1 IIT Delhi - CML 100:1 – The shortfalls of classical mechanics

Classical Physics 1) precise trajectories for particles – simultaneous specification of position and momentum 2) any amount of energy can be exchanged – implies the various modes of motion (translational, vibrational, rotational) can have continuous energies

Light electromagnetic field

frequency 휈, wavelength 휆, speed in vacuum 푐 = 3 × 108 푚푠−1.

휆휈 = 푐

휈 1 wavenumber, 휈̅ = = 푐 휆

Black Body An object capable of emitting/absorbing all frequencies of radiation uniformly

Figure 1.2: Change in the emission maxima from Figure 1.1: Representation of a blackbody the blackbody as a function of temperature

Rayleigh-Jeans Law

Classical viewpoint – electromagnetic field is a collection of oscillators of all frequencies

Equipartition principle – average energy of each oscillator = 푘푇

푑퐸 = 휌푑휆 휌 = 8휋푘푇/휆4

Ultraviolet catastrophe

As wavelength decreases, the energy density increases to infinity

2 IIT Delhi - CML 100:1 – The shortfalls of classical mechanics

Planck’s hypothesis

All possible frequencies are not allowed – energy of each oscillator is restricted to discrete values

Quantization, 퐸 = 푛ℎ휈 푛 = 0, 1, 2 … where ℎ is the Planck constant (= 6.626 × 10−34J. s)

8휋ℎ푐 푑퐸 = 휌푑휆 휌 = 휆5(푒ℎ푐/휆푘푇 − 1)

Short wavelengths: ℎ푐/휆푘푇 ≫ 1 ∴ 푒ℎ푐/휆푘푇 → ∞ faster than 휆5 → ∞. So no UV catastrophe! The hypothesis fits well with the observed data. (Actually the value of ℎ was determined by fitting)

Why this works?

Classical – heat black body – all oscillators with all frequencies excited – the excitation of high frequencies leads to UV catastrophe

Planck – oscillators can be excited only if energy ℎ휈 is available. Higher frequencies implies higher energies which is too large to supply by heating the walls of the blackbody

Heat Capacities Dulong and Petit Law: Measured value for solids = 25 J K−1 mol−1

Mean energy of an atom as it oscillates about its mean position = 푘푇 for each direction

(equipartiion principle). This gives a total of 3푘푇 for the three directions. For a mole, 푈푚 = 3푅푇

휕푈푚 −1 −1 퐶푉.푚 = ( ) = 3푅 = 24.9 J K mol 휕푇 푉

At lower temperatures, molar heat capacities are lower than 3푅.

Einstein’s explanation (1905): Each atom oscillates about its equilibrium position with frequency 휈. Used Planck’s hypothesis – energy of oscillation equals 푛ℎ휈. This gives,

3푁퐴ℎ휈 푈 = 푚 푒ℎ휈/푘푇− 1

2 휃 2 푒휃퐸/2푇 Differentiate w.r.t. 푇 to get 퐶 = 3푅푓 푓 = ( 퐸) ( ) 푉,푚 푇 푒휃퐸/푇−1

Einstein temperature 휃퐸 = ℎ휈/푘 corresponds to the frequency of oscillation.

2 2 휃퐸 1+ 휃퐸/2푇+ … At high temperatures (휃퐸 ≪ 푇) , expand 푓 = ( ) { } ≈ 1 ignoring the higher terms 푇 (1+휃퐸/푇+⋯ ) −1 which gives the high temperature result 퐶푉,푚 = 3푅

2 휃 2 푒휃퐸/2푇 휃 2 At low temperatures 휃 ≫ 푇), 푓 ≈ ( 퐸) ( ) = ( 퐸) 푒−휃퐸/푇 Exponential decays faster to 퐸 푇 푒휃퐸/푇 푇 zero than 1/푇 goes to infinity. 3 IIT Delhi - CML 100:1 – The shortfalls of classical mechanics

Reason: At low temperatures fewer atoms have the energy to oscillate.

Figure 1.3: Temperature dependence of heat capacities

Further: All atoms do not oscillate with the same frequency. There is a distribution of frequencies

ranging between 0 푡표 휈퐷. Read Debye’s formula for the heat capacities.

Atomic and Molecular Spectra

Figure 1.4: Molecular spectrum of SO2. Figure 1.5: Energy levels in a molecule

Radiation is absorbed/emitted in discrete steps and not continuously. This suggests the energy levels in an atom/molecule are quantized. The frequency of radiation absorbed/emitted is given by the Bohr frequency condition

Δ퐸 = ℎ휈

Wave-particle duality Atomic and molecular spectra show that EM radiation of frequency 휈 can possess only the energies 0, ℎ휈, 2ℎ휈 … suggesting the radiation is composed to 0, 1, 2 ... number of particles called photons. 4 IIT Delhi - CML 100:1 – The shortfalls of classical mechanics

Photoelectric effect

Ejection of electrons from a metal surface upon irradiation with light

1) No electrons are ejected unless the frequency of radiation is higher than a threshold value (work function). The intensity of radiation does not matter 2) The K.E. of ejected electron varies linearly with the frequency of the light used. It is independent of the intensity 3) All that the intensity does is to change the number of electrons ejected, i.e. the current

Figure 1.7: The photoelectric effect

Electron collides with a particle carrying sufficient energy required to eject it.

1 퐾퐸 = 푚푣2 = ℎ휈 − 휙 2 푒

Wave character of particles:

Davisson and Germer (1952) – diffraction of electrons by a crystal. The basis of TEM and SEM. Diffraction requires constructive and destructive interference of waves. Hence, particles must possess wavelike character.

de Broglie relationship (1924): 휆 = ℎ/푝. A very fundamental statement.

5 IIT Delhi - CML 100:1 – The shortfalls of classical mechanics

Hydrogen Spectra

Figure 1.6: Emission spectrum of hydrogen atom on a log scale

Figure 1.7: Atomic interpretation of the hydrogen spectrum

Rydberg Formula:

1 1 1 −1 휈̅ = = 109680 ( 2 − 2) cm (푛2 > 푛1) 휆 푛1 푛2

Bohr Theory for H-atom: Electron revolving around the nucleus

2 2 −12 2 −1 −2 Coulomb’s Law 푓 = 푒 /4휋휖0푟 휖0 = 8.85419 × 10 C . N . m

2 Centrifugal force 푓 = 푚푒푣 /푟

Bohr’s assumption – integral wavelengths fit the circular orbit for constructive interference

2휋푟 = 푛휆

which, on using the de Broglie equation gives 6 IIT Delhi - CML 100:1 – The shortfalls of classical mechanics

푛ℎ 푚 푣푟 = = 푛ℏ 푒 2휋

Equate the two forces and substitute the above expression for the first Bohr radius, 푎0

2 2 2 2 −12 2 −1 2 −34 2 휖0ℎ 푛 4휋휖0ℏ 푛 4휋(8.85419 × 10 C . N . m )(1.055 × 10 J. s) 푎0 = 2 = 2 = −31 −19 2 = 52.92 pm 휋푚푒푒 푚푒푒 (9.109 × 10 kg)(1.6022 × 10 C)

2 1 2 푒 Total energy of the electron = KE + PE = 푚푒푣 − 2 4휋휖0푟

푒2 Using the above equations, 퐸 = − 8휋휖0푟

4 푚푒푒 1 And substituting for 푟, 퐸푛 = − 2 2 2 8휖0ℎ 푛

And transitions between two states 푛1and 푛2 will be given by

4 푚푒푒 1 1 Δ퐸 = 2 2 ( 2 − 2) = ℎ휈 8휖0 ℎ 푛1 푛2

or

4 푚푒푒 1 1 ν̅ = 2 3 ( 2 − 2) 8휖0 푐ℎ 푛1 푛2

4 푚푒푒 and the Rydberg constant 푅∞ is given by 2 3 8휖0푐ℎ