An electron (mass m) with initial velocity $\vec{ν} = ν_{0} \hat{i} + ν_{0} \hat{j}$ is in an electric field $\vec{E} = - E_{0} \hat{k} .$ If $λ_{0}$ is initial de-Broglie wavelength of electron, its de-Broglie wavelength at time t is given by :

Difficult

A

$\frac{λ_{0}}{\sqrt{1 + \frac{e^{2} E_{0}^{2} t^{2}}{m^{2} ν_{0}^{2}}}}$
B

$\frac{λ_{0} \sqrt{2}}{\sqrt{1 + \frac{e^{2} {E_{0}}^{2} t^{2}}{m^{2} ν_{0}^{2}}}}$
C

$\frac{λ_{0}}{\sqrt{2 + \frac{e^{2} {E_{0}}^{2} t^{2}}{m^{2} ν_{0}^{2}}}}$
D

$\frac{λ_{0}}{\sqrt{1 + \frac{e^{2} {E_{0}}^{2} t^{2}}{2 m^{2} ν_{0}^{2}}}}$

Solution

   $\begin{array}{l} I n i t i a l l y m o m e n t u m = p = m (\sqrt{2} V_{0}) = \frac{h}{λ_{0}} \\ \Rightarrow \frac{h}{m} = \sqrt{2} V_{0} λ_{0} ......... (1) \end{array}$
$\begin{array}{l} A c c e l e r a t i o n o f e l e c t r o n = \vec{a} = \frac{e E_{0}}{m} \hat{k} \\ U \sin g e q u a t i o n o f k i n e m a t i c s \\ \vec{v} = \vec{u} + \vec{a} t \\ \Rightarrow \vec{v} = v_{0} \hat{i} + v_{0} \hat{j} + \frac{e E_{0} t}{m} \hat{k} \\ M a g n i t u d e o f v e l o c i t y = |\vec{v}| = \sqrt{{v_{0}}^{2} + {v_{0}}^{2} + {(\frac{e E_{0} t}{m})}^{2}} \end{array}$

   $S o, w a v e l e n g t h = λ = \frac{h}{m v} = \frac{h}{m \sqrt{2 v_{0}^{2} + \frac{e^{2} E_{0}^{2} t^{2}}{m^{2}}}}$
$\Rightarrow λ = \frac{\sqrt{2} v_{0} λ_{0}}{\sqrt{2} v_{0} \times \sqrt{1 + \frac{e^{2} E_{0}^{2}}{2 m^{2} v_{0}^{2}} t^{2}}} = \frac{λ_{0}}{\sqrt{1 + \frac{e^{2} E_{0}^{2}}{2 m^{2} v_{0}^{2}} t^{2}}}$

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