The failure of the classicalmechanics

We review some experimental evidences showing that severalconcepts of classical mechanics cannot be applied.

- The black-body radiation.

- Atomic and molecular spectra.

- The particle-like character of EMR.

- The photoelectric effect

This will enable us to reformulate the nature of electromagneticradiation in terms of quantum mechanics: the photon.

The black-body radiation

A hot object emits electromagnetic radiation of varyingfrequencies depending of the nature of the object and itstemperature.

Black-body = theoretical object that absorbs and emits allradiation frequencies.

As the temperatureincreases, the peakwavelength emitted bythe black bodydecreases.

As temperatureincreases, the totalenergy emittedincreases.

The Rayleigh-Jeans law

Rayleigh and Jeans considered a collection of classicalelectromagnetic oscillators of all possible frequencies .

The radiation energy distribution (or density), dE, between and d at a temperature T is given by:

4BTk8

withddE

= density of states (J/m4), kB = Boltzmann’s constant(1.3806·10-23 J/K), hence dE is expressed in J/m3 (an energyper volume).

From this, one could easily calculate the total radiation energy:

000

Tot ddλdEE

increases decreases

decreases increases !

Ultravioletcatastrophe

Planck’s law

Planck proposed that the energy of each electromagneticoscillator is limited to non-arbitrary discrete values(quantization of energy):

E = nh n = 0, 1, 2,…

where h = Planck’s constant (6.6262·10-34 J·s).

On this basis, he derived the Planck distribution:

dE = ·d with = 1eλ

hc8Tk/hc5 B

Reproduces the experimental curves for all wavelengths andresembles the Rayleigh-Jeans law apart from the all-importantexponential factor in the denominator.

For short wavelengths: 1Tk

hc

B

and TBk/hce faster

than 5 0 therefore 0 as 0 (or n )

For long wavelengths: 1Tk

hc

B

and series expansion gives:

Tk

hc1...

Tk

hc11e

BB

Tk/hc B

Classical mechanics: all oscillators share equally in the energysupplied by the walls (even highest frequencies). Quantummechanics: oscillators are excited only if they can acquire anenergy of at least h (too large at high frequencies).

Wien’s displacement law

We observed that the peak wavelength emitted, max,decreases as temperature increases. The relation betweenmax and T is known as Wien’s displacement law.

Naturally, it is deduced from the condition: d/d = 0:

0

1e

5

λ

1

1e

e

Tλk

hchc8

dλ

dρTk/hc62Tk/hc

Tk/hc

7B

BB

B

05e1

1

Tλk

hcTk/hc

B B

If we define:Tλk

hcx

B

, then: 05e1

xx

Numerically: x = 4.965114231744276… (dimensionless).

T

mK102.898

xTk

hcλ

3

Bmax

Or: mK102.898Tλ 3max

The Stefan-Boltzmann law

We observed that as temperature increases, the total energyemitted increases. The relation between the total energyemitted per unit surface per unit time (the power radiated) andT is known as the Stefan-Boltzmann law.

To define this law, we simply have to integrate the energydensity over all wavelengths:

0

Tk/hc50 1eλ

λdhc8λdρP(T)

B

The integration is easier to perform if we convert to :

2

dcd

c

Hence:

0x

3

3

4B

0Tk/h

3

30 1e

dxx

hc

Tk8

1e

d

c

h8dρP(T)

B

The last integral happens to be 4/15:

4

3

4B

5

Tc

4

hc15

Tk8)T(P

where = Stefan-Boltzmann constant (5.670·10-8 Js-1m-2K-4)

Examples of application

The temperature of a Pahoehoelava flow can be estimated byobserving its colour. The resultagrees nicely with the measuredtemperatures of lava flows at about1000 to 1200°C.

Mammals at roughly 300K emitpeak radiation at 10 μm, in the farinfrared. This is therefore the rangeof infrared wavelengths that pit vipersnakes and passive IR camerasmust sense.

The universal microwavebackground radiation, originatingfrom the Big Bang, peaks in powerat = 1 mm, and fits the Planckcurve for a black-body of T = 2.728Kto great precision.

The solar spectrum shows max atabout 500 nm, corresponding to asurface temperature of 6000K. Thiswavelength is (not by accident) fairlyin the middle of the most sensitivepart of land animal visual spectrumacuity.

Atomic and molecular spectra

Most compelling evidence for quantization of energy comesfrom spectroscopy = detection + analysis of EMR absorbed,emitted, or scattered by a substance.

A typical atomic spectrum: A typical molecular spectrum:

Obvious feature: radiation is emitted or absorbed at a series ofdiscrete frequencies. Only understood if the energy of theatoms (molecules) is also confined to discrete values.

If the energy is decreased by E, theenergy is carried away as radiation offrequency , and an emission line, asharply defined peak, appears in thespectrum. The atom (or molecule)undergoes a spectroscopic transition,a change of state, when the Bohrfrequency condition is fulfilled:

E = h

The particle-like characterof EMR

Quantization of energies h, 2h, 3h,… suggests that EMRconsists of 1, 2, 3,… particles of energy h, called photons.

As example, let’s calculate the number of photons emitted by a100 W yellow lamp, 560 nm, in 1 second:

- Each photon has an energy of h.

- Total # of photons to produce an energy E is E/h.

- Power is defined by an energy over a time: P = E/t.

- Hence:

hc

tP

/ch

tP

h

EN

- Solving with numerical inputs yields: N = 2.8·1020.

- It would take nearly 40 minutes to produce 1 mol of thesephotons.

The photoelectric effect

The photoelectric effect is the ejection of electrons from metalswhen exposed to UV radiation.

Experimental observations:1) No electrons are ejects, regardless of EMR intensity

below a -threshold, , characteristic of the metal.2) Kinetic energy of emitted electrons increases linearly

with but independent of EMR intensity.3) Even at very low intensity, electrons are ejected

immediately if > .

From classical mechanics: 20EI , hence kinetic energy of

electrons is independent. No frequency threshold.

Observations suggest a collision with a particle-like projectilecarrying enough energy to eject the electron form the metal.Conservation of energy requires:

hvm2

1 2e

is characteristic of the metal, called its work function, that is,the energy required to remove an electron to infinity.