\(U\left( r \right) = \frac{1}{2}k{r^2}\) ના કેન્દ્રીય સ્થિતિમાન ક્ષેત્રમાં એક વર્તુળાકાર કક્ષામાં \(m\) દ્રવ્યમાન વાળો એક કણ ગતિ કરે છે.જો ક્વોન્ટાઇઝેશન શરતો લગાડવામાં આવે તો શક્ય કક્ષકો અને તેના ઊર્જાસ્તરો એ ક્વોન્ટમ ક્રમ (સંખ્યા) \( n\) સાથે ______ થી ચલે છે.

A \({r_n} \propto \sqrt n ,{E_n} \propto n\)
B \({r_n} \propto \sqrt n ,{E_n} \propto \frac{1}{n}\)
C \({r_n} \propto n,{E_n} \propto n\)
D \({r_n} \propto {n^2},{E_n} \propto \frac{1}{{{n^2}}}\)

Verified Solution

Answer & Solution

Correct Answer

(A) \({r_n} \propto \sqrt n ,{E_n} \propto n\)

Step-by-step Solution

Detailed explanation

\(\mathrm{U}=\frac{1}{2} \mathrm{kr}^{2}\) Force, \(F=-\frac{d U}{d r}=-k r\) For circular motion \(\frac{\mathrm{mv}^{2}}{\mathrm{r}}=\mathrm{kr}\) .... \((i)\) And \({ mvr }=\frac{n h}{2 \pi}\) .... \((ii)\)…

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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