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.
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.
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.
General remarks As already mentioned in the Introduction, the exact solution of the main equation of quantum mechanics – the Schrödinger equation – lies beyond the potentialities of modern mathematics and computer technology. But a number of important inferences about the behaviour, structure and properties of a given quantum-mechanical many-particle system can be drawn without solving this equation, just by examining its symmetry properties. Symmetry concepts come about in physics in two ways. First, since any physical process occurs in a real space, we have to make use of one or another coordinate system. The isotropy and homogeneity of space make physically meaningful only those mathematical relationships that remain unchanged under rotations of the axes of the coordinate system, and that impose fairly rigorous constraints on the possible physical laws. Second, every physical object and process features a symmetry which should be taken into account by the physical theory. A common idea underlying particular forms of symmetry is the invariance of a system under a certain set (group) of transformations. The normally considered forms of symmetry are rotational symmetry, which is based on the equivalence of all directions in space, and permutation symmetry, which is caused by identical particles. The operations of the geometrical symmetry group are responsible for appropriate conservation laws. So, the rotational symmetry of a closed system gives rise to the law of conservation of angular momentum. The group of rotations of a three-dimensional space stands apart in atomic spectroscopy. This is mostly due to the high accuracy of the central field approximation, on which the entire modern theory of complex atoms and ions is based.
In quantum mechanics, as well as in classical mechanics, the number of variables involved in the equations of motion may be reduced by utilizing the constants of motion of the system. For a set of N particles moving in a field-free space there are six integrals if N is greater than two. These correspond to the laws of conservation of linear and of angular momentum. It is the purpose of this paper to eliminate the coordinates of rotation and of translation of the total system from the Schrodinger equation and to obtain a new set of differential equations involving only 3N-6 variables.1
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.
Currently, dynamics of a massive macroparticle is given by classical analytical mechanics (CM), while that of a massive micro one is given by quantum mechanics (QM). We propose a mechanics effective for both: We transform, under coordinate transformation, the covariant tensor of order two underlying the kinetic energy term of the Hamilton-Jacobi (H-J) eq. of CM, not with the action of the diffeomorphism group, but with an action of an extended diffeomorphism group. Then, the H-J eq., a first-order partial differential eq., is modified to a third-order one. The Euler-Lagrange eq. of CM, a second-order ordinary differential eq., related to the H-J eq. through action integral of the least action principle is accordingly modified to a fourth-order one. We thus obtain a mechanics which, because of the higher-order eqs., accommodates phenomena corresponding to quantum phenomena. Energy of a particle in a confining potential is quantized to the same values as those of QM. Particle distribution in an ensemble disagrees with that of QM, but agrees with experiments within experimental uncertainty. The mechanics therefore is a testable alternative to QM.
Abstract In order to extend the use of group theoretical arguments to the problem of accidental degeneracy in quantum mechanics, a new type of constant of the motion, known as a conditional constant of the motion, is introduced. Such a quantity, instead of commuting with the Hamiltonian H for the system, satisfies the more general relation HA = A†H, where A† denotes the hermitian conjugate (adjoint) of the conditional constant of the motion A. This expression reduces, if A is hermitian, to the usual definition of a constant of the motion. Otherwise it defines a new type of invariance, and it is this which will be referred to as conditional invariance. A discussion of the difficulties arising from the lack of hermiticity of A, which is of course essential to its definition, is given. In particular it is shown, under fairly general conditions, that the process of introducing a variable parameter in the Hamiltonian enabling it to have simultaneous eigenfunctions with A, gives rise to an eigenvalue equation in this parameter with respect to which A may be chosen to be hermitian. Conditional invariance is contrasted with both dynamical and geometric invariance. It is found to be sometimes replaceable by either of the latter forms of invariance and for such, explicit conditions are given. Some applications of conditional invariance are discussed. These include a study of the crossing of potential energy curves, a new model of symmetry breaking, a possible means of calculating the exact number of bound states for certain potentials and conditions for the existence of bound states near to the continuum.