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4.6: Energy Levels of Hydrogen Atom

  • Page ID
    1202
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    Consider a hydrogen atom, for which the potential takes the specific form

    \( R(r)\) satisfies Equation \ref{395}, which can be written $ \left[\frac{\hbar^2}{2\,\mu} \left(-\frac{1}{r^2} \frac{d}{dr}\,r...
... +\frac{l\,(l+1)}{r^2}\right) -\frac{e^2}{4\pi\,\epsilon_0\,r}- E\right] R = 0.$ \ref{397}

    Here, \( m_e\) ) and the proton (of mass \( \mu\) rotating about a fixed point. Let us write the product \( P(r)\). The above equation transforms to

    \( \mu\) moving in the effective potential \( a= \sqrt{\frac{-\hbar^2}{2\,\mu \,E}},\) \ref{400}

    and \( P(r) = f(y) \exp(-y).\)

    \ref{401}

    Here, it is assumed that the energy eigenvalue \( E\) is negative. Equation \ref{398} transforms to

    \( f(y) = \sum_{n} c_n\, y^{\,n}.\) \ref{403}

    Substituting this solution into Equation \ref{402}, we obtain

    $ \sum_n c_n \left[ n\,(n-1)\,y^{\,n-2} - 2\,n\, y^{\,n-1} - l\,(l+...
...+ \frac{2\,\mu\, e^2 \,a}{4\pi\, \epsilon_0 \,\hbar^2}\, y^{\,n-1} \right] = 0.$ \ref{404}

    Equating the coefficients of \( c_n\,[n\,(n-1) - l\,(l+1)] = c_{n-1} \left [2\,(n-1) - \frac{2\,\mu\, e^2\, a}{4\pi\, \epsilon_0\, \hbar^2}\right].\)

    \ref{405}

    Now, the power law series \ref{403} must terminate at small \( n\), otherwise \( y\rightarrow 0\). This is only possible if \( c_{n_{\rm min}}\,y^{\,n_{\rm min}}\). There are two possibilities: \( n_{\rm min} = l+1\). The former predicts unphysical behavior of the wavefunction at \( n_{\rm min} = l+1\). Note that for an \( l>0\) state there is zero probability of finding the electron at the nucleus (i.e., \( r=0\), except when \( r\rightarrow 0\) if \( y\), the ratio of successive terms in the series \ref{403} is

    \( \sum_n \frac{(2\,y)^{\,n}}{n!},\) \ref{407}

    which converges to \( f(y)\rightarrow \exp(2\,y)\) as \( y\rightarrow \infty\) . It follows from Equation \ref{401} that $ R(r) \rightarrow
\exp(r/a) /r $ as $ r\rightarrow
\infty$ . This does not correspond to physically acceptable behavior of the wavefunction, since \( n\). According to the recursion relation \ref{405}, this is only possible if

    \( c_n\, y^{\,n}\). It follows from Equation \ref{400} that the energy eigenvalues are quantized, and can only take the values \( E_0 = - \frac{\mu\, e^4}{32\pi^2\,\epsilon_0^{\,2}\, \hbar^2} = - 13.6\,{\rm eV}\) \ref{410}

    is the ground state energy. Here, \( l\), otherwise there would be no terms in the series \ref{403}.

    The properly normalized wavefunction of a hydrogen atom is written

    \( R_{n\,l}(r) = {\cal R}_{n\,l}(r/a),\) \ref{412}

    and

    \( a_0 =\frac{4\pi\, \epsilon_0\,\hbar^2}{\mu \,e^2} = 5.3\times 10^{-11}\,\,\,{\rm meters}\) \ref{414}

    is the Bohr radius, and \( \left[\frac{1}{x^2} \frac{d}{dx}\, x^2 \,\frac{d}{dx}-\frac{l\,(l+1)}{x^2} + \frac{2\,n}{x} - 1\right] {\cal R}_{n\,l} = 0\)

    \ref{415}

    that is consistent with the normalization constraint

    \( Y_{l\,m}\) are spherical harmonics. The restrictions on the quantum numbers are \( n\) is a positive integer, \( m\) an integer.

    The ground state of hydrogen corresponds to \( l=0\) and \( n=2\). The other quantum numbers are allowed to take the values \( m=0\) or \( m=-1, 0, 1\). Thus, there are \( l\), despite the fact that \( 1/r\) Coulomb potential.

    In addition to the quantized negative energy states of the hydrogen atom, which we have just found, there is also a continuum of unbound positive energy states.

    Contributors

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

      \( \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}\) \(\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}\) \(\newcommand {\btau}{\mbox{\boldmath$\tau$}}\) \(\newcommand {\bmu}{\mbox{\boldmath$\mu$}}\) \(\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}\) \(\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}\) \(\newcommand {\bomega}{\mbox{\boldmath$\omega$}}\) \(\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}\)

    This page titled 4.6: Energy Levels of Hydrogen Atom is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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