The equation above is the most general definition of p ^ ^ . In the special case of a single particle with wavefunction ψ ψ ( r ) , p ^ ^ can be written in a more specific and useful form. In one dimension: ( p ^ ^ ψ ψ ) ( r ) = i ℏ ℏ lim a → → 0 ( T ^ ^ ( a ) ψ ψ ) ( r ) − − ψ ψ ( r ) a = i ℏ ℏ lim a → → 0 ψ ψ ( r − − a ) − − ψ ψ ( r ) a = − − i ℏ ℏ ∂ ∂ ∂ ∂ r ψ ψ ( r ) or in three dimensions, p ^ ^ = − − i ℏ ℏ ∇ ∇ as an operator acting on position-space wavefunctions. This is the familiar quantum-mechanical expression for p ^ ^ , but we have derived it here from a more basic starting point.