The Hydrogen Atom

Bohr's Postulates

We briefly review the Bohr atom-- a model that was soon obsolete, but which nevertheless provided the major impetus for developing quantum mechanics. Balmer's empirical formula of 1885 had reproduced Angstrom's observations of spectral lines in hydrogen to 0.1 \AA accuracy, but it was not until 1913 that Bohr gave an explanation for this based on a quantized mechanical model of the atom. This model involved the postulates of the Bohr Atom:

• Electron and proton are point charges whose interaction is coulombic at all distances.
• Electron moves in circular orbit about the center of mass in {\it stationary states} with orbital angular momentum $L=n\hbar$.

These two postulates give the energy levels: \\

$E_{n}=-{\frac {\overbrace {\overbrace {\left({\frac {1}{2}}{\frac {me^{4}}{\hbar ^{2}}}\right.} ^{R_{\infty }}\overbrace {\left.{\frac {M}{M+m}}\right)} ^{\rm {reducedmassfactor}}} ^{R_{H}}}{n^{2}}}.$

• One quantum of radiation is emitted when the system changes between these energy levels.
• The wave number of the radiation is given by the Bohr frequency criterion\footnote{Note that the wave number is a spectroscopic unit defined as the number of wavelengths per cm, $\sigma$ = $1/\lambda$. It is important not to confuse the wave number with the magnitude of the wave vector ${\bf {k}}$ which defines a traveling wave of the form exp$i(k\cdot r-\omega t)$. The magnitude of the wave vector is $2\pi$ times the wave number.}:

$\sigma _{n\rightarrow m}=(E_{n}-E_{m})/(hc)$

The mechanical spirit of the Bohr atom was extended by Sommerfeld in 1916 using the Wilson-Sommerfeld quantization rule (valid in the JWKB approximation),

$\oint p_{i}dq_{i}=n_{i}h$

where $q_{i}$ and $p_{i}$ are conjugate coordinate and momentum pairs for each degree of freedom of the system. This extension yielded elliptical orbits which were found to have an energy nearly degenerate with respect to the orbital angular momentum for a particular principal quantum number $n$. The degeneracy was lifted by a relativistic correction whose splitting was in agreement with the observed fine structure of hydrogen. (This was a great cruel coincidence in physics. The mechanical description ultimately had to be completely abandoned, in spite of the excellent agreement of theory and experiment.) Although triumphant in hydrogen, simple mechanical models of helium or other two-electron atoms failed, and real progress in understanding atoms had to await the development of quantum mechanics.

Solution of Schrodinger Equation: Key Results

As you will recall from introductory quantum mechanics, the Schrodinger equation

$i\hbar {\frac {d\psi }{dt}}=H\psi$

can be solved exactly for the Coulomb Hamiltonian

$H=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}-{\frac {Ze^{2}}{r}}$

by separation of variables

$\psi _{n\ell m}=R_{n\ell }(r)Y_{\ell m}(\theta ,\phi ).$

The full solution is reviewed in Appendix \ref{app:schr}. Here, we will highlight some important results: \begin{itemize}

• $E_{n}=-R_{H}{\frac {Z^{2}}{n^{2}}}$, just as in the Bohr model
• $\langle r_{n\ell }\rangle ={\frac {n^{2}a_{0}}{Z}}\{1+{\frac {1}{2}}[1-{\frac {\ell (\ell +1)}{n^{2}}}]\}$
• $\langle {\frac {1}{r_{n\ell }}}\rangle ={\frac {Z}{n^{2}a_{0}}}$

\end{itemize} Here, $R_{H}={\frac {1}{2}}{\frac {\mu e^{4}}{\hbar ^{2}}}$ is the Rydberg constant and $a_{0}={\frac {\hbar ^{2}}{\mu e^{2}}}$, where $\mu ={\frac {mM}{m+M}}$ is the effective mass. Note that the relation

$E_{n}=-e^{2}\langle {\frac {1}{r_{n\ell }}}\rangle \cdot {\frac {1}{2}}$

follows directly from the Virial Theorem, which states in general that $2\langle T\rangle =\langle x{\frac {dV}{dx}}\rangle ,$ so that for a spherically symmetric potential of the form $V=r^{n}$, we have $\langle T\rangle ={\frac {n}{2}}\langle V\rangle ,$ and for the special case of the Coulomb potential (footnote: In addition to the important case of the Coulomb potential, it is worth remembering the Virial Theorem for the harmonic oscillator, $n=2\Rightarrow \langle T\rangle =\langle V\rangle$.)

$T=-{\frac {1}{2}}V.$

The same factor of ${\frac {1}{2}}$ from the Virial Theorem appears in the relationship between the Rydberg constant and the atomic unit of energy, the hartree:

$R_{H}={\frac {e^{2}}{a_{0}}}{\frac {1}{2}}=\underbrace {\frac {\mu e^{4}}{\hbar ^{2}}} _{\rm {Hartree}}{\frac {1}{2}}.$

We are working here in cgs units because the expressions are more concise in cgs than in SI. To convert to SI, simply make the replacement $e^{2}\rightarrow {\frac {e^{2}}{4\pi \varepsilon _{0}}}$. \begin{quote} Question: How does the density $|\psi _{n00}(0)|^{2}$ of an $s$-electron at the origin depend on $n$ and $Z$? A. ${\frac {Z^{3}}{a_{0}^{3}n^{6}}}$ B. ${\frac {Z}{a_{0}^{3}n^{2}}}$ C. ${\frac {Z^{3}}{a_{0}^{3}n^{3}}}$ D. ${\frac {Z^{2}}{a_{0}^{3}n^{3}}}$ \par The exact result for the density at the origin is $|\psi _{n00}(0)|^{2}={\frac {Z^{3}}{\pi a_{0}^{3}n^{3}}}$ (C). To arrive at the $Z$ scaling, we observe that the $Z$ and $e$ only appear in the Hamiltonian (\ref{eq:coulomb}) in the combination $Ze^{2}$. Thus, any result for hydrogen can be extended to the case of nuclear charge $Z$ simply by making the substiution $e^{2}\rightarrow Ze^{2}$. This leads, in turn, to the substitutions

${\begin{aligned}a_{0}&={\frac {\hbar ^{2}}{me^{2}}}\rightarrow {\frac {a_{0}}{Z}}\\R_{\infty }&={\frac {me^{4}}{\hbar ^{2}}}\rightarrow Z^{2}R_{\infty }\end{aligned}}$

Thus, the density $|\psi _{n00}(0)|^{2}\propto a_{0}^{-3}\propto Z^{3}$, so the answer can only be A or C. It is tempting to assume that the $n$ scaling can be derived from the length scale $\langle {\frac {1}{r_{n\ell }}}\rangle ={\frac {Z}{a_{0}n^{2}}}$. However, the characteristic length $l$ for the density at the origin is that which appears in the exponential term of the wavefunction $\psi _{n00}\propto e^{-r/l}$,

$l=na_{0}/Z.$

Thus,

$|\psi _{n00}(0)|^{2}\propto {\frac {Z^{3}}{n^{3}a_{0}^{3}}}$

for all $n$. Remember this scaling, as it will be important for understanding such effects as the quantum defect in atoms with core electrons and hyperfine structure. \end{quote} For the case of $\ell \neq 0$, $\psi \propto r^{\ell }$ has a node at $r=0$, but the density \textit{near} the origin is nonetheless of interest. It can be shown (Landau III, \S 36 \cite{Landau}) that, for small $r$,

$R_{n\ell }(r)\simeq r^{\ell }{\frac {2^{\ell +1}}{n^{2+\ell }(2\ell +1)!}}{\sqrt {\frac {(n+\ell )!}{(n-\ell -1)!}}}.$

Since ${\sqrt {\frac {(n+\ell )!}{(n-\ell -1)!}}}{\overrightarrow {_{n\rightarrow \infty }}}{\sqrt {n^{2\ell +1}}}$, the density near the origin scales for large $n$ as

$|\psi |^{2}\propto r^{2\ell }\times {\frac {1}{n^{3}}}.$