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If the momentum of an electron is changed by p

Question :- If the momentum of an electron is changed by P, then the de-Broglie wavelength associated with it changes by 0.5 %. The initial momentum of an electron will be :

(A) 400P

(B) P/200

(D) 200P

Solution :-

Let initial de – Broglie wavelength of electron is given by

$\displaystyle {{\lambda }_{i}}=\frac{h}{{{p}_{i}}}$ …..(1)

Let momentum is increased by P, then new de – Broglie wavelength(decreased) of electron,

$\displaystyle {{\lambda }_{f}}=\frac{h}{{{p}_{i}}+P}$ …..(2)

Equation (1) – (2) gives

$\displaystyle {{\lambda }_{i}}-{{\lambda }_{f}}=\frac{h}{{{p}_{i}}}-\frac{h}{{{p}_{i}}+P}=\frac{hP}{({{p}_{i}}+P){{p}_{i}}}$ …..(3)

Dividing (3) by (1) we get,

$\displaystyle \frac{{{\lambda }_{i}}-{{\lambda }_{f}}}{{{\lambda }_{i}}}=\frac{hP}{({{p}_{i}}+P){{p}_{i}}}\times \frac{{{p}_{i}}}{h}=\frac{P}{{{p}_{i}}+P}$