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For a real potential, the time-dependent Schrödinger equation isMultiply it by ${\psi }^{\ast }$, combine with its conjugate multiplied by $\psi$, and note that the real-potential terms cancel. This giveswhere the probability current isThe product rule gives the divergence identity because the gradient cross terms cancel.

For $F(r)=\sin (kr)/r$, the supplied radial identity gives $\Delta F=-k^{2}F$ for $r>0$. The time factor gives $i\hslash {\psi }_t=E\psi$, so the free equation holds forThe apparent origin singularity is removable: $F(0)=k$ and $F(r)=k-k^3{r}^2/6+\cdots$, so the equation extends smoothly there as well. Since the spatial factor is real and its time dependence is a common phase, ${\psi }^{\ast }\nabla \psi =F\nabla F$ is real. Thus $\mathbf{j}=0$, while $\rho =F^2$ is independent of time. Both terms in the continuity equation vanish separately. This is a standing spherical wave; the real spatial factor carries no net radial probability current.

The Huygens-Fresnel principle treats every point of an advancing wavefront as a source of secondary spherical waves; their forward envelope constructs the next wavefront. Consider two points where an oblique plane wavefront meets a plane interface a distance $AB$ apart. After its first point reaches $A$, let time $t$ elapse until the second reaches $B$. In medium 1, the incident travel distance is $v_{1}t=AB\sin i$. Meanwhile, a secondary wave from $A$ has travelled distance $v_{2}t$ in medium 2. The tangent from $B$ to that secondary wavefront gives $v_{2}t=AB\sin r$. ThereforeHere $i,r$ are the angle of incidence and angle of refraction, measured from the interface normal vector, and $n_j=c/v_j$ are the refractive indices at the common frequency. This is Snell's law.

A scattering phase shift measures the change of a partial wave's outgoing phase relative to the free radial solution. For a real central potential and elastic scattering, probability conservation gives a unit-modulus Partial-wave S-matrix $S_l=e^{2i{\delta }_l}$. Indeedso the incoming amplitude is unchanged and the outgoing one acquires $S_l$.

Expanding an incident plane wave in Legendre polynomials and comparing the outgoing spherical wave givesUse $d\sigma /d\Omega =|f{|}^2$ and $\int d\Omega P_l{P}_{l^{\prime }}=4\pi {\delta }_{l{l}^{\prime }}/(2l+1)$ to remove cross terms:This is the elastic total scattering cross-section for the one-particle central-potential problem, with no identical-particle symmetrization factor.

For time dependence $e^{-i\omega t}$, the three-dimensional radial solutions $e^{ikr}/r$ and $e^{-ikr}/r$ describe outgoing and incoming spherical waves away from the origin. Their inverse-radius amplitudes make their flux through spheres independent of radius. They form the asymptotic radial ingredients of partial-wave scattering.