Product of phase velocity and group velocity is equal to square of speed of light

Prove that $\rightarrow$
The Product of phase velocity and group velocity is equal to the square of the speed of light i.e. $\left( V_{p}.V_{g}=c^{2} \right)$

Proof → We know that

$V_{p}=\nu \lambda \qquad(1)$

And de Broglie wavelength-

$\lambda =\frac{h }{mv}\qquad(2)$

According to Einstein's mass-energy relation-

$E=mc^{2}$

$h\nu=mc^{2}$

$\nu=\frac{mc^{2}}{h}\qquad(3)$

Now put the value of $\lambda$ and $\nu$ in equation$(1)$

$V_{p}= [\frac{mc^{2}}{h}] [\frac{h}{mv}]$

$V_{p}=\frac{C^{2}}{v}$

Since group velocity is equal to particle velocity i.e. $V_{g}=v$. So above equation can be written as

$V_{p}=\frac{C^{2}}{V_{g}}$

$V_{p}.V_{g}=C^{2}$

Note →

➢ $V_{g}=V_{p}$ for a non-dispersive medium ( in a non-dispersive medium all the waves travel with phase velocity).

➢ $V_{g}< V_{p}$ for normal dispersive medium

➢ $V_{g}> V_{p}$ for anomalous dispersive media.

Dispersive medium → The medium in which the phase velocity varies with wavelength or frequency is called a dispersive medium. In such a medium, waves of different wavelengths travel with different phase velocities.

Non-dispersive medium → The medium in which the phase velocity does not vary with wavelength or frequency is called a Non-dispersive medium.

Dispersive waves → Those waves in the medium for which phase velocity varies with wavelength or frequency are called dispersive waves.

Non-dispersive waves → Those waves in which phase velocity does not vary with wavelength are called non-dispersive waves. So phase velocity independent of wavelength.