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GENERALIZED MOMENTUM OPERATORS IN QUANTUM MECHANICS. Series 3

Peter D. Robinson, Joseph O. Hirschfelder

Published Apr 16, 1962• 0 citations
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Abstract

>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.)

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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.

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Physical Review A•Oct 31, 2006•5 citations
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.)
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.
The orthogonality requirement usually specified for the eigenfunctions φ of a quantum-mechanical operator P which has a continuous spectrum of eigenvalues λ, is ∫φ*(λ)φ(λ′) dλ=δ(λ−λ′). The derivation of this result is examined; it is seen to be merely a sufficient, but not a necessary condition for the consistent expansion of a bound-state wavefunction ψ in the form ∫ a(λ)φ(λ) dλ. The situation when P is a generalized momentum operator is investigated in detail. Such operators were discussed in the preceding paper. It is shown that any two of the corresponding momentum eigenfunctions are not usually orthogonal. Nevertheless, with the help of Fourier analysis, a consistent expansion ψ=∫ a(λ)φ(λ) dλ is established. The theory is illustrated with the familiar examples of a ``particle in a box'' and a ground-state hydrogen atom. When a space coordinate has a finite range, a quantum condition can be imposed on the generalized momentum eigenvalues so that the momentum eigenfunctions form a complete discrete orthogonal set. We attempt to justify the belief that such quantization is not essential unless it is necessary to ensure single-valued eigenfunctions.
Since the conformation of physical systems is often advantageously described with the help of generalized (i.e. curvilinear) coordinates, the following questions are raised and general answers given to them: (i) What are the components of the momentum operators, those of the adjoint momentum operators and those of the hermitian momentum operators? (ii) What is the general form of the hamiltonian operator? (iii) What do the general forms of the momentum and hamiltonian operators become when expressed in terms of quasi-momentum operators (e.g. angular momentum component operators)? (iv) How are the answers to the above questions affected by an arbitrary choice of normalization convention for the total wave-function? A complete set of formulae (whatever the independent choices of coordinates, quasi-momentum operators and normalization) is given and various ‘historical’ formulae are shown to be particular instances of the general formulae proposed.
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 ihihih where hhh 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.