At an atomic — and subatomic level — we find connections hidden by larger scales: you can calculate energy from the wavelength for objects that are particles and waves at the same time, without a mass but somehow showing the effect associated to it.

In this short article we will talk about the Planck-Einstein relations. Learn:

  • Why is there a relationship between photon energy and wavelength;
  • How to calculate the wavelength from the energy: equation and examples;
  • What came next in the weird world of quantum physics.

Phtons: a quantum leap in our understanding of nature

More waves than rays: light is best described by an undulatory framework. Most of the properties witnessed by physicists — interference, diffraction, etc. — fit an undulatory light. However, when Planck tried to describe the behavior of a black body, he encountered some discrepancies that only a quantized nature of light waves could explain. It was the dawn of photons and, incidentally, of quantum physics.

The argument for the existence, and our proper understanding, of photons received a boost thanks to the work of Einstein. In his Nobel-winning description of the photoelectric effect, a corpuscular nature of light allowed the physicist to describe the previously unexplained phenomenon.

At that point, physics pretty much confirmed the existence of a relationship between a photon's energy and wavelength: the equation first developed for a generic "energy" was then adopted for light too.

How to calculate wavelength from energy in photons: the Planck-Einstein relation and the

Imagine a really distraught Planck working on his notes. The observed results of experiments on black bodies were nightmarish, to say the least. The expected ultraviolet catastrophe, an arbitrary output of energy for wavelengths in the ultraviolet region of the spectrum, was nowhere to be found. Wien's displacement law empirically described the behavior, but this was not enough. A desperate Planck resorted to what he thought was an unjustified mathematical expedient: reducing energy to a set of discrete values instead of a continuum. The quantization of energy came to light this way.

Planck hypothesis, applied to the black body radiation, translated into a relationship between energy and frequency:

E=hνE = h\cdot \nu

Where,

  • EE — The energy;
  • hh — The Planck's constant; and
  • ν\nu — The frequency.

To translate this equation into a formula for wavelength and energy, we need to know what are the objects involved. Since we are dealing with photons, we can use the relationship between wavelength and frequency:

🙋 The Planck's constant has value h=6.62607015×1034 (kgm2)/sh = 6.62607015 \times 10^{-34}\ (\text{kg}\cdot\text{m}^2)/\text{s}.

c=νλc = \nu \cdot \lambda

Here cc is the speed of light (299792458 m/s299 792 458\ \text{m}/\text{s}) and λ\lambda the wavelength.

With a simple substitution, we find the formula to calculate the energy from the wavelength and vice versa:

E=hν=hcλE = h\cdot \nu = \frac{h\cdot c}{\lambda}

In the following section, we will analyze the equation for energy and wavelength, how to calculate some of the results, deal with the measurement units, and more.

Did you know that massive objects also have a wavelength and exhibit a double undulatory-corpuscular behavior? Find more about the weirdness of quantum mechanics at our de Broglie wavelength calculator!

The electromagnetic spectrum: some results of the energy to wavelength formula

The first thing that comes to mind when looking at the wavelength to energy equation is that these quantities are inversely proportional: a shorter wavelength corresponds to higher energy, and vice versa. This fact comes with its consequences: shorter wavelengths imply longer penetration lengths, for example, which allow the harmful gamma photons to damage our internal organs. At the same time, the more energetic albeit less penetrant alpha radiation gets stopped by the skin.

We also need to raise the issue with the orders of magnitude appearing in the formula for energy and wavelength. As we see in many models for the infinitely small and the infinitely big (quantum physics and astrophysics suffer the most), the quantities appearing here vary enormously: from a sub-nanometric scale of gamma rays to the kilometers of the very low frequency (VLF) radio waves. The speed of light, with its ridiculously high value, also contributes to the confusion.

For this reason, physicists often resort to alternative measurement units. Use GHz\text{GHz} and THz\text{THz} for your frequency if you are a physicist, and change your energy units to electronvolt, Converting from eV to a wavelength in nanometers is far more comfortable than using joules and meters.

A quick example: what is the energy of a photon in the green portion of the visible spectrum? Let's choose two indicative values of wavelength:

  • λ1=500 nm\lambda_1 = 500\ \text{nm}; and
  • λ2=550 nm\lambda_2 = 550\ \text{nm}.

Apply the Planck-Einstein relation to find the radiation's eV from the wavelengths:

  • E1=2.48 eVE_1 = 2.48\ \text{eV}; and
  • E2=2.25 eVE_2 = 2.25\ \text{eV}.

A far cry from the megaelectronvolt energy of gamma rays — luckily!

And what is the color magenta's wavelength?

We are joking: magenta doesn't exist.

Davide Borchia
Energy
μeV
Wavelength
mm
Frequency
GHz
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