Planck Constant

The Planck constant, symbol $h$, is a fundamental constant of quantum mechanics that relates the energy of a photon to its frequency and sets the characteristic scale of quantum effects. Named after Max Planck, it has the exact value $6.62607015\times10^{-34}$ joule-seconds in the International System of Units (SI). Its dimensions are those of action, or energy multiplied by time, rather than energy alone. (nist.gov)

Value, dimensions, and notation

The defining SI value is

Because one joule equals one kilogram-metre squared per second squared, this can also be written as $6.62607015\times10^{-34}\ \mathrm{kg\,m^2\,s^{-1}}$. The value is exact by definition, not an experimental estimate with an uncertainty. Experimental measurements used to realize units nevertheless retain their own uncertainties. (nist.gov)

The closely related reduced Planck constant, pronounced “h-bar,” is

It is also exact, although its decimal representation does not terminate. The distinction reflects two ways of measuring frequency: ordinary frequency $\nu$ counts cycles per second, whereas angular frequency $\omega=2\pi\nu$ measures phase change in radians per second. Accordingly, photon energy may be written either $E=h\nu$ or $E=\hbar\omega$. (physics.nist.gov)

Historical origin

Planck introduced the constant in 1900 while investigating blackbody radiation, the electromagnetic radiation associated with an ideal absorbing body in thermal equilibrium. The challenge was to explain the observed distribution of radiation across frequencies. His successful treatment introduced energy elements proportional to frequency, $\varepsilon=h\nu$, into the statistical description of material resonators exchanging energy with radiation. This was a decisive departure from a wholly continuous treatment of energy exchange. (nobelprize.org)

Planck’s original work should be distinguished from the later photon interpretation of light. In 1905, Albert Einstein extended the quantum idea to light itself and used it to explain the photoelectric effect. In the elementary description, the maximum kinetic energy of an emitted electron is

where $\Phi$ is the material’s work function. Planck received the 1918 Nobel Prize in Physics for his discovery of energy quanta; Einstein’s light-quantum hypothesis was a distinct development. (nobelprize.org)

Energy, frequency, and matter waves

For electromagnetic radiation of frequency $\nu$, each photon carries energy $h\nu$. Higher-frequency photons therefore carry more energy individually. Increasing the intensity of monochromatic light can instead increase the number of photons without changing their individual energies. For radiation of vacuum wavelength $\lambda$,

where $c$ is the speed of light in vacuum. These relations underpin the conversion between measured radiation frequencies and energy differences. (ocw.mit.edu)

The constant also appears in the de Broglie relation,

which associates a wavelength with momentum $p$. Matter-wave behavior is therefore not restricted to photons: electrons and other material particles exhibit wave properties. In wave notation, momentum can equivalently be expressed as $p=\hbar k$, with wave number $k=2\pi/\lambda$. The same constant thus connects energy with temporal oscillation and momentum with spatial oscillation. (ocw.mit.edu)

These equations do not mean that every energy is an integer multiple of one universal energy unit. The spacing depends on the physical system. For example, the quantum harmonic oscillator has levels $E_n=(n+\tfrac12)\hbar\omega$, whereas a free particle can have a continuous energy spectrum. Energy quantization is therefore a property of particular states and constraints, not simply a consequence of writing down $h$. (ocw.mit.edu)

Role in quantum dynamics

The reduced constant enters the time-dependent Schrödinger equation,

where $\psi$ is the wave function and $\hat H$ the Hamiltonian operator. It sets the relationship between energy and the rate at which quantum phase evolves. For a state of definite energy $E$, time evolution contains the phase factor $\exp(-iEt/\hbar)$. (ocw.mit.edu)

It also fixes the scale of the uncertainty principle. Position and momentum along the same axis obey

Here the uncertainties are statistical spreads in the outcomes for a specified quantum state, rather than merely imperfections in measuring instruments. The inequality follows from the noncommuting position and momentum operators. Classical descriptions become useful in appropriate regimes where relevant action scales are large compared with $\hbar$; this does not require the physical constant itself to change. (ocw.mit.edu)

Measurement and the kilogram

Before the SI redefinition took effect on May 20, 2019, the numerical value of $h$ was determined experimentally in units tied to an artifact-based kilogram. The revised SI reversed that relationship: $h$ now has a fixed numerical value, and the kilogram is defined through it together with the independently defined metre and second. The selected value preserved continuity with the previous mass scale. (nist.gov)

A Kibble balance is one method of realizing this definition. It compares mechanical and electrical quantities in separate operating modes. Electrical measurements use quantum standards associated with the Josephson effect and the quantum Hall effect, connecting voltage and resistance to $h$ and elementary charge. Before redefinition, a known mass enabled a determination of $h$; afterward, fixed $h$ enables a determination of mass. Physical reference weights remain useful for routine calibration, but their traceability ultimately rests on constants rather than a uniquely designated metal artifact. (nist.gov)