An electron of mass $$m$$ with an initial velocity  $$\vec v=v_0\hat{i}\;(v_0>0)$$ enters an electric field $$\vec E=-E_0\hat{k}.$$ If the initial de Broglie wavelength is  $$\lambda_0,$$ the value after time $$t$$ would be:

Electron with $$\vec{v}=v_0\hat{i}$$ enters field $$\vec{E}=-E_0\hat{k}$$. Find de Broglie wavelength after time $$t$$.

The force on the electron is $$\vec{F}=-e\vec{E}=eE_0\hat{k}$$.

Integrating acceleration gives $$v_z=\frac{eE_0}{m}t$$ while $$v_x=v_0$$ remains constant.

Hence the speed is $$v=\sqrt{v_0^2+\frac{e^2E_0^2t^2}{m^2}}$$.

The de Broglie wavelength is $$\lambda=\frac{h}{mv}=\frac{h}{m\sqrt{v_0^2+e^2E_0^2t^2/m^2}}$$.

This can be written as $$\lambda=\frac{h}{mv_0\sqrt{1+e^2E_0^2t^2/(m^2v_0^2)}}$$.

Therefore $$\lambda=\frac{\lambda_0}{\sqrt{1+\frac{e^2E_0^2t^2}{m^2v_0^2}}}$$.