4.6: Energy Levels of Hydrogen Atom
- Page ID
- 1202
Consider a hydrogen atom, for which the potential takes the specific form
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
and \( P(r) = f(y) \exp(-y).\)
Here, it is assumed that the energy eigenvalue \( E\) is negative. Equation \ref{398} transforms to
Substituting this solution into Equation \ref{402}, we obtain
Equating the coefficients of \( c_n\,[n\,(n-1) - l\,(l+1)] = c_{n-1} \left [2\,(n-1) - \frac{2\,\mu\, e^2\, a}{4\pi\, \epsilon_0\, \hbar^2}\right].\)
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
which converges to \( f(y)\rightarrow \exp(2\,y)\) as \( y\rightarrow \infty\) . It follows from Equation \ref{401} that as . 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
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
and
is the Bohr radius, and \( \left[\frac{1}{x^2} \frac{d}{dx}\, x^2 \,\frac{d}{dx}-\frac{l\,(l+1)}{x^2} + \frac{2\,n}{x} - 1\right] {\cal R}_{n\,l} = 0\)
that is consistent with the normalization constraint
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)
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