दो कण एक दूसरे से लम्बवत् दिशाओं में गतिशील है। इन कणों की डी-ब्रोग्ली तरंग लम्बाइयाँ क्रमशः \(\lambda_{1}\) तथा \(\lambda_{2}\) है। इन कणों का पूर्णतया अप्रत्यास्थ संघट्ट होता है। परिणामी कण की डी-ब्रोग्ली तरंगदैर्ध्य \(\lambda\) इस समीकरण से दी जाती है।

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

Verified Solution

Answer & Solution

Correct Answer

(D) \(\frac{1}{{{\lambda ^2}}} = \frac{1}{{\lambda _1^2}} + \frac{1}{{\lambda _2^2}}\)

Step-by-step Solution

Detailed explanation

\(\vec{P}_{1}=\frac{h}{\lambda_{1}} \hat{i}\) and \(\vec{P}_{2}=\frac{h}{\lambda_{2}} \hat{j}\) Using momentum conservation \(\vec{P}=\vec{P}_{1}+\vec{P}_{2}=\frac{h}{\lambda_{1}} \hat{i}+\frac{h}{\lambda_{2}} \hat{j}\)…

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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