The probability distribution functions associated with the position wave function ψ (x) and the momentum wave function φ (x) have dimensions of inverse length and momentum respectively, but the entropies may be rendered dimensionless by H x = − − ∫ ∫ | ψ ψ ( x ) | 2 ln ⁡ ⁡ ( x 0 | ψ ψ ( x ) | 2 ) d x = − − ⟨ ln ⁡ ⁡ ( x 0 | ψ ψ ( x ) | 2 ) ⟩ H p = − − ∫ ∫ | φ φ ( p ) | 2 ln ⁡ ⁡ ( p 0 | φ φ ( p ) | 2 ) d p = − − ⟨ ln ⁡ ⁡ ( p 0 | φ φ ( p ) | 2 ) ⟩ where x 0 and p 0 are some arbitrarily chosen length and momentum respectively, which render the arguments of the logarithms dimensionless. Note that the entropies will be functions of these chosen parameters. Due to the Fourier transform relation between the position wave function ψ (x) and the momentum wavefunction φ (p), the above constraint can be written for the corresponding entropies as