General Formalism

A central potential does not depend on time, but rather depends only on the absolute value of the distance away from the potential's center. A central potential is rotationally invariant, not depending on the orientation. These properties can effectively reduce a three dimensional problem into a one dimensional problem.

$H={\frac {p^{2}}{2m}}+V(|r|)$

Due to the rotational symmetry, $[H,L_{z}]=0\!$ and $[H,L^{2}]=0\!$, and the eigenstates of $L^{2}\!$, $L_{z}\!$ are non-degerate. This allows us to find a complete set of states that are simultaneous eigenfunctions of $H\!$, $L_{z}\!$, and $L^{2}\!$. We can label these states by their eigenvalues of $|Elm\rangle \!$.

From this we can get a state of the same energy for a given $l\!$ with a degeneracy of $2l+1\!$. We can rewrite the Laplacian as

$\nabla ^{2}={\frac {1}{r}}{\frac {\partial ^{2}}{\partial r^{2}}}r-{\frac {L^{2}}{\hbar ^{2}r^{2}}}$

This makes the Schrödinger equation

$\left(-{\frac {\hbar ^{2}}{2m}}{\frac {1}{r}}{\frac {\partial ^{2}}{\partial r^{2}}}r+{\frac {L^{2}}{2mr^{2}}}+V(r)\right)\psi (r,\theta ,\phi )=E\psi (r,\theta ,\phi )$

Using separation of variables, $\psi (r,\theta ,\phi )=f_{l}(r)Y_{lm}(\theta ,\phi )\!$, we get:

$\left(-{\frac {\hbar ^{2}}{2m}}{\frac {1}{r}}{\frac {\partial ^{2}}{\partial r^{2}}}r+{\frac {\hbar ^{2}l(l+1)}{2mr^{2}}}+V(r)\right)f_{l}(r)Y_{lm}(\theta ,\phi )=Ef_{l}(r)Y_{lm}(\theta ,\phi )$

The term ${\frac {\hbar ^{2}l(l+1)}{2mr^{2}}}$ is referred to as the centrifugal barrier, which is associated with the motion of the particle. The classical analogue is ${\frac {l^{2}}{2mr^{2}}}$. The centrifugal barrier prevents the particle from reaching the center of force, causing the wave function to vanish at this point. Multiplying both sides by $Y_{l^{\prime }m'}\!$ and integrating over the angular dependence reduces the equation to merely a function of $r\!$.

Now if we let $u_{l}(r)=rf_{l}(r)\!$, this gives the radial Schrödinger equation:

$\left(-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial r^{2}}}+{\frac {\hbar ^{2}l(l+1)}{2mr^{2}}}+V(r)\right)u_{l}(r)=Eu_{l}(r)$

Due to the boundary condition that $f_{l}(r)\!$ must be finite the origin, $u_{l}(r)\!$ must vanish.

Often looking at the asymptotic behavior of $u_{l}(r)\!$ can be quite helpful.

As $r\rightarrow 0\!$ and $V(r)\ll {\frac {1}{r^{2}}}\!$, the dominating term becomes the centrifugal barrier giving the approximate Hamiltonian:

$-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial r^{2}}}+{\frac {\hbar ^{2}l(l+1)}{2mr^{2}}}$

which has the solutions $u_{l}(r)\sim r^{l+1},r^{-l}\!$ where only the first term is physically possible because the second blows up at the origin.

As $r\rightarrow \infty \!$ and $rV(r)\rightarrow 0$(which does not include the monopole ${\frac {1}{r}}$ coulomb potential), the Hamiltonian approximately becomes

$-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial r^{2}}}u_{l}(r)=Eu_{l}(r)$.

Letting $k=-i{\sqrt {\frac {2mE}{\hbar ^{2}}}}$ gives a solution of $u_{l}(r)=Ae^{kr}+Be^{-kr}\!$, where when $k\!$ is real, $B=0\!$, but both terms are needed when $k\!$ is imaginary.

Nomenclature

Historically, the first four (previously five) values of $l\!$ have taken on names, and additional values of $l\!$ are referred to alphabetically:

${\begin{cases}l=0&{\mbox{s-wave (sharp)}}\\l=1&{\mbox{p-wave (principal)}}\\l=2&{\mbox{d-wave (diffuse)}}\\l=3&{\mbox{f-wave (fundamental)}}\\l=4&{\mbox{g-wave (previously called t-wave for thick)}}\\l=5&{\mbox{h-wave}}\\\end{cases}}$

Worked Problem involving the energy levels in a central potential.