in the history of Physics

• A test of the Photon Theory of Light

Th l ti d d b th f

The Compton Effect

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• The proper explanation needed both of– the new theories of the 20th Century

• Special Relativity: the E = mc2 part

• Quantum Theory: the E = hf Photon part

Arthur Holly Compton (1892-1962)

• Born in Wooster, Ohio

• Father was Dean of C ll f W t

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College of Wooster

• Earned PhD from Princeton University in 1916

Arthur Holly Compton

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• Head of Department of Physics– Washington University, 1920

• Explained the Compton Effect in 1922

Arthur Holly Compton

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• Explained the Compton Effect in 1922– Awarded the Nobel Prize in 1927

• Helped Physicist Ernest Lawrence– Start the Manhattan Project during WWII

• Light changes Wavelength…– and Frequency (and Color)

• … when it bounces off an Electron

The Compton Effect

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• The Light loses some Energy– and ends up with a longer Wavelength– so it is Redder

The Compton Effect

Incident Light: λ

Scattered Light: λ′

Electron recoils′ >λ λ79

• Arthur Compton– Used SR and QM to explain this effect

theoretically

The Compton Effect

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• Then he did the experiments– and verified the results experimentally

• An important validation of both theories!

• We used the “Light as a Particle” idea– to explain the Compton and Photoelectric Effects

Y t th t “P ti l ” h W l th

There’s something weird going on here...

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• Yet that “Particle” has a Wavelength…– Light is an E&M Wave– Proven time and again in optics experiments!

• We are building to an important new idea...

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The Dual Nature of Light and Matter

Wave-Particle Duality!

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This is a very important idea…

Light is both a particle and a wave simultaneously!

• All these effects were taking place– At the Atomic level!

• We are going to answer a basic question:

The Nuclear Atom

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• What is the structure of an Atom?

• It was not until the 20th Century– That Physics began to understand the

structure and behavior of Atoms

• The idea of Atoms is an old one– “Atom” means “indivisible”

• Dating back 2500 years to the Greeks!!– Democritus and his teacher Leucippus

The Nuclear Atom

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– Democritus, and his teacher Leucippus

• Cut the amount of something– In half and in half again and in half again…– until you have the smallest piece– that cannot be further divided

• Atom = the smallest part of an Element– If you break apart an Atom– You end up with “atom parts”– You no longer have that Element

The Nuclear Atom

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• Examples of elementsH: Hydrogen Cl: ChlorineC: Carbon Na: SodiumO: Oxygen

• Atoms combine to form Molecules

• Molecule = smallest part of a Compound– If you break apart a Molecule– You end up with Atoms

The Nuclear Atom

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– You no longer have that Compound

• Examples of Compounds:2H + O → H2O (water)Na + Cl → NaCl (table salt)

12C + 22H + 11O → C12H22O11 (sucrose)

The experimental evidence shows

• Atoms are neutrally charged

N t l At t i b th

The Nuclear Atom

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• Neutral Atoms contain both – Positive and Negative charges

• The Negative charges could be ejected– Recall the Photoelectric Effect

The First Model of the Atom

• Raisin Pudding Model– J.J. Thomson: Discoverer of the Electron

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electrons++

+

+

++

+

+

+

• In 1911, Physicist Ernest Rutherford– Proposed a new model for the Atom

• It was a “planetary model”

The Nuclear Atom

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It was a planetary model …– Based loosely on the Solar System

• …that tried to explain the experimental data

Planetary Orbits

• Recall Kepler’s First Law– of planetary motion

• The Planetary orbits are ellipses

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y p– With the Sun at one focus

• The Planet is held in its orbit by– Newton’s Force of Universal Gravitation

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Planetary Orbits

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Gravity

Ernest Rutherford (1871-1937)

• Born in Nelson,New Zealand

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New Zealand

• Considered the Father of Nuclear Physics

Ernest Rutherford

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• Theories of Radioactive Decay– Showed that Uranium and Thorium

• Became different elements after decay!– An incredible, unproven idea

Ernest Rutherford

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• Discovered the Atomic Nucleus– Fired Helium atoms at thin Gold foil

• Some bounced back at him!• Could only be explained by a concentration of

charge at the center of the Gold atom– Awarded Nobel Prize in 1908 in Chemistry

• In Rutherford’s Nuclear Model of the Atom

• The negative Electrons– Orbit the positive Nucleus

The Nuclear Atom

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Orbit the positive Nucleus

• The Electron is held in its orbit– By the attractive Coulomb Electric force– Attractive Electric force ↔ opposite charges

The Nuclear Atom

Electric Force

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• Small compared to the Atom– About 100,000 times smaller

C t i l t ll f th At ’

The Nucleus

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• Contains almost all of the Atom’s mass– Much heavier than the Electrons

• Positively charged

• Much lighter than the Nucleus– And therefore easier to move

• Orbit the Nucleus

The Electrons

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– Move quite fast in their orbits– Relativistic speeds!

• Negatively charged– Exactly opposite the positive Nucleus

• The Nuclear Atom is mostly empty space

• Consider Hydrogen– The simplest Atom: 1 Proton + 1 Electron

Empty Space?

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p

• If the Proton were the size of the Sun…

• …the Electron’s orbit would be – 10 times the size of Pluto’s orbit!

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• Rutherford’s model did explain– Most of the experimental data

• But there was one serious problem

The Nuclear Atom

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– It was not stable!

• According to Classical Physics– Newton and Maxwell– This Atom would fall apart in 10-8 seconds!

• The orbiting Electron undergoes a– centripetal acceleration (Newton)

• Accelerating Electrons emit

Classically Speaking…

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– Electromagnetic Radiation (Maxwell)

• The Electron loses Energy– It spirals into the Nucleus and emits a flash of

light within a hundred-millionth of a second

• Matter couldn’t exist if this were true!

• So either the Nuclear Model is incorrect…

The Nuclear Atom

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• …or the Laws of Classical Physics do notapply to individual Atoms.

• Here we go again! Something’s wrong…

• Born Copenhagen, Denmark

• A young Danish physicist– Worked with Rutherford in 1912

Neils Bohr (1885 – 1962)

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• Tried to modify Rutherford’s model– Using the new Quantum

Hypothesis– Published his results in 1913

Neils Bohr

Same place Tycho Brahe built his observatory!

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• Studied and understood the “new” Physics– Particularly the Quantum Hypothesis – used by Planck and Einstein

Pl k Bl kb d R di ti

The Bohr Model

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• Planck: Blackbody Radiation– Produced by vibrating Atoms

• Einstein: Photoelectric Effect– Light can eject Electrons from Atoms

• Bohr needed a way to stabilize– Rutherford’s model

• After all, real Atoms are stable!

The Bohr Model

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• He reasoned there must be certain orbits– For which no Radiation is emitted– There was no proof of this at the time!

• But which Electron orbits?

• Bohr kept Rutherford’s small heavy nucleus

• And added two new rules for Atoms:

The Bohr Model

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1. A rule for the allowed Electron orbits

2. A rule for how the Atom emits Radiation

• Angular Momentum is quantized!

• Angular Momentum comes in Bundles

1. Allowed Electron Orbits

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• The allowed Orbits– Contain an integer number of Bundles

• Angular Momentum of an orbiting Electron– is its Momentum times the Radius of the Orbit

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Angular Momentum…

L mvR=Angular

MomentumMomentum Radius

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mvR n h=

2

…is Quantized! One Bundle

2πn = 0 1 2, , ,...

Number of BundlesAngularMomentum

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• Angular Momentum is a good candidate– for a rule about orbits

• It contains information about the– Size and speed of the orbit (R and v)

Angular Momentum

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Size and speed of the orbit (R and v)

• The quantization of Angular Momentum– defines an “allowed” orbit

• Bohr just made it up – because it worked!

• Bohr proposed a mechanism for an Atom– to absorb or emit Electromagnetic Radiation

• Atoms can absorb or emit Photons

2. Radiation Emission

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– as long as the Photon has a certain Energy

• What does “certain Energy” mean?

• The Electron can occupy any orbit– Any one of the allowed orbits

• Each orbit has a different Energy

The Bohr Model

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– They are even called “Energy Levels”

• The Electron can change orbits– if it can change its Energy

• The Electron can move to a higher orbit– One with more Energy

• if the Atom absorbs a Photon

The Bohr Model

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– Whose Energy exactly equals the differencein the Energy between the two orbits!

• The Electron absorbs the Photon’s Energy– And moves to a higher orbit

The Bohr Model

• The Electron can move to a lower orbit– One with less Energy

• If the Atom Emits a Photon

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– Whose Energy exactly equals the difference in the Energy between the two orbits!

• The Electron moves to a lower orbit– Atom emits a Photon with the Energy it lost

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• If the incident Photon has an Energy…– that is equal to the Energy difference– between two allowed orbits

The Bohr Model

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• …then the Atom absorbs the Energy.– and moves to the higher orbit

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photon

118

• When the Electron is in a higher orbit…– It can lose Energy in an amount that is equal

to the Energy difference between 2 orbits

The Bohr Model

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to the Energy difference between 2 orbits

• …the Atom emits a Photon– And the Electron moves to the lower orbit photon

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• In either case– The Absorption or Emission of a Photon

• Photon Energy = Energy difference

The Bohr Model

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gy gy– between the two orbits

hf E Ehi lo= −

• Bohr applied his idea to Hydrogen– The simplest Atom: 1 Proton + 1 Electron

• The results were almost exactly right!

The Bohr Model

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• Made correct predictions about– Energy levels (Orbits)– Wavelengths of emitted Light (Colors)– Some new UV emissions that were later found

An aside: Spectral Lines

Spectroscope - splits light into its component colors

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Spectral Lines

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Emission lines -single frequencies emitted by particular atoms

The emission spectrum can be used to identify specific elements:

Spectral Lines

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Emission lines can be used to study faraway objects:

Spectral Lines

The Omega Nebula:

Object M17

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Object M17

5000 light years away(30,000 million miles)

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• Made correct predictions, however, some problems existed…

• The model couldn’t determine how bright

The Bohr Model

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g(intense) the emitted radiation was

• The model predicted some light frequencies which were never observed

• Bohr then applied his model to Helium– The next simplest Atom– 2 Protons, 2 Neutrons, and 2 Electrons

It f il d l t l !

The Bohr Model

128

• It failed completely!

• It could not account for the interaction– Of the two Electrons– The repulsive electrical Force between them

• However, progress had been made– Bohr explained the Hydrogen Atom

• But there were still problems

The Bohr Model

129

– Why did Classical Physics fail for Atoms?– What is the basis for Bohr’s “rules”?

• Quantized Angular Momentum and orbits

• Let’s answer that last question next…

• Waves have lots of interesting properties– Reflection, Refraction, Diffraction, Interference

An aside…

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• Let’s discuss these properties briefly…

• The image you see of yourself in a mirror is the end result of a chain of events– Light waves reflects off you onto the mirror, then

reflects off the mirror into your eyes

Reflection of Waves

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• The speed of light in a vacuum (c) is a constant (186,000 mps)– independent of color (frequency)

Refraction of Waves

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• The speed of light in other materials is alwaysslower– and is dependent on color!

• blue light moves more slowly through air than red light

• The overall effect of this slowdown is that a white light wave bends (refracts) as it passes through materials and the colors separate– such as in glass!

Refraction of Waves

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– or water vapor!

• Light waves can bend (diffract) around the edges of objects

Diffraction of Waves

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Light Reflection/Refraction/DiffractionAll at Once!?!

135

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Interference of Waves• Light waves can add and subtract like numbers

– an effect called interference

• Two types:Constructive Interference Destructive Interference

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(Waves In Phase) (Waves Out of Phase)

Wave Properties

• We have discussed these properties for light waves– electromagnetic waves

• However, these properties all apply to sound

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However, these properties all apply to sound and water waves as well!– In fact they have been observed to occur in all types

of waves

• Now back to our regularly scheduled program…

• Born Dieppe, France

• Young French physicist– Proposed an idea in his PhD

dissertation which explained Bohr’s rules

Louis de Broglie (1892 – 1987)

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Bohr s rules

• He recognized the significance of– Einstein’s Special Relativity– Planck’s Quantum Hypothesis

Louis de Broglie

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• Quantum Theory– Light is a form of Energy: E = hf

• Special Relativity

De Broglie Waves

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– Mass is a form of Energy: E = mc2

• De Broglie suggests extending the similarity– They should be describable in the same terms

• Classical Physics– Light acts like a Wave (Macroscopically)– Matter acts like Particles (Macroscopically)

l k i i h

Matter Waves

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• Planck, Einstein, Bohr– Light acts like a Particle (Microscopically)

• Now de Broglie says finish the symmetry:– Matter acts like a Wave?

• De Broglie proposed “Matter Waves”– Moving mass has a Wavelength

• This is not Electromagnetic Radiation

Matter Waves

142

• This is a statement that Matter can– act like a Wave under the right circumstances

• It depends on the size of the Wavelength

Matter Waves

λ =h

Planck’s Constant

λ =mv

MomentumMatter Wavelength143

• For macroscopic objects…– like baseballs and cars

• …the de Broglie Wavelength is very small

Matter Waves

144

– because Planck’s constant is so small

• So we don’t notice any Wave behavior– This is Classical Physics– Matter acting like particles

17

• But for microscopic objects…– like Atoms and Electrons and Photons

• …the de Broglie Wavelength is significant

Matter Waves

145

– because the Momentum is so small

• So the Wave behavior becomes important– This is Quantum Physics– Matter acting like Waves!

• This explains Bohr’s assumptions!

• Angular momentum is Quantized.

Matter Waves

146

• The allowed orbits are the ones– That contain a whole number of Wavelengths

• These two statements are equivalent!

λ

Rn = 5 R

2πR

n = 5

147

2π λR n=

2πR n hmv

=

Integer Wavelengths

de Broglie Wavelength

mv

mvR n h=

2πAngular Momentum

is Quantized!

Bohr’s assumption!148

A: An allowed orbit (n = 4)B: A non-allowed orbit

149

All these are allowed orbits

150

• Without an integer number of Waves– exactly “fitting” into the orbit

• After each orbit the Wave would beli h l ff ( f h ) f h i bi

Matter Waves

151

– slightly offset (out of phase) from the previous orbit

• And the Wave would cancel itself out!– Destructive Interference of waves!!!– That is not an allowed orbit!

• This is called Wave-Particle Duality

• “All matter and radiation exhibit both Wave

Wave-Particle Duality

152

and Particle behavior in certain circumstances”

• Light behaves like a Wave– Refraction, Diffraction, Reflection

• Light behaves like a Particle– Photoelectric Effect, Compton Effect


153

• Matter behaves like a Particle– Baseballs, Cars, Hockey Pucks

• Matter behaves like a Wave– Atoms, Molecules, Nuclei

18

• Without Wave-Particle Duality– The wave behavior of Electrons in Atoms

• There would be no Atoms


154

– The Wave nature of the Electron– supplies the condition for the “allowed” orbits

• No Wave behavior ⇒ No Atoms!

De Broglie

• Matter waves existence proved experimentally many times since

• De Broglie awarded the Nobel Prize

155

• De Broglie awarded the Nobel Prize– In 1929

• The Bohr Model of the Atom– Worked well for Hydrogen– Failed completely for Helium

The Schrödinger Wave Theory

156

• Physics needed a more complete theory– Of the Atom– One that worked for all Atoms

• Austrian mathematical physicist

• Born Vienna, Austria

Erwin Schrödinger (1887 – 1961)

157

• Published a theory in 1926- Described how the Matter Waves moved- How they propagate in 3-Dimensions

• The Schrödinger Wave Theory- is the current modern theory of the Atom!

Erwin Schrödinger

158

• Also offered a physical interpretation– of de Broglie’s Matter Waves

• The Matter Waves are Probability Waves


159

• The Waves in Schrödinger’s Theory– are Waves of Probability

• We use the Waves to calculate Probabilities– About where the Electron is– And where is it going

• Example: the Hydrogen Atom

• Where is the electron most likely to be?– The same distance as Bohr’s 1st allowed orbit


160

– About 0.5×10-10 meters (fifty trillionths) – Called the “ground state” of the atom

• But it may be a mile away from the Nucleus– Probability: once every billion years

• The Electrons no longer follow an Orbit– The “little solar system” model is out!

• The Electrons form a “probability cloud”– that surrounds the Nucleus


161

that surrounds the Nucleus– Orbits define the “most likely” location

• The very concept of a trajectory or a path– is meaningless in QM– In QM, all we can know are probabilities

• Schrödinger developed an equation…– A wave equation

• …that describes the Wave’s motion

The Schrödinger Equation

162

• The Schrödinger Equation– does for Matter Waves what– Newton’s Laws do for Particles

19

− ∇ + =h h2 2 V iΨ Ψ

Ψ∂


− ∇ + = h2m V it

Ψ Ψ∂

Kinetic Energy PotentialEnergy

Total Energy

163

• QM is sometimes described as the science– “Where you cross your h’s and dot your o’s”

Schrödinger’s name: œ → ö

Quantum Mechanics

164

Schrödinger s name: œ → ö

The constant “h-bar”: h ≡h

2π

• The Schrödinger Equation is just– the Principle of Energy Conservation

• Written in the complicated mathematical


165

W e e co p ca ed a e a calanguage of Quantum Mechanics

• The goal of QM is to solve the equation– for the Wave Function Ψ

• The Wave Function…– the solution to the Schrödinger Equation

• …is the amplitude of the Probability Wave


Ψ

166

• From the Wave Function– We can calculate the Probabilities

• Of position, of energy, and of momentum– Which is all we can know in QM

• Ψ itself has no real physical meaning

• However, the square of the Wave Function…Ψ2


167

Function…Ψ

• …is the Probability Density– This is the “probability cloud”

• The Probability Density tell us the– Probability of finding the Electron at

a given location (orbit) at a given time


168

• It is impossible to say exactly where the Electron is at any particular time

• Now a change in terminology…

• Bohr’s “quantum jump”…– The Electron’s change of orbits


169

• … is now a “change of State”– By “state” we mean the condition of the Atom

• The State of the Atom– Determines the Probability Density

• We specify the State of the Atom– With a set of Quantum Numbers: n, l, ml

• It takes 3 Quantum Numbers…


170

Q– One for each Spatial dimension (3-D)– As opposed to Bohr’s one “n”

• …to describe the State of an Atom

• When the Atom changes State…– It has a new set of Quantum Numbers– And a new Probability Density

• the Electron has different probabilities


171

• …the Electron has different probabilities

State ⇒ Quantum Numbers ⇒ Wave Function⇒ Probability Density⇒ Probabilities

20

The Schrödinger Wave Theoryl

0 1 2

172

n

• “If it can’t be somewhere, then it probably isn’t there”

• If Classical Physics prohibits a particle– From being somewhere

Quantum Mechanics in a Nutshell

From being somewhere• A disallowed orbit

• Then in Quantum Mechanics – It will have a small probability of being there.– As long as the Wave Function is not zero!

• It can be anywhere with some probability173

• Place a marble in a cup– And leave it completely undisturbed.

• What is the probability that it gets out?

Example: a Marble in a Cup

174

• CM: zero– No forces ⇒ no acceleration ⇒ it doesn’t move

• QM: small, very very small, but not zero!– The WF outside the cup is not zero

QM Tunneling

zebu.uoregon.edu/~js/glossary/quantum_tunneling 175

• If everything is a probability…– how close can we get to knowing anything

exactly?

• How do we “see” (measure) something?

Heisenberg Uncertainty Principle

176

• Light reflects off the object into a device…– Such as our eyes or a camera

• …where the image is created.

• When we see it we know where it is

Looking is Measuring

Incident LightObject

177

Reflected LightEye

• Is there any uncertainty?– Do we really know exactly where it is?

• We when look at an object


178

j– We are measuring its position– We can specify its location

• But Light carries Momentum! p h=λ

• According to Momentum Conservation…

• Since the Light carries Momentum…– To the right in the figure

Th i i i i l M h i h


179

– There is some initial Momentum to the right

• …the Object must recoil to the right.– There must be some final Momentum to the right

• So the Object moves!

Looking is Measuring

Incident LightObject

180

Reflected LightEye Recoil

21

• For macroscopic Objects…– Like Baseballs and Cars

• …this is no problem!


181

• The Light has so little Momentum– It has no effect on the Object

• Like a mosquito hitting an elephant.

• But for microscopic objects…– Like Electrons and Atoms

• …this causes a big effect!


182

• The Light has enough Momentum– To cause the Object to significantly recoil

• By the time the Light enters our eye– The Object has moved and is still moving!

• This demonstrates the Uncertainty Principle

• It is impossible to know an Object’s Speed and Location exactly at the same time!


183

• The process of measurement– Changes the Object’s Speed and Location!


∆ ∆x p ≥ hUncertainty in

LocationUncertainty in

Momentum

184

• The more we know about Location– The less we know about Speed

• The more we know about Speed


185

e o e we ow abou Speed– The less we know about Location

• If we know one of them exactly– then we know nothing about the other!

• There is a similar Uncertainty relation– Between Time and Energy

• The more we know about Energy


186

gy– The less we know about the Time

• The more we know about Time– The less we know about Energy


∆ ∆E t ≥ hUncertainty in

EnergyUncertainty in

Time

187

• Remember the 3rd Law of Thermodynamics?

• “It is not possible to reach the absolute zero of temperature in a finite number of steps”


188

of temperature in a finite number of steps .

• Why not?

• Because of the Uncertainty Principle!

• Absolute zero is the Temperature– Of zero motion!

• The particles would be absolutely stopped!


189

p y pp

• So we would know exactly– Where they are and how fast they’re moving

• This violates the Uncertainty Principle!

22

• Born Würzburg, Germany

• German physicist

• Awarded Nobel Prize in 1932

Werner Heisenberg (1901 – 1976)

190

– for work on QM

• Headed the unsuccessful German effort in WWII– to build an atomic bomb

Werner Heisenberg

191

• Let’s go back to our marble in a cup

• When we place the marble in the cup– we know it is somewhere in the cup!

Example: a Marble in a Cup

192

• The Uncertainty in the marble’s location– is the size of the cup

• So there is an Uncertainty in its Momentum


∆∆

p ≥ h

193

∆x

∆x

• Let’s put in some numbers…


∆x = ≈5 2cm inches

194

• …and see how long we expect the marble to stay in the cup.

∆xm = ≈

5 210 1

3

cm inchesgrams ounce


px

=h

∆Estimate the Momentum:

2

195

Estimate the Energy: E pm

=2

2

Estimate the Time: tE

=h

• For the Marble:


msJ

msJp

2

3334

1012.2050.01006.1 −

×=−×

= −−

196

( )

sJ

sJt

Jkg

msJ

E

2964

34

64

233

1072.41025.2

1006.1

1025.2010.02

1012.2

×=×

−×=

×=

−

×=

−

−

−

−

• For our small but macroscopic marble

• The time is about 5 × 1029 seconds– The time it remains in the cup

A very big number


197

– A very big number…

• How big?– Approximately 1022 years

• About 1 trillion times the age of the Universe

• What’s going on here?

• The Wave Function of the marble– is very small but not zero outside the cup.


198

• So the Probability Density is not zero– It is very small but not zero!

• There is a probability the marble is outside– Is it very small but not zero!

23

• So let’s instead try to confine an Electron– Inside a Nucleus of an Atom

• a microscopic object


199

∆xm=

= ×

−

−

109 11 10

15

31

meterskilograms.

• For the Electron:


J

msJ

msJp

2

1915

34

1006.110

1006.1 −−

−

−×=

−×=

200

( )s

JsJt

Jkg

msJ

E

269

34

931

219

1072.11017.6

1006.1

1017.61011.92

1006.1

−−

−

−−

−

×=×

−×=

×=×

−

×=

• For this microscopic situation

• The time is about 2 x 10-26 seconds– That’s two hundredth-trillionth-trillionths


201

• Electrons easily escape the Nucleus!– Never been observed inside the nucleus!

• And they do escape– Just like QM says they should!

• Einstein didn’t believe the Uncertainty Principle at first…– “God does not play dice with the Universe”


202

• …but eventually even he was persuaded it was the truth

• The Light Emitting Diode:

Examples of Quantum Mechanical Devices

• Silicon pieces doped (purposely contaminated) with other elements- one with an excess of

electrons (N type)electrons (N-type)- one with a deficit of

electrons (P-type)

With no voltage applied:

203



With voltage incorrectly applied:

Depletion zone widens!

204



With voltage correctlyapplied:

Photons

Recombining holes and electrons produce photons!

205

• The Light Emitting Diode:Examples of Quantum Mechanical Devices

206

• Light Amplification by the Stimulated Emission of Radiation: The Laser


Incoming photon stimulates an already excited atom to decay releasing an identical photon 207

24

• The LaserExamples of Quantum Mechanical Devices

Pump source excites atoms. Photons reflecting off mirrors stimulates the atoms to decay over and over again releasing light! This light is all moving in the same direction with the same energy (called coherent light). 208

• The LaserExamples of Quantum Mechanical Devices

209

The End of Causality!

• Before QM…– Every effect had a specified cause– Newton: We can know everything!

A → B → C therefore A → C100% 100% 100%

210

• After QM…– Not any more!!– QM: Everything is a probability!

A → B → C therefore A → C70% 30% 21%

Introduction Chapter 7: Idea 6 You can’t predict or know...

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