An electron (mass m) with initial velocity $\overrightarrow v = {v_0}\widehat i + {v_0}\widehat j$ is in an electric field $\overrightarrow E = - {E_0}\widehat k$. If $\lambda _0$ is initial de-Broglie wavelength of electron, its de-Broglie wave length at time t is given by :

$\overrightarrow v = {v_0}\widehat i + {v_0}\widehat j$ <br><br>$\lambda$₀ = ${h \over {m\sqrt 2 {v_0}}}$ ....(1) <br><br>$\overrightarrow E = - {E_0}\widehat k$ <br><br>$\overrightarrow F = q\overrightarrow E$ = (-e)($- {E_0}\widehat k$) = $e{E_0}\widehat k$ <br><br>$\therefore$ $\overrightarrow a = {{\overrightarrow F } \over m}$ = ${{e{E_0}\widehat k} \over m}$ <br><br>Velocity at time t, <br><br>$\overrightarrow {{v_f}}$ = ${v_0}\widehat i + {v_0}\widehat j$ + ${{e{E_0}} \over m}t\widehat k$ <br><br> $\left| {\overrightarrow {{v_f}} } \right|$ = $\sqrt {v_0^2 + v_0^2 + {{\left( {{{e{E_0}t} \over m}} \right)}^2}}$ <br><br>= $\sqrt {2v_0^2 + {{{e^2}E_0^2} \over {{m^2}}}{t^2}}$ <br><br>$\therefore$ Wavelength at time t <br><br>$\lambda$ = ${h \over {m{v_f}}}$ = ${h \over {m\sqrt {2v_0^2 + {{{e^2}E_0^2} \over {{m^2}}}{t^2}} }}$ <br><br>= ${h \over {\sqrt 2 m{v_0}\sqrt {1 + {{{e^2}E_0^2} \over {2{m^2}v_0^2}}{t^2}} }}$ <br><br>= ${{{\lambda _0}} \over {\sqrt {1 + {{{e^2}E_0^2} \over {2{m^2}v_0^2}}{t^2}} }}$