The use of variational principles as a calculational tool is reviewed, with special emphasis on methods for constructing such principles. In particular, it is shown that for a very wide class of problems it is possible to construct a variational principle (VP) for just about any given quantity
Q of interest, by routine procedures which do not require the exercise of ingenuity; the resultant VP will yield an estimate of
Q correct to second order whenever the quantities appearing in the VP are known to first order. The only significant requirement for application of the routine procedures is that the entities which enter into the definition of
Q be uniquely specified by a given set of equations; the equations may involve difference or differential or integral operators, they may be homogeneous or inhomogeneous, linear or nonlinear, self-adjoint or not, and they may or may not represent time-reversible systems. No numerical calculations are presented, but procedures for the construction of VP's are illustrated for numerous quantities
Q of physical interest, particularly those
Q arising in quantum-mechanical scattering and transition probability calculations. For pedagogical purposes VP's are also derived for several problems in classical and (simple) mathematical physics which the authors hope will prove instructive and perhaps even amusing. The quantum-mechanical quantities
Q whose VP's are examined include various matrix elements and the quantum-mechanical eigenfunctions themselves. Topics examined include some points which have not always been appreciated in the literature, such as the necessity for properly specifying the phase when complex eigenfunctions are involved, and the importance of avoiding, wherever possible, formulations requiring the inversion of singular operators. The basic element of the technique is the recognition that the defining equations of a system can be incorporated into the VP as constraints through the use of (generalized) Lagrange multipliers; these can be constants, scalar, vector, or tensor functions of one or more variables, operators, etc. In the typical problem, these Lagrange multipliers
L serve as a new set of adjoint functions or entities, and the construction of the VP simultaneously provides well-defined equations for the
L. Moreover, these
L often have ready physical significance themselves; for example, they often may be regarded as generalized Green's functions. The construction of the VP also readily yields so-called variational identities for the quantity
Q of interest; these identities give explicit (if formal) expressions for the error in the variational estimate of
Q. In some cases this error can be shown to have a definite sign, so that the VP actually is an extremum principle, that is, that it yields an upper or lower variational bound for
Q; however, our routine procedures for constructing VP's do not routinely yield extremum principles.
For many years we have advocated a form of quantum mechanics based on the application of sum rule methods (completeness) to the equations of motion and to the commutation relations, i.e., to Heisenberg matrix mechanics. Sporadically we have discussed or alluded to a variational foundation for this method. In this paper we present a series of variational principles applicable to a range of systems from one-dimensional quantum mechanics to quantum fields. The common thread is that the stationary quantity is the trace of the Hamiltonian over Hilbert space (or over a subspace of interest in an approximation) expressed as a functional of matrix elements of the elementary operators of the theory. These parameters are constrained by the kinematical relations of the theory introduced by the method of Lagrange multipliers. For the field theories, variational principles in which matrix elements of the density operators are chosen as fundamental are also developed. A qualitative discussion of applications is presented.
The variation principle has not been well defined in quantum mechanics for a Lagrangian containing a velocity dependent potential. We investigate the method of variation to derive an Euler-Lagrange equation in connection with Noether's theorem when the Lagrangian is invariant under some point transformation. Obtained are the Euler-Lagrange equation retaining the above invariance by means of the variation method and the Hamiltonian with which the canonical equation of motion leads to the Euler-Lagrange equation.
The events observable at the atomic scale cannot be adequately explained by classical physics, hence a new conceptual framework for physics must be developed. "Quantum mechanics" is the name given to this new theory of the physical cosmos. The basic formula of relativistic quantum mechanics is the problem of Klein-Gordon. In relativistic quantum physics, the wave function is crucial since it contains all the data describing a quantum system, in relativity quantum physics, they are crucial. How to solve Klein Gordon's equation has been the topic of intense debate in recent years. To determine the system's spectrum using numerous potentials and methodologies. To obtain the Klein-Gordon issues solution Through determining the harmonic and perturbed Coulomb potentials' energies, The recent study employed an entirely distinct methodology. using the Semi-Inverse Variational Technique to aid in the process, we were able to identify the eigen energies for Harmonic- and Perturbed Coulomb Potentials for a variety of quantum numbers. A further benefit of the technique was that it enabled us to obtain the wave eigenfunctions for various potentials, demonstrating the viability of this approach. We are excited to deal with additional challenges in our upcoming works, the first being the use of nonlinear potentials to solve the Dirac equation due to the effectiveness of this approach.
We review the development and practical uses of a generalized Maupertuis least action principle in classical mechanics in which the action is varied under the constraint of fixed mean energy for the trial trajectory. The original Maupertuis (Euler–Lagrange) principle constrains the energy at every point along the trajectory. The generalized Maupertuis principle is equivalent to Hamilton's principle. Reciprocal principles are also derived for both the generalized Maupertuis and the Hamilton principles. The reciprocal Maupertuis principle is the classical limit of Schrödinger's variational principle of wave mechanics and is also very useful to solve practical problems in both classical and semiclassical mechanics, in complete analogy with the quantum Rayleigh–Ritz method. Classical, semiclassical and quantum variational calculations are carried out for a number of systems, and the results are compared. Pedagogical as well as research problems are used as examples, which include nonconservative as well as relativistic systems. '... the most beautiful and important discovery of Mechanics.' Lagrange to Maupertuis (November 1756)