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11.7: Fine Structure of Hydrogen Energy Levels

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    1259
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    For the case of a hydrogen atom,

    \( \left(\frac{1}{a_1} - \frac{\alpha}{y}\right)\psi_a - \left(\frac{d}{dy} + \frac{k+1}{y}\right)\psi_b\) \( \left(\frac{1}{a_2} +\frac{\alpha}{y}\right)\psi_b - \left(\frac{d}{dy} - \frac{k-1}{y}\right)\psi_a\) \( y=r/a_0\), and \( = \frac{\alpha}{1-{\cal E}},\) \ref{1263} \( = \frac{\alpha}{1+{\cal E}},\) \ref{1264}

    with \( a_0=4\pi\,\epsilon_0\,\hbar^2/(m_e\,e^2)\) is the Bohr radius, and \( \psi_a(y) = \frac

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    {y}\,f(y),\)

    \ref{1265} \( a = (a_1\,a_2)^{1/2} = \frac{\alpha}{\sqrt{1-{\cal E}^{\,2}}},\) \ref{1267}

    we obtain

    \( = 0,\) \ref{1268} \( = 0.\) \ref{1269}

    Let us search for power law solutions of the form

    \( = \sum_{s} c_s\,y^s,\) \ref{1270} \( = \sum_s c_s'\,y^s,\) \ref{1271}

    where successive values of \( \frac{c_{s-1}}{a_1} -\alpha\,c_s - (s+k)\,c_s' + \frac{c_{s-1}}{a}\) \( \frac{c_{s-1}'}{a_2}+\alpha\,c_s' -(s-k)\,c_s + \frac{c_{s-1}}{a}\) \( a\) , and the second by \( c_{s-1}\) and \( a/a_1=a_2/a\) . We are left with

    \( y=0\) require that \( y\,\psi_b\rightarrow 0\) as \( f\rightarrow 0\) and \( y\rightarrow 0\) . Consequently, the series \ref{1270} and \ref{1271} must terminate at small positive \( s_0\) is the minimum value of \( c_s\) and \( s=s_0\) and \( \alpha\,c_{s_0}+(s_0+k)\,c_{s_0}'\) \( \alpha\,c_{s_0}' - (s_0-k)\,c_{s_0}\) \( \alpha^2 = - s_0^{\,2} + k^2.\) \ref{1277}

    Since the boundary condition requires that the minimum value of \( s_0 = (k^2-\alpha^2)^{1/2}.\) \ref{1278}

    To investigate the convergence of the series \ref{1270} and \ref{1271} at large \( c_s/c_{s-1}\) for large \( s\) , Equations \ref{1273} and \ref{1274} yield

    \( \simeq \frac{c_{s-1}}{a}+\frac{c_{s-1}'}{a_2},\) \ref{1279} \( \simeq a\,c_s',\) \ref{1280}

    since \( \frac{c_s}{c_{s-1}}\simeq \frac{2}{a\,s}.\) \ref{1281}

    However, this is the ratio of coefficients in the series expansion of \( y\) unless they terminate at large \( c_s\) and \( c_{s+1}=c_{s+1}'=0\) . It follows from \ref{1272} and \ref{1273}, with \( s\) , that

    \( = 0,\) \ref{1282} \( = 0.\) \ref{1283}

    These two expressions are equivalent, because \( a_1\left[a\,\alpha-a_2\,(s-k)\right] = a\left[a_2\,\alpha+a\,(s+k)\right],\) \ref{1284}

    which reduces to

    \( {\cal E} = \left(1+\frac{\alpha^2}{s^2}\right)^{-1/2}.\) \ref{1286}

    Here, \( s_0\) by some non-negative integer \( s = i+ (k^2-\alpha^2)^{1/2}= i+[(j+1/2)^2-\alpha^2]^{1/2}.\) \ref{1287}

    where \( J^{\,2}\) . Hence, the energy eigenvalues of the hydrogen atom become

    \( \alpha \simeq 1/137\) , we can expand the above expression in \( \frac{E}{m_e\,c^2} = 1 - \frac{\alpha^2}{2\,n^2}- \frac{\alpha^4}{2\,n^4}\left(\frac{n}{j+1/2}-\frac{3}{4}\right)+{\cal O}(\alpha^6),\) \ref{1289}

    where \( n\) playing the role of the radial quantum number (see Section 4.6). Finally, the third term corresponds to the fine structure correction to these energy levels (see Exercise 4). Note that this correction only depends on the quantum numbers \( j\) . Now, we showed in Section 7.7 that the fine structure correction to the energy levels of the hydrogen atom is a combined effect of spin-orbit coupling and the electron's relativistic mass increase. Hence, it is evident that both of these effects are automatically taken into account in the Dirac equation.

    Contributors

    • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

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    This page titled 11.7: Fine Structure of Hydrogen Energy Levels is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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