An electron (mass m ) with an initial velocity v → = v 0 i ^ v 0 0 is moving in an electric field E → = - E 0 i ^ E 0 0 where E 0 is constant. If at t = 0 , de-Broglie wavelength is λ 0 = h m v 0

Question

An electron (mass m ) with an initial velocity v → = v 0 i ^ v 0 > 0 is moving in an electric field E → = - E 0 i ^ E 0 > 0 where E 0 is constant. If at t = 0 , de-Broglie wavelength is λ 0 = h m v 0 , then its de-Broglie wavelength after time t is given by

TestHub · Accepted Answer

Initial de-Broglie wavelength of electron,
$λ_{0} = \frac{h}{m v_{0}}$ ....(i)
Force on electron in electric field,
$F = - e E = - e [- E_{0} \hat{i}] = e E_{0} \hat{i}$
Acceleration of electron, $a = \frac{F}{m} = \frac{e E_{0} \hat{i}}{m}$
Velocity of electron after time $t$ ,
$v = v_{0} \hat{i} + (\frac{e E_{0} \hat{i}}{m}) t = (v_{0} + \frac{e E_{0}}{m} t) \hat{i} = v_{0} (1 + \frac{e E_{0}}{m v_{0}} t) \hat{i}$
de-Broglie wavelength associated with electron at time $t$ is
$λ = \frac{h}{m v}$
$\Rightarrow λ = \frac{h}{m [v_{0} (1 + \frac{e E_{0}}{m v_{0}} t)]} = \frac{λ_{0}}{[1 + \frac{e E_{0}}{m v_{0}} t]}$

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