Central force problems and separation of variables

The Coulomb potential

The hydrogen atom consists of an electron bound to a proton by the Coulomb attraction. To a good approximation, the electron moves in a central potential depending only on distance from the proton:

V(r)=-rac{e^2}{4piepsilon_0 r}

Because the potential depends only on $r$, the problem has spherical symmetry. This symmetry makes angular momentum important and allows separation of variables in spherical coordinates.

Schrödinger equation in three dimensions

The time-independent Schrödinger equation is

abla^2psi+V(r)psi=Epsi$$ Here $mu$ is the reduced mass of the electron-proton system: $$mu=rac{m_em_p}{m_e+m_p}$$ Using reduced mass accounts for the fact that both electron and proton move around their common center of mass. ## Spherical coordinates In spherical coordinates, position is described by $r$, $ heta$, and $phi$. The Laplacian separates into radial and angular parts. Because the potential is central, solutions can be written as $$psi(r, heta,phi)=R(r)Y( heta,phi)$$ More specifically, $$psi_{nlm}(r, heta,phi)=R_{nl}(r)Y_l^m( heta,phi)$$ where $R_{nl}$ is the radial part and $Y_l^m$ is a spherical harmonic. ## Angular and radial equations Separation produces an angular equation related to angular momentum and a radial equation involving an effective potential. The angular solutions are labeled by quantum numbers $l$ and $m$. The radial equation depends on $l$ because angular momentum contributes a centrifugal-like term. A useful transformed radial function is $$u(r)=rR(r)$$ The radial equation can be written in a one-dimensional-like form with an effective potential containing $$rac{hbar^2l(l+1)}{2mu r^2}$$ This term acts like an angular momentum barrier. ## Quantum numbers Hydrogen bound states are labeled by three quantum numbers: $$n=1,2,3,ldots$$ $$l=0,1,2,ldots,n-1$$ $$m=-l,-l+1,ldots,l$$ The principal quantum number $n$ primarily sets the energy. The orbital angular momentum quantum number $l$ sets the angular momentum magnitude. The magnetic quantum number $m$ sets one component of angular momentum. ## Boundary conditions Physical solutions must be finite, single-valued, and normalizable. The wavefunction must not diverge at the origin and must decay at infinity for bound states. These requirements lead to discrete energies and allowed quantum numbers. Normalization requires $$int |psi|^2 d^3r=1$$ In spherical coordinates, $$d^3r=r^2sin heta,dr,d heta,dphi$$ ## The big idea The hydrogen atom is solvable because its Coulomb potential has spherical symmetry. Separation of variables splits the wavefunction into radial and angular parts. Boundary conditions and angular momentum quantization produce quantum numbers $n$, $l$, and $m$, which organize atomic orbitals and spectra.