De Broglie Hypothesis

Today we know that every particle exhibits both matter and wave nature. This is called wave-particle duality. The concept that matter behaves like wave is called the de Broglie hypothesis, named after Louis de Broglie, who proposed it in 1924.

De Broglie gave the following equation which can be used to calculate de Broglie wavelength, $\lambda$, of any massed particle whose momentum is known:

$$\lambda = \frac{h}{p},$$

where $h$ is the Plank's constant and $p$ is the momentum of the particle whose wavelength we need to find.

With some modifications the following equation can also be written for velocity $(v)$ or kinetic energy $(K)$ of the particle (of mass $m$):

$$\lambda = \frac{h}{mv} = \frac{h}{\sqrt{2mK}}.$$

Notice that for heavy particles, the de Broglie wavelength is very small, in fact negligible. Hence, we can conclude that though heavy particles do exhibit wave nature, it can be neglected as it's insignificant in all practical terms of use.

Calculate the de Broglie wavelength of a golf ball whose mass is 40 grams and whose velocity is 6 m/s.

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We have

$$\lambda = \frac{h}{mv} = \frac{6.63 \times 10^{-34}}{40 \times 10^{-3} \times 6} \text{ m}=2.76 \times 10^{-33} \text{ m}.\ _\square$$

One of the main limitations of Bohr's atomic theory was that no justification was given for the principle of quantization of angular momentum. It does not explain the assumption that why an electron can rotate only in those orbits in which the angular momentum of the electron, $mvr,$ is a whole number multiple of $\frac{h}{2\pi}$.

De Broglie successfully provided the explanation to Bohr's assumption by his hypothesis.