Bohmian and quantum phase space distribution expansions and approximations

• Based on work by Dias and Prata, we give a new expansion of the Wigner distribution.
• This expansion includes the Dias and Prata expansion as a special case.
• The expansion coefficients are related to the Wigner moments and cumulants.
• The general expansion allows for different approximations for higher-order terms.
• We show how the results are applicable to dispersive pulse propagation.

Dias and Patra derived an expansion of the Wigner distribution and related it to the de Broglie–Bohm model. We show that the coefficients of the expansion are related to the conditional central moments and cumulants of the Wigner distribution. The even order cumulants depend only on the amplitude of the wave function and the odd order cumulants depend only on the phase. In addition, we give a different expansion of the Wigner distribution from which their expansion can be derived as a special case. Our expansion allows for different approximations for higher order terms. We also give expansions for the momentum representation. We show how the results are applicable to pulse propagation in a dispersive medium.