Question 1: (a) What is Planck’s Quantum theory? What are the postulates of this theory?
(b) Prove that E = hcν̅, where E = Energy, h= Planck’s constant, c = velocity of light and ν̅ is wave number.
(c) What will be the energy of radiation with λ = 2 × 10-8 m.
Planck's Quantum Theory
Towards the end of 19th century, Physicist were quite comfortable that they had solved all the problems related to Physics. However, there came some phenomena that could not be explained on the basis of existing theories. One of them was the energy from a black body.
The German Physicist, Max Planck, proposed his quantum theory of radiations to explain the energy from the black bodies. He said that the high energy was discontinuously emitted by the particles of a black body. (Classically, the assumption was of the continuous emission of energy). He said that each radiation is associated with a specific frequency which increases with temperature.
Postulates of Planck's Quantum Theory
Main postulates of his theory are summarized below.
- Energy is not emitted or absorbed in a continuous way. Rather the emission or absorption of energy takes place in a discontinuous manner. It is always in the form of “quanta” (i-e, small amounts). These quanta come from small pulses of energy or wave packets emitted during radiation. In case of light energy, they are called photons.
- Each quantum or wave packet has an energy E, which is directly proportional to the frequency ν of the radiation. Symbolically, E ∝ ν OR E = hν. ‘h’ is called Planck’s constant and its value is 6.6262 × 10-34
- The energy emitted or absorbed by a body is always in whole number multiple of quantum. Mathematically, E = nhν, where n = 1, 2, 3, … . This shows the quantization of energy.
(b) To prove that E = hcν̅
We know that wave number is the number of waves in a unit length (1 m, 1 cm etc). If λ is the wavelength of a wave, then there would be 1/λ waves in a unit length. Therefore, ν̅ = 1/λ m-1
Similarly, the relation between frequency and wavelength of a photon is ν = c/λ , where c is the velocity of photon (radiation) and its value is 2.99 × 108 ms-1.
We know that Planck’s relation for the energy E of a photon of frequency ν is E = hν. Put the value ν from the above equation, we hve, E = h × c/λ = hc× 1/λ . But 1/λ = ν̅, as derived above. Therefore, E = hcν̅. This proves the given statement.
(c) Energy of the radiation for λ = 2 × 10-8 m.
We know, from Planck’s Quantum theory, that E = hν.
Here h is the Planck’s constant = 6.6262 × 10-34 Js and ν = c/λ , where c is the velocity of electromagnetic radiation = 2.99 × 108 ms-1. So put the values of h, ν and the given value of λ in the above equation.