Which of the following transitions for quantum numbers (n, l, ml, ms) are allowed for the hydrogen atom, and for those allowed, what is the energy involved? (a) (2, 0, 0, 1/2) → (3, 1, 1, 1/2) (b) (2, 0, 0, 1/2) → (3, 0, 0, 1/2) (c) (4, 2, -1, -1/2) → (2, 1, 0, 1/2) Strategy We want to compare Δl

Question

Which of the following transitions for quantum numbers (n, [latex]\ell, m_{\ell}, m_{s}[/latex]) are allowed for the hydrogen atom, and for those allowed, what is the energy involved?

(a) [latex](2,0,0,1 / 2) ightarrow(3,1,1,1 / 2)[/latex]

(b) [latex](2,0,0,1 / 2) ightarrow(3,0,0,1 / 2)[/latex]

(c) [latex](4,2,-1,-1 / 2) ightarrow(2,1,0,1 / 2)[/latex]

Strategy We want to compare [latex]\Delta \ell ext { and } \Delta m_{\ell}[/latex] with the selection rules of Equation (7.36). If allowed, the energies may be obtained from Equation (7.21) with [latex]E_{0}=13.6[/latex] eV.

[latex]E_{n}=\frac{-\mu}{2}\left(\frac{e^{2}}{4 \pi \epsilon_{0} \hbar} ight)^{2} \frac{1}{n^{2}}=-\frac{E_{0}}{n^{2}}[/latex] (7.21)

[latex]\Delta \ell=\pm 1[/latex] (7.36)

Accepted Answer

(a) [latex]\Delta \ell=+1, \Delta m_{\ell}=1[/latex]; allowed.

[latex]\Delta E=E_{3}-E_{2}=-13.6 eV \left(\frac{1}{3^{2}}-\frac{1}{2^{2}} ight)[/latex]

= 1.89 eV, corresponding to absorption of a 1.89-eV photon

(b) [latex]\Delta \ell=0, \Delta m_{\ell}=0[/latex]; not allowed, because [latex]\Delta \ell eq \pm 1[/latex].

(c) [latex]\Delta \ell=-1, \Delta m_{\ell}=1[/latex]; allowed. Notice that [latex]\Delta n=-2[/latex] and [latex]\Delta m_{s}=+1[/latex] does not affect whether the transition is allowed.

[latex]\Delta E=E_{2}-E_{4}=-13.6 eV \left(\frac{1}{2^{2}}-\frac{1}{4^{2}} ight)[/latex]

= -2.55 eV, corresponding to emission of a 2.55-eV photon

Question 7.10: Which of the following transitions for quantum numbers (n, l...

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