© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Journal of Mathematical Physics

Generalized Momentum Operators in Quantum Mechanics

Peter D. Robinson, Joseph O. Hirschfelder

Published Mar 1, 1963• 30 citations

Abstract

The usual form P0 for the quantum-mechanical operator P conjugate to a generalized coordinate q1 is, in atomic units, P0=−ig−12 ∂/∂q1(g12),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 P=−i(A·grad+12 div 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 P=−i[g−12 ∂/∂q1(g12)+12 δ+(q1−α)]. 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.

DOI: 10.1063/1.1703959

Related papers

Integral Forms for Quantum-Mechanical Momentum Operators

Peter D. Robinson

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 (g½), 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 Pf(q)=(2π)−12g−12(q) ∫ −∞∞exp (ikq)kF(k) dk, α≤q≤β,where F(k)=(2π)−12 ∫ αβexp (−ikξ)f(ξ)g12(ξ) dξ, f∈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.

Journal of Mathematical Physics•Nov 1, 1966•8 citations

GENERALIZED MOMENTUM OPERATORS IN QUANTUM MECHANICS. Series 3

Peter D. Robinson, Joseph O. Hirschfelder

>The usual form for the quantum-mechanical operator P conjugate to a generalized coordinate q is noted to be nonHermitian with respect to physically acceptable bound state wave functions in some cases. A corrected form is shown to be always Hermitian with respect to acceptable bound state wave functions. Eigenfunctions for the new form are not necessartly orthogonal as with the usual form. (D.C.W.)

•Apr 16, 1962•0 citations

Derivation of the coordinate-momentum commutation relations from canonical invariance

Moshe Shapiro

Based on (i) the spectral resolution definition of the momentum operator, (ii) the linearity of correspondence between physical observables and quantum Hermitian operators, (iii) the definition of conjugate coordinate-momentum variables in classical mechanics, and (iv) the invariance of the classical Hamiltonian to canonical transformations (transformations that change from one set of conjugate variables to another), we show that the $[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x},\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}]$ coordinate-momentum commutator must have the value $ih$ where $h$ is a real nonzero number (which we can identify with the experimentally determined $\ensuremath{\hbar}$). The results are then extended to include all generalized coordinates and their conjugate momenta as well as the Cartesian special-relativistic case.

Physical Review A•Oct 31, 2006•5 citations

Canonical Transformations, Quantization, Mutually Unbiased and Other Complete Bases

Donald Jack Kouri, Cameron L. Williams, Nikhil Pandya

Using ideas based on supersymmetric quantum mechanics, we design canonical transformations of the usual position and momentum to create generalized “Cartesian-like” positions, W, and momenta, Pw , with unit Poisson brackets. These are quantized by the usual replacement of the classical , x Px by quantum operators, leading to an infinite family of potential “operator observables”. However, all but one of the resulting operators are not Hermitian (formally self-adjoint) in the original position representation. Using either the chain rule or Dirac quantization, we show that the resulting operators are “quasi-Hermitian” relative to the x-representation and that all are Hermitian in the W-representation. Depending on how one treats the Jacobian of the canonical transformation in the expression for the classical momentum, Pw , quantization yields a) continuous mutually unbiased bases (MUB), b) orthogonal bases (with Dirac delta normalization), c) biorthogonal bases (with Dirac delta normalization), d) new W-harmonic oscillators yielding standard orthonormal bases (as functions of W) and associated coherent states and Wigner distributions. The MUB lead to W-generalized Fourier transform kernels whose eigenvectors are the W-harmonic oscillator eigenstates, with the spectrum (±1,±i) , as well as “W-linear chirps”. As expected, W, Pw satisfy the uncertainty product relation: ΔWΔPw ≥1/2 , h=1.

Applied Mathematics•Jan 1, 2017•1 citation

Quantum-Mechanically Correct Form of Hamiltonian Function for Conservative Systems

Boris Podolsky

Dirac showed that, if in the Hamiltonian $H$ momenta ${\ensuremath{\eta}}_{r}$ conjugate to the co-ordinates ${\ensuremath{\xi}}_{r}$ are replaced by $(\frac{h}{2\ensuremath{\pi}i})\frac{\ensuremath{\partial}}{\ensuremath{\partial}{\ensuremath{\xi}}_{r}}$, the Schr\"odinger equation appropriate to the coordinate system ${\ensuremath{\xi}}_{r}$ is $(H\ensuremath{-}E){\ensuremath{\psi}}_{\ensuremath{\xi}}=0$. Applied to coordinate systems other than cartesian this usually leads to incorrect results. The difficulty is here traced partially to the way in which ${\ensuremath{\psi}}_{\ensuremath{\xi}}$ is normalized and partly to the choice of $H$. In $H$ expressions such as $\mathrm{qp}{q}^{\ensuremath{-}1}p$ and ${p}^{2}$ are not equivalent, and the simplified form is generally incorrect. A formula satisfying all the requirements of quantum mechanics for a Hamiltonian of a conservative system, in an arbitrary coordinate system, is therefore developed $H=\frac{1}{2\ensuremath{\mu}}\ensuremath{\Sigma}\stackrel{r=n}{r=1}\ensuremath{\Sigma}\stackrel{s=n}{s=1}{g}^{\ensuremath{-}\frac{1}{4}}{p}_{r}{g}^{\frac{1}{2}}{g}^{\mathrm{rs}}{p}_{s}{g}^{\ensuremath{-}\frac{1}{4}}+U$ This formula is applied to a case of plane polar coordinates and leads to correct results.

Physical Review•Nov 1, 1928•460 citations

Some intricacies of the momentum operator in quantum mechanics

Utpal Roy, Suranjana Ghosh, Bhattacharya, Kaushik

In quantum mechanics textbooks the momentum operator is defined in the Cartesian coordinates and rarely the form of the momentum operator in spherical polar coordinates is discussed. Consequently one always generalizes the Cartesian prescription to other coordinates and falls in a trap. In this work we introduce the difficulties one faces when the question of the momentum operator in general curvilinear coordinates arises. We have tried to elucidate the points related to the definition of the momentum operator taking spherical polar coordinates as our specimen coordinate system and proposed an elementary method in which we can ascertain the form of the momentum operator in general coordinate systems.

arXiv:0706.0924•Jun 6, 2007•4 citations

The usual form P0 for the quantum-mechanical operator P conjugate to a generalized coordinate q1 is, in atomic units, P0=−ig−12 ∂/∂q1(g12),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 P=−i(A·grad+12 div 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 P=−i[g−12 ∂/∂q1(g12)+12 δ+(q1−α)]. 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.

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 (g½), 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 Pf(q)=(2π)−12g−12(q) ∫ −∞∞exp (ikq)kF(k) dk, α≤q≤β,where F(k)=(2π)−12 ∫ αβexp (−ikξ)f(ξ)g12(ξ) dξ, f∈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.

Based on (i) the spectral resolution definition of the momentum operator, (ii) the linearity of correspondence between physical observables and quantum Hermitian operators, (iii) the definition of conjugate coordinate-momentum variables in classical mechanics, and (iv) the invariance of the classical Hamiltonian to canonical transformations (transformations that change from one set of conjugate variables to another), we show that the $[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x},\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}]$ coordinate-momentum commutator must have the value

ih

where

h

is a real nonzero number (which we can identify with the experimentally determined $\ensuremath{\hbar}$). The results are then extended to include all generalized coordinates and their conjugate momenta as well as the Cartesian special-relativistic case.

Using ideas based on supersymmetric quantum mechanics, we design canonical transformations of the usual position and momentum to create generalized “Cartesian-like” positions, W, and momenta, Pw , with unit Poisson brackets. These are quantized by the usual replacement of the classical , x Px by quantum operators, leading to an infinite family of potential “operator observables”. However, all but one of the resulting operators are not Hermitian (formally self-adjoint) in the original position representation. Using either the chain rule or Dirac quantization, we show that the resulting operators are “quasi-Hermitian” relative to the x-representation and that all are Hermitian in the W-representation. Depending on how one treats the Jacobian of the canonical transformation in the expression for the classical momentum, Pw , quantization yields a) continuous mutually unbiased bases (MUB), b) orthogonal bases (with Dirac delta normalization), c) biorthogonal bases (with Dirac delta normalization), d) new W-harmonic oscillators yielding standard orthonormal bases (as functions of W) and associated coherent states and Wigner distributions. The MUB lead to W-generalized Fourier transform kernels whose eigenvectors are the W-harmonic oscillator eigenstates, with the spectrum (±1,±i) , as well as “W-linear chirps”. As expected, W, Pw satisfy the uncertainty product relation: ΔWΔPw ≥1/2 , h=1.

Dirac showed that, if in the Hamiltonian

H

momenta ${\ensuremath{\eta}}_{r}$ conjugate to the co-ordinates ${\ensuremath{\xi}}_{r}$ are replaced by $(\frac{h}{2\ensuremath{\pi}i})\frac{\ensuremath{\partial}}{\ensuremath{\partial}{\ensuremath{\xi}}_{r}}$, the Schr\"odinger equation appropriate to the coordinate system ${\ensuremath{\xi}}_{r}$ is $(H\ensuremath{-}E){\ensuremath{\psi}}_{\ensuremath{\xi}}=0$. Applied to coordinate systems other than cartesian this usually leads to incorrect results. The difficulty is here traced partially to the way in which ${\ensuremath{\psi}}_{\ensuremath{\xi}}$ is normalized and partly to the choice of

H

. In

H

expressions such as $\mathrm{qp}{q}^{\ensuremath{-}1}p$ and

{p}^{2}

are not equivalent, and the simplified form is generally incorrect. A formula satisfying all the requirements of quantum mechanics for a Hamiltonian of a conservative system, in an arbitrary coordinate system, is therefore developed $H=\frac{1}{2\ensuremath{\mu}}\ensuremath{\Sigma}\stackrel{r=n}{r=1}\ensuremath{\Sigma}\stackrel{s=n}{s=1}{g}^{\ensuremath{-}\frac{1}{4}}{p}_{r}{g}^{\frac{1}{2}}{g}^{\mathrm{rs}}{p}_{s}{g}^{\ensuremath{-}\frac{1}{4}}+U$ This formula is applied to a case of plane polar coordinates and leads to correct results.

In quantum mechanics textbooks the momentum operator is defined in the Cartesian coordinates and rarely the form of the momentum operator in spherical polar coordinates is discussed. Consequently one always generalizes the Cartesian prescription to other coordinates and falls in a trap. In this work we introduce the difficulties one faces when the question of the momentum operator in general curvilinear coordinates arises. We have tried to elucidate the points related to the definition of the momentum operator taking spherical polar coordinates as our specimen coordinate system and proposed an elementary method in which we can ascertain the form of the momentum operator in general coordinate systems.