A) \[\frac{{{\lambda }_{1}}}{{{\lambda }_{2}}}>1\]
B) \[\frac{{{\lambda }_{1}}}{{{\lambda }_{2}}}=1\]
C) \[\frac{{{\lambda }_{1}}}{{{\lambda }_{2}}}>\sqrt{2}\]
D) \[\frac{{{\lambda }_{1}}}{{{\lambda }_{2}}}<1\]
Correct Answer: D
Solution :
When an excited atom at rest in lab frame emits a photon of energy hf. the photon also carries a momentum \[p=\frac{hf}{c}.\]Conservation of momentum requires that atom also recoils with same momentum. So, energy carried by atom is \[\frac{{{p}^{2}}}{2m}.\]Hence, in emission we can write,\[\Delta E=hf+\] recoil energy of atom. So, emission energy of photon is smaller in case of recoil of atom. As \[E\propto \frac{1}{\lambda },\]emitted wavelength is larger. So, \[{{\lambda }_{2}}>{{\lambda }_{1}}\]and \[\frac{{{\lambda }_{1}}}{{{\lambda }_{2}}}<1.\]
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