Planck’s Quantum Theory

Max Planck's quantum theory, which is fundamental to quantum mechanics, explains that energy in electromagnetic waves is quantized. This theory helps to explain phenomena like the photoelectric effect and the behaviour of radiated emissions, which classical mechanics couldn’t fully account for.

1.0Introduction

In 1900, Max Planck introduced his groundbreaking theory that energy is quantised, existing in discrete packets known as quanta. Central to Planck's quantum theory, this principle asserts that power can only be transferred in specific, quantised amounts. Initially developed for radiation, Planck’s theory was extended in the 1920s and 1930s to other areas, including heat and electricity. 

A key constant derived from this theory is the Planck constant, 6.62×10⁻³⁴J⋅s, essential in measuring quantum properties. Planck also discovered that the frequency of radiation emitted by an object is inversely proportional to its wavelength at a given temperature, meaning higher-frequency light has a shorter wavelength.

2.0What is Planck’s Quantum Theory?

Planck’s Quantum Theory provides a framework for understanding the emission and absorption of radiation. The postulates of the theory are:

1. Discrete Energy Packets: Matter emits or absorbs energy in discrete amounts, not continuously, but in small packets or bundles called quanta.
2. Quantum Definition: The smallest unit of energy is called a quantum. For light, this unit is specifically referred to as a photon.
3. Energy-Frequency Relationship: The energy of each quantum is directly proportional to the frequency of the radiation. Mathematically, this is expressed as:
   E=hv 

Where E is the energy of the quantum,

ℎ is Planck’s constant, and v is the frequency of the radiation.

4. Whole-Number Multiples: Energy can only be emitted or absorbed in whole-number multiples of a quantum (nhv), where n is a positive integer. This means energy changes occur in units like hv, 2hv, 3hv, etc., not in fractional amounts like 1.5hv or 2.5hv.

This discovery was pivotal in the development of quantum mechanics. Planck’s findings revealed that electromagnetic waves display both particle-like and wave-like properties when interacting with matter. This dual nature concept inspired further exploration by de Broglie, who derived an equation linking a particle's wavelength to its mass and momentum. The relationship between a body’s wavelength and its momentum is expressed as:

$\lambda =\frac{h}{mv}$

Where:

• h is Planck’s constant
• m is the mass of the particle
• v is the velocity of the particle

3.0Black Body Radiation

A black body is an ideal object that absorbs all the radiation falling on it, regardless of frequency or angle. It also emits radiation across the full spectrum of frequencies. The amount and type of electromagnetic radiation a black body emits depends on its temperature. This emission is described by Planck’s law, which illustrates how the intensity of emitted radiation varies with frequency for a given temperature. 

This phenomenon, known as Planck’s radiation, is a type of thermal radiation. As the temperature of a black body increases, it emits more radiation at all wavelengths, with the peak intensity shifting to higher frequencies.

4.0Relation of Black Body Radiation with Planck’s Law

Planck’s law explains the relationship between the temperature of a black body and the energy it radiates. As the temperature of a black body rises, the intensity of radiation emitted at every wavelength also increases. Planck's formula mathematically expresses this relationship:

$B(\nu ,T)=\frac{2h{\nu }^3}{c^2}.\frac{1}{e^{\frac{h\nu }{K_{b}T}}-1}$

to be written in latex 

Where:

• B(v, T) represents the spectral radiance, which quantifies the energy emitted by a black body per unit area, unit time, unit solid angle, and per unit frequency.
• v is the frequency of the radiation.
• h is Planck’s constant.
• c is the speed of light.
• k_b​ is Boltzmann's constant.
• T is the absolute temperature of the black body.

This equation shows that radiation intensity depends on both the frequency and temperature, with higher temperatures resulting in increased radiation across all wavelengths.

5.0Electromagnetic Radiation

Electromagnetic radiation is the transmission of electromagnetic waves through space. These waves do not need a medium to travel. The radiation is driven by the interaction between the electric and magnetic fields, which continually induce each other as they propagate. These fields are oriented in perpendicular planes to each other.

Characteristics of electromagnetic waves include:

• Wavelength: The distance between two successive crests (high points) or troughs (low points) in a wave.
• Frequency:  The number of wave cycles passing a given point within a specified time frame.

Electromagnetic radiation is categorised according to its wavelength and frequency. Waves with frequencies higher than visible light include X-rays, gamma rays, and ultraviolet rays. Those with lower frequencies encompass infrared rays, radio waves, and microwaves. The portion of electromagnetic waves visible to the human eye is called visible light.

6.0Applications of Planck’s Quantum Theory

Planck’s quantum theory forms the foundation of quantum mechanics, leading to its use across various fields. Key applications include:

• Electronics: Quantum theory is crucial in developing semiconductors and transistors, the core components of modern electronic devices.
• Medical Field: It is used in MRI (Magnetic Resonance Imaging) and PET (Positron Emission Tomography) scanners, which rely on quantum principles to visualise internal body structures.
• Quantum Computing: Quantum mechanics enables the development of quantum computers, which use qubits to perform complex calculations much faster than classical computers.
• Lasers: Lasers' functioning is based on the principles of quantum mechanics, including energy quantisation and photon emission.
• Quantum Cryptography: Quantum theory is applied to create secure communication systems, where encryption relies on the properties of quantum particles to detect eavesdropping.