An electron of mass m moving with initial velocity $\overrightarrow{\mathrm{v}}={\mathrm{v}}_0\widehat{\mathrm{i}}+{\mathrm{v}}_0\widehat{\mathrm{j}}$ is in an electric field $\overrightarrow{\mathrm{E}}=-{\mathrm{E}}_0\widehat{\mathrm{k}}\text{. If }{\lambda }_0$ is initial de-Broglie wavelength of electron, its de-Broglie wave length at time t is given by:

Electric force on electron due to the electric field is,

$\overrightarrow{F}=m\overrightarrow{a}=e\overrightarrow{E}$

For velocity, using $v_f=u+at$

$\begin{array}{l}\Rightarrow {\overrightarrow{v}}_f=v_0\widehat{\mathrm{i}}+v_0\widehat{\mathrm{j}}+\left(\frac{-e{E}_0}{m}t\right)\widehat{\mathrm{k}}\\ \left|\overrightarrow{v_f}\right|=\sqrt{2v_0^2+\frac{e^2{E}^2}{m^2}{t}^2}\end{array}$

Now applying equation for de-broglie wavelength,

$\begin{array}{l}\mathrm{\lambda }=\frac{\mathrm{h}}{\mathrm{m}\sqrt{2{\mathrm{v}}_0^2+\frac{{\mathrm{e}}^2{\mathrm{E}}_0^2}{{\mathrm{m}}^2}{\mathrm{t}}^2}}\\ \mathrm{\lambda }=\frac{{\mathrm{\lambda }}_0}{\sqrt{1+\frac{{\mathrm{e}}^2{\mathrm{E}}_0^2}{2{\mathrm{m}}^2{\mathbf{v}}_0^2}{\mathrm{t}}^2}}\end{array}$