Resonances in scattering from potentials

In <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>, resonance cross section occurs in the context of quantum <a href="/facts/Scattering_theory/79Nx76Ub">scattering theory</a>, 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 <a href="/facts/Energy%E2%80%93momentum_relation/HPXngJQF">momentum/energy</a>) of the incident particle. For a free quantum particle incident on the potential, the plane wave solution to the time-independent <a href="/facts/Schr%C3%B6dinger_equation/3csgSIWF">Schrödinger wave equation</a> is:

ψ
        (
        
          
            
              r
              →
            
          
        
        )
        =
        
          e
          
            i
            (
            
              
                
                  k
                  →
                
              
            
            ⋅
            
              
                
                  r
                  →
                
              
            
            )
          
        
      
    
    {\displaystyle \psi ({\vec {r}})=e^{i({\vec {k}}\cdot {\vec {r}})}}

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

T
        =
        
          
            
              
                |
              
              
                
                  
                    
                      J
                      →
                    
                  
                
                
                  
                    t
                    r
                    a
                    n
                    s
                  
                
              
              
                |
              
            
            
              
                |
              
              
                
                  
                    
                      J
                      →
                    
                  
                
                
                  
                    i
                    n
                    c
                  
                
              
              
                |
              
            
          
        
      
    
    {\displaystyle T={\frac {|{\vec {J}}_{\mathrm {trans} }|}{|{\vec {J}}_{\mathrm {inc} }|}}}

where 
 
 
 
 
 
 
 J
 →
 
 
 
 
 
 {\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 <a href="/facts/Scattering_cross-section/lo014h2D">scattering cross-section</a> 
 
 
 
 σ
 
 
 {\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, 
 
 
 
 
 σ
 
 l
 
 
 
 
 {\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 
 
 
 
 
 
 
 k
 →
 
 
 
 
 
 {\displaystyle {\vec {k}}}
 
, the wave-vector of the incident wave, which is related to its energy by:

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

The values of these quantities of interest, the <a href="/facts/Transmission_coefficient/893q1h2K">transmission coefficient</a> 
 
 
 
 T
 
 
 {\displaystyle T}
 
 (in case of one dimensional potentials), and the partial cross-section 
 
 
 
 
 σ
 
 l
 
 
 
 
 {\displaystyle \sigma _{\text{l}}}
 
 show peaks in their variation with the incident energy 
 
 
 
 E
 
 
 {\displaystyle E}
 
. These phenomena are called resonances.

Resonances in scattering from potentials open-in-new

Resonances in scattering from potentials