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Physical Science International Journal

Simple Generalization of the Quantum Mechanical Virial Theorem for Nonrelativistic Systems with Rotational Symmetry

Domagoj Kuić

Published Jan 10, 2016• 0 citations
Publisher

Abstract

In this paper we show that the generalization of the virial theorem can be achieved for nonrelativistic quantum mechanical systems under the conditions of rotational symmetry and the constancy of the trace of moment of inertia tensor. Under these conditions the matrix elements of the commutator of the generator of dilations G and the Hamiltonian H are equal to zero on the subspace of the Hilbert space generated by the simultaneous eigenvectors of the particular maximal set of commuting self-adjoint operators which contains H, J2, Jz, the trace of the moment of inertia tensor TrI and additional operators. The result obtained is relevant for an important class of N-particle nonrelativistic quantum mechanical systems.

DOI
10.9734/psij/2016/27935

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In this paper we show that the generalization of the virial theorem can be achieved for nonrelativistic quantum mechanical systems under the conditions of rotational symmetry and the constancy of the trace of moment of inertia tensor. Under these conditions the matrix elements of the commutator of the generator of dilations G and the Hamiltonian H are equal to zero on the subspace of the Hilbert space generated by the simultaneous eigenvectors of the particular maximal set of commuting self-adjoint operators which contains H, J2, Jz, the trace of the moment of inertia tensor TrI and additional operators. The result obtained is relevant for an important class of N-particle nonrelativistic quantum mechanical systems.
In this second paper in a series, we show that the the general statistical approach to nonrelativistic quantum mechanics developed in the first paper yields a representation of quantum spin and magnetic moments based on classical nonrelativistic spinning top models, using Euler angle coordinates. The models allow half-odd-integer spin and predict supraluminal speeds only for electrons and other leptons, which must be treated relativistically. The spin operators in the space-fixed frame satisfy the usual commutation rules, while those in the rotating body-fixed frame satisfy "left-handed" rules. The commutation rules are independent of the structure of the top, so all nonrelativistic rigidly rotating objects must have integer or odd-half-integer spin. Physical boundary conditions restrict all mixed spin states to involve only half-odd-integer or only integer spin eigenstates. For spin 1/2, the theory automatically yields a modified Pauli-Schrödinger equation. The Hamiltonian operator in this equation contains a rigid rotator term and a term involving the square of the magmetic field, as well as an interaction term having the usual form in spherically symmetric and some cylindrically symmetric models, valid for any magnetogyric ratio.
Properties of quantum-mechanical time-dependent correlation functions of pairs of irreducible tensor operators are derived from the transformation properties of the tensor operators under rotations, the assumed rotational invariance of the Hamiltonian of the system, and the Wigner-Eckart theorem. It is shown that many of the possible correlation functions are zero, and that the nonzero correlation functions can be expressed in terms of reduced matrix elements. Some examples of irreducible tensor operators which occur in applications are given, and their reduced matrix elements are calculated.
We prove that any nonrelativistic classical system must obey a statistical wave equation that is exactly the same as the Schrödinger equation for the system, including the usual “canonical quantization” and Hamiltonian operator, provided an unknown constant is set equal to . We show why the two equations must have exactly the same sets of solutions, whereby this classical statistical theory (CST) and nonrelativistic quantum mechanics may differ only in their interpretations of the same quantitative results. We identify some of the different interpretations. We show that the results also imply nonrelativistic Lagrangian classical mechanics and the associated Newtonian laws of motion. We prove that the CST applied to a nonrelativistic rigid rotator yields spin angular momentum operators that obey the quantum commutation rules and allow both integer and half-odd-integer spin. We also note that the CST applied to systems of identical massive particles is mathematically equivalent to nonrelativistic quantum field theory for those particles.
Generalisations of the virial theorm in Classical Mechanics and Quantum Mechanics are examined. It is shown that the generalised virial theorem in Quantum Mechanics leads to certain relations between matrix elements. The differences between the generalisations in Classical and Quantum Mechanics are identified. Some results arising from the radial Schrödinger equation in Quantum Mechanics are discussed. It is also shown that the generalisations of the virial theorem may be extended to arbitrary number of dimensions.
Programm (1) Classical mechanics: Principle of the least action. Lagrangian mechanics. Symmetries and Noether’s theorem. The Kepler problem. The Legendre transform. Hamiltonian mechanics. The classical action. Poisson brackets. Canonical transformations and generating functions. Symplectic manifolds. Hamilton’s and Liouville’s representations. [1][Chapter 1] (2) Hilbert spaces: Basic denitions. Bounded operators. Compact operators. The spectral theorem for self-adjoint compact operators. Trace-class and HilbertSchmidt operators. Closed operators. Symmetric and self-adjoint operators. Resolvent and spectrum. The spectrum of self-adjoint operators. The spectral theorem for (bounded and unbounded) self-adjoint operators. Stone’s theorem. [2, 3] (3) Basic principles of quantum mechanics: Observables, states, and dy
We prove explicitly that to every discrete, semibounded Hamiltonian with constant degeneracy and with finite sum of the squares of the reciprocal of its eigenvalues and whose eigenvectors span the entire Hilbert space there exists a characteristic self-adjoint time operator which is canonically conjugate to the Hamiltonian in a dense subspace of the Hilbert space. Moreover, we show that each characteristic time operator generates an uncountable class of self-adjoint operators canonically conjugate with the same Hamiltonian in the same dense subspace. Keywords: Time operators; Quantum canonical pairs; Pauli’s theorem 1.