The Max Plank Law and the Constant

Learning Objectives

• Describe the Historical Context: Explain the discovery of Planck's constant by Max Planck and its significance in the context of blackbody radiation.
• Explain Fundamental Concepts: Articulate the role of Planck's constant in quantum mechanics, particularly in the equation \( E = h\nu \) which relates the energy of a photon to its frequency.
• Analyze Wave-Particle Duality: Discuss the dual nature of photons, demonstrating both wave-like and particle-like properties.
• Apply Knowledge to Practical Phenomena: Illustrate the practical implications of Planck's constant in understanding phenomena such as interference, diffraction, and the photoelectric effect.

Deriving Planck's constant involves understanding its role in quantum mechanics and its historical context. Planck's constant, denoted as h, was introduced by Max Planck in 1900 to explain black-body radiation. He proposed that energy is quantized and can be emitted or absorbed in discrete amounts called quanta. Before we proceed with the derivation process there are bunch of areas that we need to learn first. Let us start with the Plank's law.

Planck's Law

Planck's law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature T. The law is given by:

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

Planck's Law Components

\(B(\nu, T)\) is the spectral radiance (energy emitted per unit area per unit time per unit solid angle per unit frequency).

\(\nu\) is the frequency of the radiation.

\(T\) is the absolute temperature of the black body.

\(h\) is Planck's constant.

\(c\) is the speed of light.

\(k\) is the Boltzmann constant.

Planck's law implies that energy is emitted in discrete packets, known as quanta, rather than continuously. This concept of quantization of energy is fundamental to understanding various phenomena in quantum mechanics. Additionally, Planck's law explains the spectrum of radiation emitted by a black body, which is an idealized physical body that absorbs all incident electromagnetic radiation. The black-body radiation spectrum provides insights into the thermal properties and behavior of such bodies at different temperatures. Let us dive into the specifics of this law.

Spectral Radiance

Spectral radiance is a measure of the amount of energy emitted by a source, such as a black body, per unit area, per unit time, per unit solid angle, and per unit frequency. It essentially describes how much radiation is emitted at a specific frequency or wavelength. When we talk about energy emitted, we're referring to the total amount of energy that a source, like a black body, gives off. Per unit area means we're looking at the energy coming from a specific part of the source's surface. Per unit time tells us how much energy is emitted in a certain amount of time, usually measured in seconds. A solid angle is like a three-dimensional angle that helps us understand the direction in which the energy is emitted. Finally, per unit frequency means we're focusing on the energy emitted at a specific frequency of the electromagnetic spectrum.

The relationship between energy (E) and wavelength (λ)

In the electromagnetic spectrum, different wavelengths carry varying amounts of energy. Specifically, shorter wavelengths correspond to higher energy levels. This means that as the wavelength decreases, the energy it carries increases. For example, gamma rays have very short wavelengths and therefore possess high energy, while radio waves have much longer wavelengths and carry lower energy. This relationship is inverse.

Energy and Wavelength Relationship

The relationship between energy and wavelength is given by the equation:

$$ E = \frac{hc}{\lambda} $$

where:

• E is the energy
• h is Planck's constant
• c is the speed of light
• λ is the wavelength

The image illustrates the electromagnetic spectrum, which encompasses all types of electromagnetic radiation. It shows how different wavelengths correspond to varying frequencies and energy levels. Radio waves, with the longest wavelengths, have the lowest frequencies and energies and are used in communication systems like radio and television. Microwaves, which have shorter wavelengths than radio waves, are used in radar technology and for cooking food. Infrared radiation, shorter than microwaves, is associated with heat and used in thermal imaging and remote controls. Visible light, the range of wavelengths that the human eye can see, spans from red to violet and includes all the colors of the rainbow. Ultraviolet (UV) radiation, shorter than visible light, can cause sunburn and is used in sterilization and fluorescent lighting. X-rays, with even shorter wavelengths and higher energies, are used in medical imaging. Gamma rays, having the shortest wavelengths and highest energies, are produced by radioactive atoms and certain nuclear reactions, and are used in cancer treatment and sterilizing medical equipment. The image highlights the inverse relationship between wavelength and energy, fundamental to understanding the properties and uses of different types of electromagnetic radiation.

Frequency

Frequency refers to the number of complete cycles or oscillations that occur in a given period of time. In the context of waves, such as sound waves or electromagnetic waves, a cycle is one full wave pattern, which includes a crest and a trough. The frequency is typically measured in Hertz (Hz), where one Hertz equals one cycle per second. For example, if a wave completes 10 cycles in one second, its frequency is 10 Hz. Higher frequencies mean more cycles per second, while lower frequencies mean fewer cycles per second. 

Absolute Temperature

Absolute temperature is a measure of temperature relative to absolute zero, the point at which all molecular motion ceases. It is measured in Kelvin (K), which is the SI unit for temperature. The Kelvin scale starts at absolute zero (0 K), which is equivalent to -273.15°C. Unlike Celsius or Fahrenheit, the Kelvin scale does not use degrees; it simply uses Kelvins.

Planck's Constant

Planck's constant, denoted as h, is a fundamental constant in physics that plays a crucial role in quantum mechanics. Its value is approximately $$ 6.62607015 \times 10^{-34} \, \text{joule-seconds (J·s)} $$ Planck's constant relates the energy of a photon to its frequency through the equation E=hν, where E is the energy, h is Planck's constant, and ν (nu) is the frequency of the photon. Planck's constant h was determined by fitting this theoretical model to experimental data on black-body radiation.

Planck's Constant

Planck's constant \( h \) is fundamental in quantum mechanics and has several important applications:

• Energy of Photons: The energy of a photon is given by \( E = hf \), where \( f \) is the frequency of the photon.
• Quantum Mechanics: Used in the Schrödinger equation: \( i\hbar \frac{\partial \psi}{\partial t} = \hat{H} \psi \).
• Heisenberg Uncertainty Principle: States that \( \Delta x \Delta p \geq \frac{\hbar}{2} \).
• Atomic Spectra: Explains discrete energy levels in atoms with \( \Delta E = hf \).
• Defining the Kilogram: Used to define the kilogram in SI units.

Speed of Light

The speed of light in a vacuum is a fundamental constant of nature, denoted by the symbol c. Its value is approximately 299,792,458 meters per second (m/s). 

Explanation Questions

Question 1:

Explain the significance of Planck's constant in quantum mechanics. How does it relate to the quantization of energy?

Question 2:

Describe the components of Planck's law and their roles in determining the spectral radiance of a blackbody. Why is the concept of spectral radiance important in understanding blackbody radiation?

Question 3:

Discuss the relationship between energy and wavelength in the electromagnetic spectrum. How does Planck's law help explain this relationship?