# Physical significance of frequency and wavelength in Planck's equation?

Max Planck theorized that electromagnetic radiation is transferred in chunks known as quanta, whose energies are proportional to $h\approx 6.63\cdot 10^{-34}$ J s, known as Planck's constant. He summarized the result in form

$$E=h\nu\hspace{0.5cm}\text{or}\hspace{0.5cm}E= \frac{hc}{\lambda},$$

where $c$ is the speed of light, $\nu$ is the frequency of the light, and $\lambda$ is its wavelength.

The radical implication of Planck’s equation was that light came in finite packets multiple of $h\nu$ but what do the frequency and wavelength signify in this equation? How can particles be assumed to have wavelength and frequency? Typically, these two quantities are associated with waves. Moreover, Planck didn't know about the dual nature of light!

• Particles do have a wavelength and frequency, as show by De Broglie Wavelength: $\lambda = \frac{h}{p}$ where $\lambda$ is the wavelength of the matter wave, h is Planck's constant, and p is the matter's momentum. physics.bu.edu/ulab/modern/Electron_Diffraction.pdf – iammax Jun 27 '17 at 22:19
• It is worth noting that Planck wasn't actually motivated by thinking of light as discrete chunks. It was originally introduced as just a mathematical trick to explain the observed spectrum of blackbody radiation (in much the same way that Lorentz transformations had been invented as just a mathematical trick to explain the Michelson Morley experiment, before Einstein came up with relativity). – Bob Knighton Jun 27 '17 at 22:53

The fundamental thing to realize is that a "particle" can have a wavelength and momentum classically (assuming an appropriate definition of particle). If we define a "particle" of light as being a localized wavepacket (see the figure below) whose electric field is given by

$$\textbf{E}=E_0f(x-ct)e^{ikx-i\omega t}\hat{\textbf{z}},$$

where $f$ is some function that is localized around zero, then the wavelength of this particle is simply $\lambda=2\pi/k$ and its frequency is simply $\nu=\omega/2\pi$. The wavelength and frequency are read off in the phase of the electromagnetic field. A highly localized wavepacket is about as "particle" as a wave can get, so this is a pretty good definition of what it classically means to have a localized "photon." This is kind of what the Planck relation is hinting at.

This is an intuitive way to think of what a photon might "look like" as a particle (in the localized sense). The ideas extend in a similar way when we do quantum mechanics.

Also, I want to destroy a misconception. The wave-particle-duality is a really misunderstood and outdated concept. Quantum particles are not "both particles and waves until observed." Quantum particles are waves. Sometimes the waves are spread out and sometimes they are highly localized and sometimes they do all kinds of weird stuff. But they are, first and foremost (at least in the sense of non-relativistic quantum mechanics) waves of probability amplitudes.

I hope this helped!

• Should one say that a quantum particle is a particle when a detector observed it? – Yaman Sanghavi Mar 26 '18 at 18:25
• My point is that one shouldn't rely on strict definitions like "particle" and "wave" when describing things that very intrinsically have properties of both. To be more specific to your question, if my detector is measuring momentum, then any wavefunction collapses to a very non-localized wave that has a very precise momentum (wavelength), in which case one might be obliged to call it a "wave". However, if the detector is measuring position, the oposite occurs, and one might be obliged to call it a "particle". The terms themselves are simply are too vague to fully handle quantum states. – Bob Knighton Mar 26 '18 at 20:57

First, let's establish that the implication of Planck's Law was not that light was no longer considered a wave. What it established was that light was an object that (like all other quantum objects) can be represented as either a wave or a particle. Each interpretation reveals different facets of light's behavior, and in certain regimes, one interpretation or the other may be more appropriate. For example, at low frequencies/photon energies, like in the radio band, the wave interpretation is far more useful, whereas at short frequencies/photon energies, the particle interpretation is more appropriate. In addition, situations involving high photon flux are often more amenable to the wave interpretation, while situations with low flux reveal the quantized, particle-like nature of light.

If you're thinking in the particle frame, the wavelength/frequency of the photon can be thought of as a quantity that partially tells you how accurate it is to consider the photon as a particle. The higher the number, the better the particle view is.

It also didn't matter that Planck didn't consider the dual nature of light, because in order to derive the above laws, he had to quantize electromagnetic radiation anyway. This naturally leads to the existence of individual quanta, or photons.

The "chunks" (quanta) of energy show up best in very tiny, simple systems like a pair of nitrogen atoms in the atmosphere (N2). They can only rotate at discrete energy levels, not just any random amount of rotational energy. And they transfer that energy in discrete amounts as well, when they interact with other particles or molecules.