# CCSU Origin of Quantum Theory Physics Study Material

CCSU Origin of Quantum Theory Physics Study Material

## CCSU Origin of Quantum Theory : Planck’s Radiation Law : Photon

### Spectral Distribution of Energy in Black-Body Radiation :

A (perfectly) black-body is a full radiator, that is, it emits radiation of all possible wavelengths. Lummer and Pringsheim, in 1899, studied the spectral distribution of energy (that is, energy distribution among the various wavelengths) in the radiation of a black body at different temperatures. They used an electrically-heated chamber with a narrow hole as a black body, and measured temperature by a thermocouple. The experimental arrangement is shown in Fig.

The radiation coming from the black body O is allowed to fall on a slit S by means of a concave mirror M_{1}. The slit S is in the focal plane of another concave mirror M_{2}, and so the radiation falling on M_{2} is reflected in the form of a parallel beam and falls on a fluorospar prism P. After refraction through the prism P, the radiation falls on a third mirror M_{3 }which focusses it on a Lummer-Kurlbaum linear bolometer T placed in its focal-plane. On rotating the mirror M_{3} about a vertical axis, the radiations of different wavelengths fall on the bolometer one after the other. Each time the deflection in the galvanometer G connected in the bolometer circuit is read. These deflections are the measure of radiant energies E_{λ} of different wavelengths falling on the bolometer. E_{λ}, called as ‘spectral radiancy’, is defined such that the quantity E_{λ}dλ is the energy, For wavelengths lying between λ and λ + dλ, emitted per second per unit surface area of the black body. Now, a graph is plotted between energy and wavelength. The graph so obtained is called the ‘spectral distribution curve’.

Lummer and Pringsheim heated the black body to different temperatures and drew distribution curves for all temperatures on the same graph paper. The curves so obtained are shown in Fig. These curves whose general shape is same for all temperatures give the following information regarding the characteristics of black-body radiation.

(i) At a particular temperature T, the spectral radiancy E_{λ} first increases with increase in wavelength λ, attains a peak value, and then decreases. This means that for a given temperature, the radiant energy emitted by a black-body is maximum for a particular wavelength λ_{m} (say). Most of the energy is emitted at wavelengths not very different from λ_{m}.

(ii) As the temperature is raised, the peaks of the curves shift towards shorter wavelengths, that is, the maximum value of E_{λ} is obtained at smaller values of λ. In 1896, Wien established the following relation between temperature T and λ_{m }(for which Eλ is maximum),

λ_{m}T = constant.

This is called ‘When’s displacement law’. It means that as the temperature of the black body is raised, the body emits radiation of shorter wavelengths in increasing quantity.

(iii) Further, as the temperature is raised, the areas enclosed by the curves go on increasing. The area enclosed by a curve represents the total radiant energy E (of all wavelengths) emitted by the black body at that temperature. When the areas enclosed by different curves are measured, they are found to be proportional to the fourth power of the corresponding absolute temperatures, that is,

Thus, these curves verify Stefan’s law.

**Success and Failure of Classical Theory in explaning Energy Distribution :** Wien, in 1893, had shown from pure thermodynamic reasoning that the spectral energy-density u_{λ} in wavelength interval λ to λ+ dλ emitted by a black body at temperature T is of the form

u_{λ}dλ = A/λ^{5} f(λT)dλ,

Where A is a constant and f(λT) is an undetermined function. This is in agreement with the experimental result that the product λT at the peak of the E_{λ}– λ curve is same at all temperatures.

In order to find the form of f (λT), Wien assumed that the black-body radiation inside a cavity may be supposed to be emitted by resonators of molecular dimensions having Maxwellian velocity distribution, and the frequency of emitted radiation is proportional to the kinetic energy of the corresponding resonator. On the basis, When established the following distribution formula :

u_{λ}dλ = A/λ^{5} e^{-B/λT}dλ,

Where A and B are constants.

Wien’s formula was found to agree with experiment at short wavelengths but did not fit well at long wavelengths (Fig.) According to the formula, u_{λ }= 0 for λ=0 and also for λ=∞ as it should be, but it keeps u_{λ } finite for T = ∞ which is unlikely.

Rayleigh and Jeans considered the black-body radiator (cavity) full of electromagnetic waves of all wavelengths between 0 and ∞ which, due to reflection at the walls, form standing waves. They calculated the number of possible waves having wavelengths between λ and λ + dλ, and using law of equipartition of energy, established the following formula :

u_{λ}dλ = 8πkT/λ^{4}dλ,

Where k is the Boltzmann’s constant.

Rayleigh-Jeans formula was found to agree with experiment at long wavelengths only (Fig.). It is, however, open to a serious objection. As we move towards shorter wavelengths (that is, towards ultraviolet), the predicted energy would increase *without limit*, thus diverging enormously from experiment. This completely erroneous prediction is known as the ‘ultraviolet catastrophe’.

Furthermore, at any temperature T, the total energy of radiation, is predicted to be infinite which is against the Stefan’s law.

The shortcomings of classical theory were overcome by Planck’s quantum hypothesis of radiation.

**Planck’s Quantum Hypothesis of Radiation : ** Planck, in 1900, introduced an entirely new idea to explain the distribution of energy among the various wavelengths of the black-body (cavity) radiation. He assumed that **the atoms of the walls of a uniform temperature enclosure (known as cavity radiator) behave as oscillators, each with a characteristic frequency of oscillation.** These oscillators emit electromagnetic radiant energy into the cavity and also absorb the same from it, and maintain an equilibrium state. Planck made two rather revolutionary assumptions regarding these atomic oscillators:

(i) **An oscillator can have only discrete energies given by**

ε = nhv,

Where v is the frequency of the oscillator, h is a constant known as ‘Planck’s constant’, and n is an integer known as ‘quantum number’. This means that the oscillator can have energies hv, 2hv, 3hv, …., and not any energy in between. In other can have energy of the oscillator is quantized.

**The oscillators do not emit (or absorb) energy continuously but only in ‘jumps’.** That is, an oscillator emits (or absorbs) energy in the form of small bundles or packets of definite amount of energy which are called ‘quanta’. Later on, these bundles were called ‘photons’. The energy associated with each photon (quantum) is hv. Thus, emitted (or absorbed) energies may be hv, 2hv, 3hv, ….. but not in between.

**Average Energy of Planck’s Oscillator :** Let us now calculate the average energy of a Planck’s oscillator of frequency v. The relative probability that an oscillator has energy hv at temperature T is given by the Boltzmann factor e^{-hv/kT}. Now, let N_{0}, N_{1}, N_{2}….N_{r}……be the number of oscillators having energies 0, hv, 2hv, …. rhv… respectively. Then, we have

N_{r} = N_{0 }e^{-rhv/kT}

The total number of oscillators is

N = N_{0} + N_{1} + N_{2} + ………………..

= N_{0} (1 + e^{-hv/kT} + e^{-2hv/kT} + ……………….)

= N_{0}/1- e^{-hv/kT} …………….(i)

The total energy of the oscillators is given by

ε = (N_{0} x 0) + (N_{1} x hv) + (N_{2} x 2hv) + ……………..

= (N_{0} x 0) + (N_{1} e^{-hv/kT} x hv) + (N_{2} e^{-2hv/kT }x 2hv) + ……………..

= N_{0 }e^{-hv/kT} hv (1+ 2 e^{-hv/kT }+ 3 e^{-2hv/kT} + …………….)

= N_{0 }e^{-hv/kT }hv/(1- e^{-hv/kT})^{2* }…………….(ii)

Dividing eq. (ii) by eq. (i), we obtain the average energy of an oscillator as given by

έ = ε/N = e^{-hv/kT}(hv)/1- e^{-hv/kT }= hv/ e^{hv/kT }-1 …………………(iii)

**Planck’s Radiation Formula :** The energy-density of radiation, Uv, in the frequency range v to v+dv is related to the average energy of an oscillator emitting v radiation by

U_{v} dv = 8πv^{2}dv/c^{3} x έ.

The term 8πv^{2}dv/c^{3} represents the numbers of electromagnetic standing waves per unit volume in the frequency range v to v + dv in the cavity radiation, as computed by Rayleigh and Jeans.

Substituting the value of έ from eq. (iii), we have

U_{v} dv = 8πv^{2}dv/c^{3} x (hv/ e^{hv/kT }-1)

Or U_{v} dv = 8πhv^{3}/c^{3} x (dv/ e^{hv/kT }-1)

This is Planck’s radiation formula in terms of frequency v. To express it in terms of wavelength, we observe that, since v = c/λ,

dv = -c/λ^{2} dλ

and since an increase in frequency corresponds to a decrease in wavelength,

u_{λ }dλ = – u_{v} dv.

Therefore u_{λ }dλ = 8πh(c/λ)^{3} (c/ λ^{2}d λ)/c^{3} .e^{hc/} ^{λ}^{kT }-1

Or u_{λ }dλ = 8πhc .d λ)/ λ ^{5} .e^{hc/} ^{λ}^{kT }-1 ……………..(iv)

This is Planck’s radiation formula in terms of wavelength.

**Explanation of Energy Distribution by Planck’s Formula : Wien’s Law and Rayleigh-Jeans Law are Special Cases :** The Planck’s formula is found to be in complete agreement with experiment for the entire wavelength range at all temperatures. This can be seen in the following way :

- When λ is very small, then e
^{hc/}^{λ}^{kT }>> 1, so that Planck’s Formula given by eq. (iv) can be written as

u_{λ }dλ = 8πhc / λ ^{5} .e^{-hc/} ^{λ}^{kT }d λ.

Putting 8πhc = A and hc/k = B, we have

u_{λ }dλ = A / λ ^{5} .e^{-B/} ^{λ}^{T }d λ.

This is Wien’s law which agrees with experiment at short wavelengths.

2. When λ is very large, then e^{hc/} ^{λ}^{kT }= 1 + hc/λkT, so that eq. (iv) can be written as

u_{λ }dλ = 8πhc / λ ^{5 }(1 + hc/λkT – 1) dλ = 8πkT/λ^{4} dλ.

This is Rayleigh-Jeans law which agrees with experiment at long wavelengths.

Further, both Wien’s displacement law and Stefan’s law can be derived from the Planck’s formula.

**Photon :** The different forms of energy like radio waves, heat rays, ordinary light, X-rays, ϒ-rays etc., emitted by atoms under different situations are all electromagnetic radiation of different frequencies (or wavelengths). We call them electromagnetic waves because under suitable circumstances they exhibit refraction, interference and diffraction.

The wave-like character of radiation, however, failed to explain the observed energy distribution in the continuous spectrum of black-body radiation. To meet this serious problem, Planck, in 1900, presented his quantum theory of radiation. According to this theory, radiation is emitted (or absorbed) discontinuously in indivisible packets of energy, named as ‘photons’ or ‘quanta’. Each photon of radiation of a given frequency v has the same energy hv, where h is Planck’s constant. Planck did not suggest anything new regarding the propagation of radiation in space.

Einstein extended Planck’s quantum hypothesis by assuming that** radiation not only is emitted (or absorbed) as indivisible photons, but also continues to propagate through space as photons.** On this hypothesis, Einstein in 1905 explained photoelectric effect and in 1907 explained the specific heat of solids. In 1913, Bohr used the quantum theory to explain hydrogen spectrum, and in 1922, Compton applied it to the scattering of X-rays.

### CCSU Properties of Photons :

- Photons are indivisible packets of electromagnetic energy.
- They retain their identity until completely absorbed by some atom (as happens in photoelectric effect and pair production).
- The size (energy content) of a photon (hv) is proportional to the frequency of radiation so that photons of different radiations are different sizes. For example, blue photons are larger than red photons; X-ray photons are considerable larger than visible light photons.
- The intensity of radiation (I) is equal to the number of photons (N) crossing unit area per second multiplied by the size (hv) of the photon, that is,

I = Nhv

Thus, for a given frequency, the intensity depends simply upon the number of photons.

5. All photons travel with the speed of light in vacuum (c) , and have a zero rest mass. Hence there is no frame in which a photon is at rest.

6. Although the rest-mass of a photon is zero, but its kinetic mass is not zero. According to Einstein’s mass-energy relation (E =mc^{2}), the energy of a photon is

E (= hv) = mc^{2 }

Hence the kinetic mass of photon is

m = hv/c^{2}

But v = c/λ, where λ is the wavelength of the photon radiation.

m = h/c^{2 }(c/λ) = h/cλ.

Thus, the kinetic mass of photon is

m = hv/c^{2 }= h/cλ.

The momentum of photon is

p = kinetic mass of photon x velocity of photon (light) = hv/c2 x c = hv/c.

Again, v = c/λ.

Thus, the momentum of photon is

p = hv/c = h/λ

The Compton effect is a direct evidence of the existence of momentum for a photon.

### See more of CCSU Study Material on Facebook

Like Our CCSU Study Material Facebook Page

## Leave a Reply