Generalized Momentum Operators in Quantum Mechanics

The usual form P₀ for the quantum‐mechanical operator P conjugate to a generalized coordinate q₁ is, in atomic units, $$P_0{\mathrm{=-ig}}^{-\frac{1}{2}}{\mathrm{\hspace{0.17em}\partial /\partial q}}_1{\mathrm{(g}}^{\frac{1}{2}})$$ , where g is the Jacobian of the transformation from Cartesian coordinates to the generalized coordinates. However, in some cases, this plusible form for P is not Hermitian with respect to physically acceptable bound‐state wavefunctions, as it must be if it is to represent a real observable quantity. In this paper, the general form $$\mathrm{P=-i(}\mathbf{A}·\mathrm{grad}+\frac{1}{2}\hspace{0.17em}\mathrm{div}\hspace{0.17em}\mathbf{A})$$ is justified. Here A = h₁q̂₁, where h₁ is the metric scale factor corresponding to q₁, and q̂₁ is the unit vector in the direction of increasing q₁. This general form reduces to P₀ if the usual formula for the divergence is applied. In the cases where P₀ is not Hermitian, it transpires that q̂₁ is ill‐defined at one or both of the end points α and β of the range of q₁, and the divergence formula is thus invalid at such points. It is shown that, in order to obtain a Hermitian form for P, certain terms involving delta functions similar to Dirac's must be added to the usual formula for div A. These terms can be regarded as implicit in div A. If q̂₁ is ill‐defined at the lower limit q₁ = α, then the resulting new Hermitian form for P, which we propose as the correct one, is $${\mathrm{P=-i[g}}^{-\frac{1}{2}}{\mathrm{\hspace{0.17em}\partial /\partial q}}_1{\mathrm{(g}}^{\frac{1}{2}}\mathrm{)+}\frac{1}{2}{\mathrm{\hspace{0.17em}\delta }}_{+}{\mathrm{(q}}_1\mathrm{-\alpha )]}$$ . If q̂₁ is, in addition, ill‐defined at the upper limit q₁ = β, then the extra term +½i δ_(β − q₁) must be added. Corresponding new forms are obtained for the Laplacian operator. In addition, the new formulas for P are applied to hypervirial relations. In the Appendix, Charles Goebel obtains a similar expression for the momentum operators by replacing the metric scale function by θg where θ is a step function, unity inside and zero outside the range of definition of the generalized coordinates. The differentiation of θ then produces the delta functions.