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Particle in a Box

At the expense of any pretensions of historical accuracy, I am going to see how many interesting conclusions we can draw from one simple hypothesis posed by Louis Victor Pierre Raymond duc de Broglie in his 26-page doctoral thesis in 1924. It had been shown two decades earlier that light, which is certainly a wave, comes quantized in clumps like particles (called photons) with the energy of each photon equal to Planck's constant times its frequency: $E = h \nu$, where $h = 6.626 \times 10^{-34}$ J-s is Planck's constant. (It was the explanation of this phenomenon in 1905 that won Albert Einstein the Nobel prize. Relativity was just gravy.) It had already been shown earlier still (in the late Nineteenth Century) that an electromagnetic wave carries both energy E and momentum p, in the ratio E = pc where c is the speed of light. This ratio holds also for quantized photons, which therefore have momentum $p = h \nu / c$. But for any wave, $c = \lambda \nu$, so

$$\lambda \; = \; {h \over p} .$$ (24.1)

Louis' hypothesis was amazingly simple: he reasoned that if waves are like particles, then maybe particles are like waves. In particular, an electron is in some mysterious sense a wave with its wavelength $\lambda$ given in terms of its momentum p by Eq. (1). This simple suggestion was the basis for the WAVE/PARTICLE DUALITY that has perplexed generations of Physics students ever since (and formed the basis for all QUANTUM MECHANICS). But suppose we just take it at face value and examine a few "obvious" consequences.