Planck's Constant: Definition & Equation (w/ Chart of Useful Combinations)

Planck's constant is one of the most fundamental constants describing the universe. It defines the quantization of electromagnetic radiation (the energy of a photon) and underpins much of quantum theory.

Who Was Max Planck?

Max Planck was a German physicist who lived from 1858-1947. In addition to many other contributions, his notable discovery of energy quanta earned him the Nobel Prize in physics in 1918.

When Planck attended the University of Munich, a professor advised him against going into physics since supposedly everything was already discovered. Planck didn’t heed this suggestion, and in the end turned physics on its head by originating quantum physics, the details of which physicists are still trying to understand today.

Value of Planck's Constant

Planck's constant h (also called the Planck constant) is one of several universal constants that define the universe. It is the quantum of electromagnetic action and relates photon frequency to energy.

The value of h is exact. Per NIST_, h_ = 6.62607015 × 10-34 J Hz-1. The SI unit of the Planck constant is the joule-second (Js). A related constant ℏ ("h-bar") is defined as h/(2π) and is used more often in some applications.

How Was Planck’s Constant Discovered?

The discovery of this constant came about as Max Planck was trying to resolve a problem with black-body radiation. A black body is an idealized absorber and emitter of radiation. When in thermal equilibrium, a black-body continuously emits radiation. This radiation is emitted in a spectrum that is indicative of the body’s temperature. That is to say, if you plot the radiation intensity vs. wavelength, the graph will peak at a wavelength associated with the object’s temperature.

Black-body radiation curves peak at longer wavelengths for cooler objects and shorter wavelengths for hotter objects. Before Planck came into the picture, there was no overall explanation for the shape of the black-body radiation curve. Predictions for the shape of the curve at lower frequencies matched, but diverged significantly at higher frequencies. In fact, the so called "ultraviolet catastrophe" described a feature of the classical prediction where all matter should instantaneously radiate all of its energy away until it was near absolute zero.

Planck solved this problem by assuming the oscillators in the black body could only change their energy in discrete increments that were proportional to the frequency of the associated electromagnetic wave. This is where the notion of quantization comes in. Essentially, the allowed energy values of the oscillators had to be quantized. Once that assumption is made, then the formula for the correct spectral distribution could be derived.

While initially it was thought that Planck’s quanta were a simple trick to make the math work, later it became clear that energy did indeed behave in this way, and the field of quantum mechanics was born.

Planck Units

Other related physical constants, such as the speed of light c, the gravitational constant G, the Coulomb constant ke and Boltzmann’s constant kB can be combined to form Planck units. Planck units are a set of units used in particle physics where the values of certain fundamental constants become 1. Not surprisingly, this choice is convenient when performing calculations.

By setting c = G = ℏ = ke = kB = 1, the Planck units can be derived. The set of base Planck units are listed in the following table.

TABLE

From these base units, all other units can be derived.

Planck's Constant and Quantized Energy

In an atom, the electrons are only allowed to exist in very specific quantized energy states. If an electron wants to be in a lower energy state, it can do so by emitting a discrete packet of electromagnetic radiation to carry off the energy. Conversely, in order to jump into an energy state, that same electron must absorb a very specific discrete packet of energy.

The energy associated with an electromagnetic wave depends on the wave’s frequency. As such, atoms can absorb and emit only very specific frequencies of electromagnetic radiation consistent with their associated quantized energy levels. These energy packets are called photons and they can only be emitted with values of energy E that are a multiples of Planck’s constant, giving rise to the relationship:

E=h\nu

Where ν (the Greek letter nu) is the photon’s frequency

Planck’s Constant and Matter Waves

In 1924 it was shown that electrons can act like waves in the same way photons do – that is, by exhibiting particle-wave duality. By combining the classical equation for momentum with the quantum mechanical momentum, Louis de Broglie determined that the wavelength for matter waves is given by the formula:

\lambda = \frac{h}{p}

where λ is wavelength and p is momentum.

Soon scientists were using wave functions to describe what electrons or other similar particles were doing with the help of the Schrodinger equation – a partial differential equation that can be used to determine the evolution of the wave function. In its most basic form, the Schrodinger equation can be written as follows:

i\hbar \frac{\partial}{\partial t}\Psi(r,t)=\Big[\frac{-\hbar^2}{2m}\nabla^2+V(r,t)\Big]\Psi(r,t)

Where Ψ is the wave function, r is the position, t is time and V is the potential function.

Quantum Mechanics and the Photoelectric Effect

When light, or electromagnetic radiation, hits a material such as a metal surface, that material sometimes emits electrons, called photoelectrons. This is because the atoms in the material are absorbing the radiation as energy. Electrons in atoms absorb radiation by jumping to higher energy levels. If the energy absorbed is high enough, they leave their home atom entirely.

What was most special about the photoelectric effect, however, is that it did not follow classical predictions. The way in which the electrons were emitted, the number that were emitted and how this changed with intensity of light all left scientists scratching their heads initially.

The only way to explain this phenomenon was to invoke quantum mechanics. Think of a beam of light not as a wave, but as a collection of discrete wave packets called photons. The photons all have distinct energy values that correspond to the frequency and wavelength of the light, as explained by wave-particle duality.

In addition, consider that the electrons are only able to jump between discrete energy states. They can have only specific energy values, and never any values in between. Now the observed phenomena can be explained. Electrons are released only when they absorb very specific sufficient energy values. None are released if the frequency of the incident light is too low regardless of intensity because none of the energy packets are individually big enough.

Once the threshold frequency is exceeded, increasing intensity only increases the number of electrons released and not the energy of the electrons themselves because each emitted electron absorbs one discrete photon. There is also no time delay even at low intensity as long as the frequency is high enough because as soon as an electron gets the right energy packet, it is released. Low intensity only results in fewer electrons.

Planck’s Constant and Heisenberg’s Uncertainty Principle

In quantum mechanics, the uncertainty principle might refer to any number of inequalities that give a fundamental limit to the precision with which two quantities can simultaneously be known with precision.

For example, a particle’s position and momentum obey the inequality:

\sigma_x\sigma_p \geq\frac{\hbar}{2}

Where σx and σp are the standard deviation of position and momentum respectively. Note that the smaller one of the standard deviations becomes, the larger the other one must become in order to compensate. As a result, the more precisely you know one value, the less precisely you know the other.

Additional uncertainty relationships include uncertainty in orthogonal components of angular momentum, uncertainty in time and frequency in signal processing, uncertainty in energy and time, and so on.

References

About the Author

Gayle Towell is a freelance writer and editor living in Oregon. She earned masters degrees in both mathematics and physics from the University of Oregon after completing a double major at Smith College, and has spent over a decade teaching these subjects to college students. Also a prolific writer of fiction, and founder of Microfiction Monday Magazine, you can learn more about Gayle at gtowell.com.