Integral Forms for Quantum-Mechanical Momentum Operators

The usual differential form P₀ for the quantum‐mechanical momentum operator P which is conjugate to a generalized coordinate q (α ≤ q ≤ β) is, in atomic units, P₀ = − i(g^−½) ∂/∂q (g^½), where g is the Jacobian of the transformation from Cartesian to generalized coordinates. However, P₀ is not always self‐adjoint on the domain D of physically acceptable bound‐state wavefunctions, as a proper quantum‐mechanical operator should be. An integral form is proposed for P, defined by $${\text{Pf(q)=(2π)}}^{-\frac{1}{2}}{g}^{-\frac{1}{2}}\mathrm{(q)\hspace{0.17em}}{\displaystyle {\int }_{\mathrm{-\infty }}^{\infty }}\exp \mathrm{\hspace{0.17em}(ikq)kF(k)\hspace{0.17em}dk,\hspace{0.17em}\hspace{0.17em}\alpha \le q\le \beta }$$ , where $${\text{F(k)=(2π)}}^{-\frac{1}{2}}\hspace{0.17em}{\displaystyle {\int }_{\alpha }^{\beta }}\exp {\mathrm{\hspace{0.17em}(-ik\xi )f(\xi )g}}^{\frac{1}{2}}\mathrm{(\xi )\hspace{0.17em}d\xi ,\hspace{0.17em}\hspace{0.17em}f\in }\mathcal{D}$$ . The effect of this integral operator (which is suggested by the ideas of Fourier transforms) differs from that of P₀ only at the end‐points of the range of q. In a sense, it is formally equivalent to an operator (suggested by Robinson and Hirschfelder) which is obtained by adding certain delta‐function terms to P₀, but it suffers from none of the defects, since delta‐functions do not appear explicitly. Various properties of the integral operator are derived. Some discussion of the domain D is presented as an appendix.