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Quantum Theory Quantum Theory 11

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TopicsTopics

Discovery of the Electron

Millikan’s Experiment

Blackbody Radiation

An Act of Desperation

Summary

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The Discovery of the ElectronThe Discovery of the Electron

1838 – M. Faraday Discovery of arc light in dilute gas

1857 – H. Geissler Discovery of fluorescence in dilute gas

1880s – W. Crookes Development of cathode ray tube

1895 – J. Perrin Cathode rays found to be negatively charged

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The Discovery of the ElectronThe Discovery of the Electron

1896 – P. Zeeman First evidence of atomic particles with a specific

charge-to-mass ratio

1897 – J.J. Thomson Measurement of charge-to-mass ratio of cathode

ray particles. In effect, this was the discovery of the electron and

the dawn of modern physics, in particular, particle physics (also known as high energy physics)

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The Discovery of the ElectronThe Discovery of the Electron

C cathodeA,B slitsD,E deflection

plates

,

/M EF quB F qE

u E B

q u

m RB

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The Discovery of the ElectronThe Discovery of the Electron

C cathodeA,B slitsD,E deflection

plates

,

/M EF quB F qE

u E B

Thomson found that whatevergas was used he always got the same q/m ratio ofabout 0.7 x 1011 C/Kg. The cathode particles were called corpuscles by Thomson and later electrons by Lorentz

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Millikan’s ExperimentMillikan’s Experiment

Millikan began experiments in 1909He measured e to be e = 1.601 x 10-19 C

Charged oil drops

Suspendedoil drops

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Blackbody RadiationBlackbody Radiation

Josef Stefan 1835 – 1893

In 1879, Josef Stefanfound an empirical formulato describe the powerradiated by an ideal black body

P =σT 4

σ = 5.6703 x 10-8 W/m2K4

Five years later, Ludwig Boltzmann was able toderive this formula from thermodynamics

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Blackbody RadiationBlackbody RadiationThe radiation energydensity per unitwavelength u(λ) wasfound to depend only on the absolutetemperature T

Wikemedia Commons

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Blackbody RadiationBlackbody Radiation

In the late 19th century, the race was on to derive, from first principles,the black bodyspectrum

The task was to compute theenergy densityof radiationwithin a cavity

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Rayleigh-Jeans LawRayleigh-Jeans Law

Lord Rayleigh 1842 – 1919

In 1900, Lord Rayleighshowed that the radiation energy per unit volume per unit wavelengthin a cavity must have the form

4( )

Tu a

In 1905, Sir James Jeansshowed that a = 8πk, wherek is Boltzmann’s constant

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Rayleigh-Jeans LawRayleigh-Jeans Law

Lord Rayleigh 1842 – 1919

Unfortunately, when integrated overall wavelengths, this law predicted an absurdity: the electromagneticenergy density in a cavity is infinite!

0

( )u d

This result, called theultraviolet catastrophe, revealed a serious flaw in classical physics

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An Act Of DesperationAn Act Of Desperation

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The October Revolution

In early October 1900, Max Planck found through trial an error that the formula

could reproduce the experimental results

u() =

8πT4

hc / kTehc/ kT −1

⎛

⎝⎜⎞

⎠⎟

An Act of DesperationAn Act of Desperation

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An Act of DesperationAn Act of Desperation

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Planck was to make a presentation at the German Physical Society meeting on October 19, 1900.

But since he had arrived at his formula through inspired guesswork, he was rather anxious to derive it in a more respectable way!

An Act of DesperationAn Act of Desperation

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Planck’s Model A cavity is modeled as a system of N oscillators

(presumably, atoms) that can emit and absorb electromagnetic radiation

The radiation can be distributed amongst the oscillators in a large number of ways Ω

Ludwig Boltzmann had earlier introduced the formula S = k ln Ω, for the total entropy of a system

An Act of DesperationAn Act of Desperation

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Planck’s Model Planck computed the entropy per oscillator

assuming that the radiation is absorbed and emitted in discrete amounts

He regarded this assumption as nothing more than a mathematical trick, an “act of desperation”. But, try as he might, he could not make headway without this crazy assumption

An Act of DesperationAn Act of Desperation

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Planck’s Model For a given , Planck assumed that the energy in

the cavity is in the form of M indistinguishable quanta of energy εdistributed over N indistinguishable oscillators

The total energy of the radiation is E = Mε. So the average energy per oscillator is e = E/N = Mε/N

Planck derived s, the entropy per oscillator, i.e., s = S / N

An Act of DesperationAn Act of Desperation

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Planck’s Model To compute the entropy per oscillator

Planck had to compute the number of ways Ω to distribute M indistinguishable quanta amongst the N indistinguishable oscillators

He then had to determine the value of the quantum of energy ε

s =k 1+

Eε

⎛

⎝⎜⎞

⎠⎟ln 1+

Eε

⎛

⎝⎜⎞

⎠⎟−EεlnEε

⎡

⎣⎢

⎤

⎦⎥

An Act of DesperationAn Act of Desperation

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The Reluctant Revolutionary To determine ε, Planck started with his formula for

the spectral density u(λ) and computed the from it the entropy per oscillator, using the thermodynamic relation 1/T = ∂S/∂E, which , which relates the relates the absolute temperature T, entropy S and average energy E of N objects.

When he compared his two expressions for the entropy he found they agreed only if he assumed each quantum had energy

/hcε

An Act of DesperationAn Act of Desperation

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/hcε

An Act of DesperationAn Act of Desperation

And thus did this reluctant revolutionary start, in an “act of desperation”,

the quantum revolution

The constant, h, which does notappear in classical physics is calledPlanck’s constant in his honor

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Extra Credit (5)Extra Credit (5)

1. Using Planck’s model, derive Planck’s formula for the entropy per oscillator

2. Then use 1/T = ∂S/∂E = ∂s/∂e to show that the average energy per oscillator is given by

e = ε /(eε/kT - 1)

Hint:

for large N, ln N! ~ N ln N – N

and recall that e = εM / N Due 10/06/09

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Planck’s Model

The number of ways of distributing M indistinguishable things amongst N indistinguishable boxes is

)!1(!

)!1(

ΩNM

NM

Extra Credit (5)Extra Credit (5)

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SummarySummary

Discovery of Electron J.J. Thomson measured e/m ratio in 1897 Millikan measured e in 1909

Ultraviolet Catastrophe Rigorous application of the laws of

thermodynamics, Newton and Maxwell by Lord Rayleigh led to an absurd prediction: a hot oven should emit an infinite amount of energy!

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SummarySummary

The Birth of Quantum Physics In 1900, Planck derived the correct formula for the

blackbody spectral density under the assumption that electromagnetic energy changes in discrete amounts given by ε = hc/, an assumption he regarded as “an act of desperation”

He spent many years, thereafter, trying to evade this assumption; but failed!