Canonical commutation relation

In <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>, the canonical commutation relation is the fundamental relation between <a href="/facts/Canonical_conjugate/NQca4AKf">canonical conjugate</a> quantities (quantities which are related by definition such that one is the <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a> of another). For example,

[
        
          
            
              x
              ^
            
          
        
        ,
        
          
            
              
                p
                ^
              
            
          
          
            x
          
        
        ]
        =
        i
        ℏ
        
          I
        
      
    
    {\displaystyle [{\hat {x}},{\hat {p}}_{x}]=i\hbar \mathbb {I} }

between the position operator x and momentum operator px in the x direction of a point particle in one dimension, where [x , px] = x px − px x is the <a href="/facts/Commutator/eYxzFmFY">commutator</a> of x and px , i is the <a href="/facts/Imaginary_unit/1SCllQlU">imaginary unit</a>, and ℏ is the <a href="/facts/Reduced_Planck_constant/qGHAe3qq">reduced Planck constant</a> h/2π, and 
 
 
 
 
 I
 
 
 
 {\displaystyle \mathbb {I} }
 
 is the unit operator. In general, position and momentum are vectors of operators and their commutation relation between different components of position and momentum can be expressed as

[
        
          
            
              
                x
                ^
              
            
          
          
            i
          
        
        ,
        
          
            
              
                p
                ^
              
            
          
          
            j
          
        
        ]
        =
        i
        ℏ
        
          δ
          
            i
            j
          
        
        ,
      
    
    {\displaystyle [{\hat {x}}_{i},{\hat {p}}_{j}]=i\hbar \delta _{ij},}

where 
 
 
 
 
 δ
 
 i
 j
 
 
 
 
 {\displaystyle \delta _{ij}}
 
 is the <a href="/facts/Kronecker_delta/gcGAy0bm">Kronecker delta</a>.
This relation is attributed to <a href="/facts/Werner_Heisenberg/n4qzYvj4">Werner Heisenberg</a>, <a href="/facts/Max_Born/6VnUDWhA">Max Born</a> and <a href="/facts/Pascual_Jordan/YNLoGJ5d">Pascual Jordan</a> (1925), who called it a "quantum condition" serving as a postulate of the theory; it was noted by <a href="/facts/Earle_Hesse_Kennard/k37kqkAo">E. Kennard</a> (1927) to imply the <a href="/facts/Werner_Heisenberg/n4qzYvj4">Heisenberg</a> <a href="/facts/Uncertainty_principle/UUSoSp4T">uncertainty principle</a>. The <a href="/facts/Stone%E2%80%93von_Neumann_theorem/4C64mKYR">Stone–von Neumann theorem</a> gives a uniqueness result for operators satisfying (an exponentiated form of) the canonical commutation relation.

Canonical commutation relation open-in-new

Canonical commutation relation