16.5: Wave-Particle Duality

Einstein had shown that the momentum of a photon is

\[ p = \dfrac{h}{\lambda}. \label{17.5.1} \]

This can be easily shown as follows. Assuming \(E = h \nu\) for a photon and \(\lambda \nu = c\) for an electromagnetic wave, we obtain

\[ E = \dfrac{h c}{\lambda} \label{17.5.2} \]

Now we use Einstein’s relativity result, \(E = m c^2\), and the definition of mementum \(p=mc\), to find: \[ \lambda = \dfrac{h}{p}, \label{17.5.3} \]

which is equivalent to Equation \ref{17.5.1}. Note that \(m\) refers to the relativistic mass, not the rest mass, since the rest mass of a photon is zero. Since light can behave both as a wave (it can be diffracted, and it has a wavelength), and as a particle (it contains packets of energy \(h \nu\)), de Broglie reasoned in 1924 that matter also can exhibit this wave-particle duality. He further reasoned that matter would obey the same Equation \ref{17.5.3} as light. In 1927, Davisson and Germer observed diffraction patterns by bombarding metals with electrons, confirming de Broglie’s proposition.\(^1\)

Rewriting the previous equations in terms of the wave vector, \(k=\dfrac{2\pi}{\lambda}\), and the angular frequency, \(\omega=2\pi\nu\), we obtain the following two equations

\[ \begin{aligned} p &= \hbar k \\ E &= \hbar \omega, \end{aligned} \label{17.5.4} \]

which are known as de Broglie’s equations. We will use those equation to develop wave mechanics in the next chapters.

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1. The previous 3 sections were adapted in part from Prof. C. David Sherrill’s A Brief Review of Elementary Quantum Chemistry Notes available here.