Question

Relativistic corrections become necessary when the expression for the kinetic energy $\frac{1}{2}m{v}^2$, becomes comparable with $m{c}^2$, where $m$ is the mass of the particle. At what de Broglie wavelength will relativistic corrections become important for an electron?

• A. $\lambda ={10}^{-1}nm$
  
  
• B. $\lambda ={10}^{-4}nm$
  
  
  
• C. $\lambda =10nm$
  
  
  
• D. $\lambda ={10}^{-6}nm$

Answer 1

Answer: (B), (D)

$\lambda ={10}^{-6}nm$

Formula used: $\lambda =\frac{h}{mv}$

The de-Broglie wavelength at whoich relativistic corrections becomes important when phase velocity of the matter waves can be greater than the speed of the light $(3\times {10}^{8}m/s)$.
We know the de Brogile wavelength is given by,
$\lambda =\frac{h}{mv}$
$h$ is the Planck's constant,
$m$ is the mass and $v$ is the velocity.
$v=\frac{h}{m\lambda }$
use hit and trail method

(a) For, $\lambda =10nm$
$v=\frac{h}{m\lambda }=\frac{6.6\times {10}^{-34}}{9.1\times {10}^{-31}\times 10nm}$
$v\approx {10}^{5}m/s$


(b) For, $\lambda ={10}^{-1}nm$
$v=\frac{h}{m\lambda }=\frac{6.6\times {10}^{-34}Js}{9.1\times {10}^{-31}\times {10}^{-1}nm}$
$v\approx {10}^{7}m/s$

(c) For, $\lambda ={10}^{-4}nm$
$v=\frac{h}{m\lambda }=\frac{6.6\times {10}^{-34}Js}{9.1\times {10}^{-31}\times {10}^{-4}nm}$
$v\approx {10}^{10}m/s$

(d) For, $\lambda ={10}^{-6}nm$
$v=\frac{h}{m\lambda }=\frac{6.6\times {10}^{-34}Js}{9.1\times {10}^{-31}\times {10}^{-6}nm}$
$v\approx {10}^{12}m/s$


Hence, in cases (c) and (d) velocities are greater than the speed of light and hence relativistic corrections will become important for an electron.
Final Answer: (c)(d)