General Formalism
A central potential is a potential that depends only on the absolute value of the distance away from the potential's center. A central potential is rotationally invariant. We may use these properties to reduce this otherwise three-dimensional problem to an effective one-dimensional problem. The general form of the Hamiltonian for a particle immersed in such a potential is
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{H}=\frac{\hat{p}^2}{2m}+V(|\hat{r}|).}
Due to rotational symmetry, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\hat{H},\hat{L}_z]=0\!} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\hat{H},\hat{L}^2]=0.\!} This allows us to find a complete set of states that are simultaneous eigenstates of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{H},\!} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{L}_z,\!} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{L}^2.\!} We will label these eigenstates as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |n,l,m\rangle,\!} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l\!} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\!} are the orbital and magnetic quantum numbers, as defined in the previous chapter, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\!} represents the quantum numbers that define the radial dependence of the wave function; this is the only part of the state that depends on the exact form of the potential, as we will see shortly.
Let us now write the Schrödinger equation for this system and solve for the angular dependence of the wave function, thus reducing the problem to an effective one-dimensional problem. The equation is
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{\hbar^2}{2m}\nabla^2\psi+V(r)\psi=E\psi.}
The Laplacian in spherical coordinates may be written as
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla^2=\frac{1}{r}\frac{\partial^2}{\partial r^2}r-\frac{\hat{\mathbf{L}}^2}{\hbar^2 r^2},}
so that the equation becomes
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(-\frac{\hbar^2}{2m}\frac{1}{r}\frac{\partial^2}{\partial r^2}r+\frac{\hat{\mathbf{L}}^2}{2mr^2}+V(r)\right)\psi=E\psi.}
We already know the eigenfunctions of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\hat{\mathbf{L}}^2}{2mr^2}} from the previous chapter, and thus the entire angular dependence of the wave function. We may therefore use separation of variables and write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(r,\theta,\phi)=f_l(r)Y_l^m(\theta,\phi),\!} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_l^m(\theta,\phi)\!} are the spherical harmonics. Substituting this into the Schrödinger equation, we obtain
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(-\frac{\hbar^2}{2m}\frac{1}{r}\frac{d^2}{dr^2}r+\frac{\hbar^2 l(l+1)}{2mr^2}+V(r)\right)f_l(r)=Ef_l(r).}
The term Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 the term, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{l^2}{2mr^2},} that arises in treating central forces classically. The centrifugal barrier prevents the particle from reaching the center of force, causing the wave function to vanish at this point.
Now, if we let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_l(r)=rf_l(r)\!} , we finally arrive at the effective one-dimensional Schrödinger equation for the radial dependence of the wave function, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(-\frac{\hbar^2}{2m}\frac{d^2}{dr^2}+\frac{\hbar^2 l(l+1)}{2mr^2}+V(r)\right)u_l(r)=Eu_l(r).}
Due to the boundary condition that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_l(r)\!} must be finite the origin, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_l(r)\!} must vanish.
In many cases, looking at the asymptotic behavior of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_l(r)\!} can be quite helpful, as we will see in later sections.
Nomenclature
Historically, the first four (previously five) values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l\!} have taken on names, and additional values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l\!} are referred to alphabetically:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.