Radial Schrodinger equation for the hydrogen atom

For zero orbital angular momentum and Coulomb potential $-q^2/r$, the radial equation is

For the usual three-dimensional Coulomb potential Hamiltonian operator, a separated radial wavefunction $R(r)$ uses the reduced function $u=rR$, whose radial norm has measure $dr$. The regular physical operator domain imposes $u(0)=0$. Near the origin the leading radial equation gives the indicial equation $c(c-1)-\ell (\ell +1)=0$ for $u\sim r^c$, with roots $c=\ell +1$ and $c=-\ell$.

The regular root gives $R\sim r^{\ell }$. The other root gives $R\sim r^{-\ell -1}$, and its radial probability integral ${\int }_0^{\epsilon }{r}^2|R{|}^{2}dr\sim {\int }_0^{\epsilon }{r}^{-2\ell }dr$ diverges for $\ell \ge 1$. At $\ell =0$, $R\sim 1/r$ is locally square-integrable in three dimensions, but violates $u(0)=0$ and has divergent kinetic energy integral ${\int }_0^{\epsilon }{r}^2|R^{\prime }{|}^{2}dr\sim {\int }_0^{\epsilon }{r}^{-2}dr$. Thus local wavefunction normalization alone is insufficient to select the physical branch in this exceptional case; the standard Coulomb energy domain supplies the missing condition.

For three-dimensional radial wavefunctions with measure $r^{2}dr$, this formal symmetric momentum expression obeys $[f(r),p_r]=i\hslash f^{\prime }(r)$ on appropriate test functions and has square $-{\hslash }^2({\partial }_r^2+2{\partial }_r/r)$. It is useful in separating kinetic energy as $p^2=p_r^2+L^2/r^2$. Formal symmetry and these identities do not establish a self-adjoint momentum observable on the half-line; domains and origin conditions must be specified when taking adjoints or defining products.

For a normalized maximal-angular-momentum hydrogen state, the probability inside a small radius $r_p$ is the integral of $|R{|}^2{r}^2$. Since $R$ behaves as $r^l$ near zero, it equals $2^{2l+3}(r_p/a{)}^{2l+3}/[(2l+3)!(l+1{)}^{2l+3}]$ times $1+O(r_p/a)$. Increasing angular momentum strongly suppresses penetration toward the Coulomb centre.

A hydrogen atom state with principal quantum number $n=l+1$ has no radial nodes and radial factor $R(r)=A{r}^l{e}^{-r/[a(l+1)]}$, where $a$ is the Bohr radius. Its energy is $-{\hslash }^2/[2m{a}^2(l+1{)}^2]$. With unit-normalized spherical harmonics, the radial normalization gives $|A{|}^2=2^{2l+3}/[a^{2l+3}(l+1{)}^{2l+3}(2l+2)!]$.

After the logarithmic radial coordinate $r=e^x$, the Langer replacement $l(l+1)\mapsto (l+1/2{)}^2$ turns the radial Coulomb problem into a two-turning-point WKB approximation problem. Its action integral giveswhich in this Coulomb case agrees with the exact dependence on the principal quantum number.

For a radial solution regular at the origin, writing $R=e^{-\gamma r}{\sum }_{n\ge 0}{a}_n{r}^n$ givesNormalizability at infinity forces the series to terminate, so $2\gamma N=\beta$ for a positive integer $N$ and $E_N=-m{q}^4/(2{\hslash }^2{N}^2)$.

The lowest Coulomb bound-state energy is

In atomic units, use the radial trial wavefunction $\psi (r)=\max (1-r/\beta ,0)$ with $\beta >0$. Its radial norm integral is $\int r^2{\psi }^{2}dr={\beta }^3/30$, its gradient contribution is $\int r^2{\psi }^{\prime 2}dr=\beta /3$, and its Coulomb contribution is $-2\int r{\psi }^{2}dr=-{\beta }^2/6$. In the hydrogen radial Rayleigh quotient, this gives the displayed energy, minimized at $\beta =4$. It is above the exact hydrogen ground-state energy $-1/2$, as required by the quantum variational principle. The derivative jumps at $r=\beta$, but the function has a square-integrable weak first derivative and is an admissible trial for the energy quadratic form. One must use that form rather than assume that a classical second derivative exists at the join.