Two electrons are moving with non-relativistic speeds perpendicular to each other. If corresponding de Broglie wavelengths are \({\lambda _1}\) and \({\lambda _2}\), their de Broglie wavelength in the frame of reference attached to their centre of mass is

A \({\lambda _{CM}} = {\lambda _1} = {\lambda _2}\)
B \(\frac{1}{{{\lambda _1}}} = \frac{1}{{{\lambda _1}}} + \frac{1}{{{\lambda _2}}}\)
C \({\lambda _{CM}} = \frac{{2{\lambda _1}{\lambda _2}}}{{\sqrt {\lambda _1^2 + \lambda _2^2} }}\)
D \({\lambda _{CM}} = \left( {\frac{{{\lambda _1} + {\lambda _2}}}{2}} \right)\)

Verified Solution

Answer & Solution

Correct Answer

(C) \({\lambda _{CM}} = \frac{{2{\lambda _1}{\lambda _2}}}{{\sqrt {\lambda _1^2 + \lambda _2^2} }}\)

Step-by-step Solution

Detailed explanation

Momentum \((p)\) of each electron \(\frac{\mathrm{h}}{\lambda_{1}} \hat{\mathrm{i}}\) and \(\frac{\mathrm{h}}{\lambda_{2}} \hat{\mathrm{j}}\) Velocity of centre of mass \({{\text{V}}_{{\text{cm}}}} = \frac{{\text{h}}}{{2{\text{m}}{\lambda _1}}}\widehat {\text{i}} + \)…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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