Resonances in scattering from potentials

In quantum mechanics, resonance cross section occurs in the context of quantum scattering theory, which deals with studying the scattering of quantum particles from potentials. The scattering problem deals with the calculation of flux distribution of scattered particles/waves as a function of the potential, and of the state (characterized by conservation of momentum/energy) of the incident particle. For a free quantum particle incident on the potential, the plane wave solution to the time-independent Schrödinger wave equation is:

${\displaystyle \psi ({\vec {r}})=e^{i({\vec {k}}\cdot {\vec {r}})}}$

For one-dimensional problems, the transmission coefficient ${\displaystyle T}$ is of interest. It is defined as:

${\displaystyle T={\frac {|{\vec {J}}_{\mathrm {trans} }|}{|{\vec {J}}_{\mathrm {inc} }|}}}$

where ${\displaystyle {\vec {J}}}$ is the probability current density. This gives the fraction of incident beam of particles that makes it through the potential. For three-dimensional problems, one would calculate the scattering cross-section ${\displaystyle \sigma }$, which, roughly speaking, is the total area of the incident beam which is scattered. Another quantity of relevance is the partial cross-section, ${\displaystyle \sigma _{\text{l}}}$, which denotes the scattering cross section for a partial wave of a definite angular momentum eigenstate. These quantities naturally depend on ${\displaystyle {\vec {k}}}$, the wave-vector of the incident wave, which is related to its energy by:

${\displaystyle E={\frac {\hbar ^{2}|{\vec {k}}|^{2}}{2m}}}$

The values of these quantities of interest, the transmission coefficient ${\displaystyle T}$ (in case of one dimensional potentials), and the partial cross-section ${\displaystyle \sigma _{\text{l}}}$ show peaks in their variation with the incident energy ${\displaystyle E}$. These phenomena are called resonances.