An electron of mass 'm' with an initial velocity vec v = v0i (v0 > 0) enters an electric field

Question

An electron of mass 'm' with an initial velocity \(\vec {\mathrm{v}}=\mathrm{v}_{0} \hat{\mathrm{i}}\left(\mathrm{v}_{0}>0\right)\) enters an electric field \(\vec{\mathrm{E}}=-\mathrm{E}_{0} \hat{\mathrm{k}}\). If the initial de Broglie wavelength is \(\lambda_{0}\), the value after time t would be :-

(1) \(\frac{\lambda_{0}}{\sqrt{1+\frac{\mathrm{e}^{2} \mathrm{E}_{0}^{2} \mathrm{t}^{2}}{\mathrm{~m}^{2} \mathrm{v}_{0}^{2}}}}\)

(2) \(\frac{\lambda_{0}}{\sqrt{1-\frac{e^{2} E_{0}^{2} t^{2}}{m^{2} v_{0}^{2}}}}\)

(3) \(\lambda_{0}\)

(4) \(\lambda_{0} \sqrt{1+\frac{e^{2} E_{0}^{2} t^{2}}{m^{2} v_{0}^{2}}}\)

Answer 1

Correct option is : (1) \(\frac{\lambda_{0}}{\sqrt{1+\frac{\mathrm{e}^{2} \mathrm{E}_{0}^{2} \mathrm{t}^{2}}{\mathrm{~m}^{2} \mathrm{v}_{0}^{2}}}}\) 

\(\vec{\mathrm{v}}=\mathrm{v}_{0} \hat{\mathrm{i}}-\frac{\mathrm{E}_{0} \mathrm{e}}{\mathrm{m}} t \hat{\mathrm{k}}\) 

\(|\vec{\mathrm{v}}|=\sqrt{\mathrm{v}_{0}^{2}+\frac{\mathrm{E}_{0}^{2} \mathrm{e}^{2} \mathrm{t}^{2}}{\mathrm{~m}^{2}}}\) 

\( \lambda_{0}=\frac{\mathrm{h}}{\mathrm{mv}_{0}}\) 

\( \lambda^{\prime}=\frac{h}{\operatorname{mv}_{0} \sqrt{1+\frac{\mathrm{E}_{0}^{2} \mathrm{e}^{2} \mathrm{t}^{2}}{\mathrm{v}_{0}^{2} \mathrm{~m}^{2}}}}\) 

\(\lambda^{\prime}=\frac{\lambda_{0}}{\sqrt{1+\frac{\mathrm{E}_{\mathrm{0}}^{2} \mathrm{e}^{2} \mathrm{t}^{2}}{\mathrm{v}_{0}^{2} \mathrm{~m}^{2}}}}\)