Electron's Wave Nature

Special relativity #

An object has an intrinsic mass/rest mass m.

Such an object is said to posses a rest-mass energy E=mc²

If an object has momentum p, then its energy is given by

$$ E=\sqrt{p^2c^2+m^2c^4}\approx mc^2+\frac{p^2}{2m}\\ for\\ p«mc $$

Momentum is related to the velocity by

$$
\begin{align} p=\gamma mv\\ \gamma=\frac{1}{\sqrt{1-v^2/c^2}} \end{align} $$

Substituting the p, in the expression for E we get: $$ \begin{align} E=\gamma mc^2\\ \frac{E}{pc}=\frac{c}{v} \end{align} $$

Photons #

Einstein’s view of light is a stream of particles called ‘photons’ which travel at the speed of light.

Therefore, $E=pc$ and the rest mass=0

Using planck’s relation $E=hv=hc/\lambda$: $p=h/\lambda$

These relations are applicable in describing the 6. Compton Effect

Radiation: Wave or particle #

• Interference and diffraction could be explained only by assuming light is a wave
  • Properties that we talk about: Frequency, Amplitude, Wavelength, Speed
• Photoelectric effect and the Compton effect
  • Properties that we talk about: Energy, momentum

De-Broglie hypothesis #

De-broglie suggested that if photons have wave light property, all matter should do too.

We know that light of wavelength λ has momentum $p=h/\lambda$

De-broglie suggested that matter of moment p have a wavelength $\lambda = h/p$

Consequences #

Bohr’s quantization condition can be explained with the De-broglie hypothesis.

In terms of De-Broglie wavelength, it can be expressed as:

$$ \begin{align} (h/\lambda)r=n\hbar\\ n\lambda=2\pi r \end{align} $$ i.e. Orbits are stable if the orbit length contains integer multiples of De-Broglie wavelength. Phases line up, and there is sustained oscillation.