Question

What is the speed of an electron whose Debroglie wavelength is equal to its compton wavelength?

Answer 1

Solution

Let rest mass of electron$=m_0$

Let Speed of election =v

Then moving mass of electron can be expressed as

$m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$ . ... eq.1

Where c is speed of light.

De-broglie wavelength is given as

$\lambda=\frac{h}{p}=\frac{h}{mv}$ ............. eq. 2

Compton wavelength for electron is

$\lambda_c=\frac{h}{m_0c}$ ............ eq. 3

Now according to the question de-broglie wavelength is equal to the compton wavelength for electron

therefore using equations 1,2,3 we got as below

$\frac{h}{mv}=\frac{h}{m_0c}$

$\frac{h\sqrt{1-\frac{v^2}{c^2}}}{m_0v}=\frac{h}{m_0c}$

$\sqrt{1-\frac{v^2}{c^2}}=\frac{v}{c}$

taking square at both sides we get as

${1-\frac{v^2}{c^2}}=\frac{v^2}{c^2}$

$\frac{2v^2}{c^2}=1$

$​ \fcolorbox{red}{yellow}{$v=\frac{c}{\sqrt{2}}$}$

Therefore speed of electron is $\frac{c}{\sqrt{2}}$ .