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American Journal of Physics

A Quantum Equation of Motion with Higher Derivatives

Marius Borneas

Published Feb 1, 1972• 9 citations
Publisher

Abstract

The problem of how to obtain a quantum equation of motion with higher derivatives is discussed. First the basic ideas of generalized classical mechanics with higher derivatives are reviewed. The generalized canonical conjugate momenta and a function N, corresponding to the Hamiltonian function, are constructed. Then certain reduced momenta are introduced, which in absence of higher derivatives are identical to the canonical conjugate momenta. The function N is expressed in terms of the reduced momenta. In order to obtain a quantum equation of motion, the reduced momenta in N are replaced by operators fulfilling some given conditions; thus the corresponding operator N̂ can be identified with the Hamiltonian operator. With its help, and introducing a wave function, the quantum equation of motion with higher derivatives is obtained.

DOI
10.1119/1.1986500

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The problem of how to obtain a quantum equation of motion with higher derivatives is discussed. First the basic ideas of generalized classical mechanics with higher derivatives are reviewed. The generalized canonical conjugate momenta and a function N, corresponding to the Hamiltonian function, are constructed. Then certain reduced momenta are introduced, which in absence of higher derivatives are identical to the canonical conjugate momenta. The function N is expressed in terms of the reduced momenta. In order to obtain a quantum equation of motion, the reduced momenta in N are replaced by operators fulfilling some given conditions; thus the corresponding operator N̂ can be identified with the Hamiltonian operator. With its help, and introducing a wave function, the quantum equation of motion with higher derivatives is obtained.
A generalized canonical form of action of dynamic theories with higher derivatives is proposed, which does not require the introduction of additional dynamic variables. This form is the initial point for the construction of quantum theory, in which the state of motion of the system is described by the wave functional on its trajectories in the configuration space, and the functional itself is an eigenvector of the action operator. The Pais-Uhlenbeck oscillator is considered as the simplest model. For this case, the correspondence between the new form of quantum theory and "ordinary" quantum mechanics has been established in the local limit, when the Pais-Uhlenbeck oscillator is reduced to the "ordinary" harmonic oscillator.
A general theory of quantum mechanical constants of motion of arbitrary order in the momenta is developed. Because of rotational invariance the theory assumes an elegant tensorial form. The fundamental equations for the nth-order constants of motion in m variables are set up in Cartesian coordinates. The first two equations of the set do not contain the potential function and their solutions are shown to be polynomials of degree n and n-1 with coefficients satisfying certain symmetry relations. The next two equations contain the first and the second derivatives of the potential function. From these equations all unknown functions are eliminated in the special cases n=3 and n=4. The elimination gives partial differential equations for the potential function, of second and third order for n=3, and third and fourth order for n=4. The possibility of extending the results to higher values of n is indicated.
The solution of the quantum dynamical equation $\frac{i\ensuremath{\hbar}\mathrm{dT}}{\mathrm{dt}}=HT$, for the time-displacement operator TTT, is given, when the Hamiltonian HHH is a polynomial of the second degree in canonically conjugate variables, with arbitrary time-dependent coefficients. Heisenberg's equations of motion are then solved, and the general integral of Schr\"odinger's equation in coordinate space is expressed by the Green's function corresponding to TTT. An example is given.
Universal quantum integrals of motion are introduced, and their relation with the universal quantum invariants is established. The invariants concerned are certain combinations of the second- and higher-order moments (variances) of quantum-mechanical operators, which are preserved in time independently of the concrete form of the coefficients of the Schrödinger equation, provided the Hamiltonian is either a generic quadratic form of the coordinate and momenta operators, or a linear combination of generators of some finite-dimensional algebra (in particular, any semisimple Lie algebra). Using the phase space representation of quantum mechanics in terms of the Wigner function, the relations between the quantum invariants and the classical universal integral invariants by Poincaré and Cartan are elucidated. Examples of the `universal invariant solutions' of the Schrödinger equation, i.e. self-consistent eigenstates of the universal integrals of motion, are given. Applications to the physics of optical and particle beams are discussed.
A quantum mechanical model with one bosonic degree of freedom is discussed in detail. Conventionally, when a quantum mechanical model is constructed, one must know the corresponding classical model. And by applying the correspondence between the classical Poisson brackets and the canonical commutator, the canonical quantization condition can be obtained. In the quantum model, study of the corresponding classical model is needed first. In this model, the Lagrangian is an operator gauge invariant. After localization, in order to keep gauge invariance, the operator gauge potential must be introduced. The Eular-Lagrange equation of motion of the dynamical argument gives the usual operator equation of motion. And the operator gauge potential just gjves a constraint. This constraint is just the usual canonical quantization condition.