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Planck constant

Fundamental physical constant linking a photon's energy to frequency; central to quantum mechanics and the modern SI system, often used in the form ħ = h/2π.

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Overview

The Planck constant, denoted h, is a fundamental physical constant that relates the energy of a photon to the frequency of its electromagnetic wave via the relation E = h·f. The constant sets the scale for quantum effects and appears in many foundational formulas of quantum mechanics. It has dimensions of action — the product of energy and time — and is commonly expressed in SI units of joule·second. The microscopic role of h distinguishes classical from quantum descriptions of physical systems.

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Key relations and properties

Two frequently used relations that involve h are the photon energy formula E = h·f and the de Broglie relation p = h/λ linking momentum and wavelength. Quantum mechanical operators and commutation relations typically involve the reduced Planck constant ħ (pronounced “h-bar”), defined by ħ = h/(2π). Many uncertainty relations, such as Heisenberg’s position–momentum inequality, are written in terms of ħ. The Planck constant carries units equivalent to kg·m2·s−1 in SI and can also be interpreted as a quantum of action or angular momentum.

Historical background

The constant is named after the German physicist Max Planck, who introduced the idea of quantized energy exchanges in 1900 while studying blackbody radiation. Planck’s hypothesis — that energy is exchanged in discrete packets proportional to frequency — led to a formula that matched experimental spectra and helped launch the development of quantum theory. The concept was later extended and exploited by other physicists, including Albert Einstein in his explanation of the photoelectric effect.

Modern definition and standards

Since the 2019 revision of the International System of Units, the numeric value of the Planck constant in SI units is defined exactly: joule·second. Fixing h provides a foundation for realizing the kilogram and other units through quantum experiments such as the Kibble balance. Instruments that determine mass, electrical standards, and precision metrology often rely on measurements traceable to the defined value of h.

Applications and examples

  • Photon energy: the energy of light at frequency f is E = h·f, a relation used across spectroscopy and photonics (photon, frequency).
  • Wave–particle duality: de Broglie’s formula p = h/λ connects momentum and wavelength, important in electron microscopy and matter-wave experiments (momentum, distance).
  • Quantum standards: electrical and mass metrology use h to relate measured quantities to SI definitions (newton, metre).

Notable facts and distinctions

Because of its central role, h appears in the definitions of natural units called Planck units; combinations of h (or ħ) with the speed of light and the gravitational constant define scales such as the Planck length and the Planck time. The reduced constant ħ is often preferred in wave mechanics and field theory because it simplifies angular factors. In everyday contexts h is extremely small compared with classical action scales, which is why quantum effects are most apparent at atomic and subatomic sizes.

For further technical information and experimental details, authoritative resources and metrology organizations provide measurements, derivations, and practical realizations of the Planck constant and its role in the modern system of units and quantum physics (energy concepts and measurement methods are treated extensively) — see specialized literature and standards bodies for in-depth treatment.

Commemorative plaque - Humboldt-Universität zu Berlin
Commemorative plaque - Humboldt-Universität zu Berlin

Definition

Planck's quantum of action h is, for every physical system that can oscillate harmonically, always the same ratio of the smallest possible energy conversion to the oscillation frequency. Larger energy conversions are only possible if they are integer multiples of this smallest energy amount. However, this quantization of energy is practically not noticeable in macroscopic systems.

Moreover, for any physical system, h is the ratio of its total energy content to the frequency of its quantum mechanical phase.

Planck's quantum of action has the dimension of energy times time, which is called the effect. However, the effect is not quantized, as one might conclude from the name.

The quantum of action gets its universal meaning from its appearance in the basic equations of quantum physics (Schrödinger equation, Heisenberg equation of motion, Dirac equation).

Some general consequences

  • Any harmonic oscillation (with frequency f, angular frequency ω \omega =2\pi f) can absorb or release energy only in discrete amounts that \Delta E = h f \equiv \hbar \omegaare integer multiples of the quantum of oscillation Δ
  • Any physical system can \hbar \equiv \tfrac{h}{2\pi}change its angular momentum J(more precisely, the projection of the angular momentum vector {\vec {J}}onto any straight line) only by integer multiples of
  • Each physical system with momentum pis associated with a matter wave of wavelength λ \lambda = \tfrac{h}{p}
  • For any physical system, energy Eand angular frequency ω \omega of its quantum mechanical phase satisfy the equation E = \hbar \omega.
  • Each two variables of a physical system which are canonically conjugate to each other (e. g., location xand momentum p of a particle, or generalized location and generalized momentum, e. g., angle of rotation and angular momentum) satisfy an uncertainty relation according to which they cannot both have well-defined values simultaneously in any state of the system. Rather, for the scatterers σ of \sigma_x,\, \sigma_pthe values of both variables: σ \sigma_x\cdot\sigma_p\ge\tfrac{\hbar}{2}.

Questions and answers

Q: What is the Planck constant?

A: The Planck constant is a fundamental physical constant that says how much the energy of a photon increases when the frequency of its electromagnetic wave increases by 1. It is written as h and expressed in joule seconds (J⋅s) or (N⋅m⋅s) or (kg⋅m2⋅s−1).

Q: Who was it named after?

A: The Planck constant was named after physicist Max Planck.

Q: What are the dimensions of physical action for this constant?

A: The dimensions of physical action for the Planck constant are energy multiplied by time, or momentum multiplied by distance.

Q: How is it expressed in SI units?

A: In SI units, the Planck constant is expressed in joule seconds (J⋅s) or (N⋅m⋅s) or (kg⋅m2⋅s−1).

Q: What measurements can be calculated using this quantity?

A: Scientists have used this quantity to calculate measurements like the Planck length and the Planck time.

Q: What equation describes magnetron W and electron L?

A: Magnetron W=Wb/2P Electron L=4C/3X = 25e/3 =(13U1d).

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