Highly excited states for hydrogen-like atoms | vedclass

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Question

(A) For a hydrogen-like atom,the radius of the $n$-th orbit is $r_n = a_0 \frac{n^2}{Z}$.The relative change in radius for two consecutive orbitals is

Vedclass · Accepted Answer

$A, B, D$

Vedclass · Answer

(A) For a hydrogen-like atom,the radius of the $n$-th orbit is $r_n = a_0 \frac{n^2}{Z}$.The relative change in radius for two consecutive orbitals is $\frac{\Delta r}{r} \approx \frac{dr}{r} = \frac{2n \, dn}{n^2} = \frac{2 \Delta n}{n}$. Since $\Delta n = 1$,$\frac{\Delta r}{r} \propto \frac{1}{n}$. This is independent of $Z$. Thus,$(A)$ and $(B)$ are correct.The energy of the $n$-th orbit is $E_n = -E_0 \frac{Z^2}{n^2}$.The relative change in energy is $\frac{\Delta E}{E} \approx \frac{dE}{E} = \frac{2n^{-3} \, dn}{n^{-2}} = \frac{2 \Delta n}{n}$. Since $\Delta n = 1$,$\frac{\Delta E}{E} \propto \frac{1}{n}$. Thus,$(C)$ is incorrect.The angular momentum is $L = \frac{nh}{2\pi}$.The relative change is $\frac{\Delta L}{L} = \frac{\Delta n}{n}$. Since $\Delta n = 1$,$\frac{\Delta L}{L} \propto \frac{1}{n}$. Thus,$(D)$ is correct.Therefore,the correct statements are $(A)$,$(B)$,and $(D)$.

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