Planck, Einstein, and de Broglie

Max Planck was the first to make a connection between the energy of a particle and the frequency of the associated wave. Planck developed his formula in his successful attempt to explain the emission and absorption of electromagnetic radiation from warm material surfaces. It suffices here to convey Planck's formula, which was originally intended to apply only to parcels of electromagnetic radiation, or photons:

$$E = h f = \hbar \omega ~~~~ \mbox{(Planck relation)} ,$$ (8.2)

where $E$ is the energy of the photon, $f$ is the rotational frequency of the associated light wave, and $\omega$ is its angular frequency. The constant $h = 6.63 \times 10^{-34} \mbox{ kg} \mbox{ m}^2 \mbox{ s}^{-1}$ is called Planck's constant. The related constant $\hbar = h/2 \pi = 1.06 \times 10^{-34} \mbox{ kg} \mbox{ m}^2 \mbox{ s}^{-1}$ is also referred to as Planck's constant, but to avoid confusion with the original constant, we will generally refer to it as ``h bar''.

Notice that a new physical dimension has appeared, namely mass, with the unit kilogram, abbreviated ``kg''. The physical meaning of mass is much like our intuitive understanding of the concept, i. e., as a measure of the resistance of an object to its velocity being changed. The precise scientific meaning will emerge shortly.

Einstein showed that Planck's idea could be used to explain the emission of electrons which occurs when light impinges on the surface of a metal. This emission, which is called the photoelectric effect, can only occur when electrons are supplied with a certain minimum energy $E_B$ required to break them loose from the metal. Experiment shows that this emission occurs only when the frequency of the light exceeds a certain minimum value. This value turns out to equal $\omega_{min} = E_B / \hbar$, which suggests that electrons gain energy by absorbing a single photon. If the photon energy, $\hbar \omega$, exceeds $E_B$, then electrons are emitted, otherwise they are not. It is much more difficult to explain the photoelectric effect from the classical theory of light.

Louis de Broglie proposed that Planck's energy-frequency relationship be extended to all kinds of particles. In addition he hypothesized that the momentum $\mbox{\boldmath$\Pi$}$ of the particle and the wave vector $\mbox{\bf k}$ of the corresponding wave were similarly related:

$$\mbox{\boldmath$\Pi$} = \hbar \mbox{\bf k} ~~~~ \mbox{(de Broglie relation)} .$$ (8.3)

Note that this can also be written in scalar form in terms of the wavelength as $\Pi = h/ \lambda$.

De Broglie's hypothesis was inspired by the fact that wave frequency and wavenumber are components of the same four-vector according to the theory of relativity, and are therefore closely related to each other. Thus, if the energy of a particle is related to the frequency of the corresponding wave, then there ought to be some similar quantity which is correspondingly related to the wavenumber. It turns out that the momentum is the appropriate quantity. The physical meaning of momentum will become clear as we proceed.

We will also find that the rest frequency, $\mu$, of a particle is related to its mass, $m$:

$$E_{rest} \equiv mc^2 = \hbar \mu .$$ (8.4)

The quantity $E_{rest}$ is called the rest energy of the particle.

From our perspective, energy, momentum, and rest energy are just scaled versions of frequency, wave vector, and rest frequency, with a scaling factor $\hbar$. We can therefore define a four-momentum as a scaled version of the wave four-vector:

$$\underline{\Pi} = \hbar \underline{k} .$$ (8.5)

The spacelike component of $\underline{\Pi}$ is just $\mbox{\boldmath$\Pi$}$, while the timelike part is $E/c$.

Planck, Einstein, and de Broglie had extensive backgrounds in classical mechanics, in which the concepts of energy, momentum, and mass have precise meaning. In this text we do not presuppose such a background. Perhaps the best strategy at this point is to think of these quantities as scaled versions of frequency, wavenumber, and rest frequency, where the scale factor is $\hbar$. The significance of these quantities to classical mechanics will emerge bit by bit.