Planck’s Quantum Theory: Formula, Postulates, Relation of black body radiation with Planck’s law, Black body and Electromagnetic Radiation

Planck’s quantum theory is a natural phenomenon of quantum mechanics. The theory was put forward by German physicist Max Planck. It explains the quantum nature of the energy of electromagnetic waves. Planck’s quantum theory deals with phenomena such as the photoelectric effect and the nature of radiated emission which were not explained by the laws of classical mechanics.

The black body spectrum, which is the radiation emitted by a body that absorbs all the light that falls on it, shows significant divergence at higher frequencies. This divergence was at odds with hypothetical results predicted by the classical model of physics. Planck resolved this conundrum by theorizing that a particle in oscillation only absorbs radiation with a minimum amount of energy. Radiation with energy less than this minimum amount is not absorbed. The value of this minimum energy was discovered by Planck to be proportional to the frequency of oscillation of the oscillating particle. The relation between the energy required to excite the particle to a higher energy level and the frequency of the particle is given by the relation:

E = hν

Here,

E is the minimum energy

h is Planck’s constant

ν is the frequency of the particle

This discovery proved significant in the later development of quantum mechanics. Planck’s result showed that electromagnetic waves behave as both particles and waves when interacting with matter. This dual nature of electromagnetic radiation led to further investigations by de Broglie. He was successful in obtaining a formula for the wavelength of any particle with a mass and momentum. The relation between the wavelength of a body with its momentum is given by:

Here,

h is Planck’s constant

m is the mass of the body

v is the velocity of the body

Electromagnetic Radiation

Electromagnetic radiation refers to the phenomenon of electromagnetic waves travelling through space. Electromagnetic waves do not need a medium to propagate. The radiation is sustained by the electric component and the magnetic component of the wave giving rise to each other through space. Both the components lie in mutually perpendicular planes. All electromagnetic waves have certain features associated with them. These can be summarised as below:

Wavelength: Wavelength is the distance between two consecutive crests (elevation in the wave) or two consecutive troughs (depression in wave) of a wave.
Frequency: Frequency can be defined as the number of waves occurring in a given interval of time.

Electromagnetic radiation can be classified based on its frequency and wavelength. Waves of higher frequency than the visible light come into the categories of x-rays, gamma rays and ultraviolet rays. While the waves of lower frequency than that of the visible light are infrared rays, radio waves and microwaves. Humans can only perceive electromagnetic waves that fall in a certain frequency range. The waves that fall in this range are called visible light.

Black body radiation

A black body is a body or object which absorbs all the radiation falling on it. An ideal black body absorbs and emits radiations of all frequencies. The emission of electromagnetic radiation from a black body depends on temperature. The frequency variation of emitted radiation from a black body can be described by Planck’s law. This type of radiation is also known as Planck’s radiation which is a type of thermal radiation. Varying with temperature, the higher the temperature of the body, the more is the emission of radiation of all wavelengths.

Relation of black body radiation with Planck’s law

Planck’s radiation law describes the relation of temperature with radiated energy. With the increase in temperature, the radiation of every wavelength emitted from a black body also increases. The relation is given by:

B (v, T) = (2hv³/c²) . (1/e^hv/k_b^T – 1)

Where,

v is for frequency,

k_b is Boltzmann constant

h is Planck’s constant.

Postulates of Planck’s quantum theory

Planck’s quantum theory states the following postulates:
1. The energy is not radiated or emitted continuously. It is emitted in small proportions in the form of energy packets called quanta.
2. Radiation when in the form of light, each particle is known as a photon. Photons are energy particles of small proportion in the case of light.
3. The energy of a photon or one quantum of energy is directly proportional to the frequency of the radiation. E = hν where h is Planck’s constant and v is the frequency of radiation.
4. The total energy of radiation is represented as a whole number multiple of energy of a quantum as hν, 2hν, 3hν and so on.

Frequently asked questions related to Planck’s Quantum Theory:

Q1. What is Planck's Quantum Theory?
Answer: Planck's Quantum Theory is a fundamental theory in physics that explains the behavior of energy at the atomic and subatomic levels. It was proposed by Max Planck in 1900 to resolve the problem of black body radiation. The theory states that energy is not continuous but is quantized, meaning it can only be emitted or absorbed in discrete amounts called quanta.

Q2. What is the formula associated with Planck's Quantum Theory?
Answer: The central formula of Planck's Quantum Theory is E=hνE , where E is the energy of a quantum, h is Planck's constant is the frequency of the radiation.

Q3. What are the main postulates of Planck's Quantum Theory?
Answer: The main postulates of Planck's Quantum Theory are:

Energy is emitted or absorbed in discrete units called quanta.
The energy of each quantum is directly proportional to the frequency of the radiation, expressed by the formula E=hνE = h \nuE=hν.
The quantization of energy explains the black body radiation spectrum.

Q4. How is black body radiation related to Planck's law?
Answer: Black body radiation refers to the electromagnetic radiation emitted by a perfect black body, an idealized object that absorbs all incident radiation. Planck's law provides a formula to describe the spectral density of this radiation, which depends on the temperature of the black body. The law accurately explains the observed spectrum of black body radiation and resolved the ultraviolet catastrophe predicted by classical physics.

Q5. What is the significance of Planck's law in understanding black body and electromagnetic radiation?
Answer: Planck's law is significant because it was the first to accurately describe the spectral distribution of radiation emitted by a black body at different temperatures. It marked the beginning of quantum mechanics by introducing the concept of energy quantization. This law helps in understanding the behavior of electromagnetic radiation and has applications in various fields such as quantum mechanics, thermal radiation, and astrophysics.