Abstract This chapter introduces Schrödinger’s equation and the action integral from which the differential equation is derived by the principle of stationary action. This principle is central to almost all of physics. The discretization of the integral with finite elements and interpolation with polynomials scaled by interpolation coefficients at the ends of the elements (nodes) allows us to integrate out the spatial dependence leaving the action as a binomial expression in the coefficients. Now the action is varied with respect to these coefficients to obtain a discretized matrix version of the Schrödinger’s equation. This typically leads to matrix equation or a generalized eigenvalue problem to solve. In simple terms, this is the essence of the finite element method. The boundary conditions for the wavefunction and the derivative of the wavefunction can be very easily imposed at the nodes prior to the solution of the matrix equations. The probability current has to be continuous at interfaces in the interior of the physical domain. The form of the current is obtained by a gauge-variational method in an unambiguous manner. Going from the differential equation to the action integral, and vice versa, are treated in great detail so that the reader can gain confidence in working with variational principles.
Abstract This chapter introduces Schrödinger’s equation and the action integral from which the differential equation is derived by the principle of stationary action. This principle is central to almost all of physics. The discretization of the integral with finite elements and interpolation with polynomials scaled by interpolation coefficients at the ends of the elements (nodes) allows us to integrate out the spatial dependence leaving the action as a binomial expression in the coefficients. Now the action is varied with respect to these coefficients to obtain a discretized matrix version of the Schrödinger’s equation. This typically leads to matrix equation or a generalized eigenvalue problem to solve. In simple terms, this is the essence of the finite element method. The boundary conditions for the wavefunction and the derivative of the wavefunction can be very easily imposed at the nodes prior to the solution of the matrix equations. The probability current has to be continuous at interfaces in the interior of the physical domain. The form of the current is obtained by a gauge-variational method in an unambiguous manner. Going from the differential equation to the action integral, and vice versa, are treated in great detail so that the reader can gain confidence in working with variational principles.
This chapter focuses on widely used approximate method: the numerical integration of the Schrodinger equation, restricting ourselves to one dimension for simplicity. There is a plethora of approaches that yield highly accurate numerical eigenvalues and eigenfunctions. In quantum mechanics one sometimes seeks after bound stationary states that satisfy given physical boundary conditions. Boundary conditions at infinity play a relevant role in quantum mechanics, in which case one looks for bound states that vanish asymptotically at the end points of coordinate interval. Solving the propagation matrix equation exactly for every coordinate pair in the physical interval is as difficult as solving the original differential equation. However, sometimes the former is preferable to develop approximate numerical algorithms for the accurate calculation of eigenvalues and eigenfunctions. The perturbation method based on the Taylor expansion only requires derivatives of the potential energy function, which one obtains easily in most cases and appears to be preferable for the systematic improvement of the approximate propagation matrix.
Abstract Chapter 4 introduces the Schwinger Action Principle, along with associated particle and potential sources. While the methods described here originally arose in the relativistic quantum field theory of elementary particle physics, they have also profoundly advanced our understanding of non-relativistic many-particle physics. The Schwinger Action Principle is a quantum-mechanical variational principle that closely parallels the Hamilton Principle of Least Action of classical mechanics, generalizing it to include the role of quantum operators as generalized coordinates and momenta. As such, it unifies all aspects of quantum theory, incorporating Hamilton equations of motion for those operators and the Heisenberg equation, as well as producing the canonical equal-time commutation/anticommutation relations. It yields dynamical coupled field equations for the creation and annihilation operators of the interacting many-body system by variational differentiation of the Hamiltonian with respect to the field operators. Also, equations for the development of matrix elements (underlying Green’s functions) are derived using variations with respect to particle and potential “sources” (and coupling strength). Variational calculus, involving impressed potentials, c-number coordinates and fields, also quantum operator coordinates and fields, is discussed in full detail. Attention is given to the introduction of fermion and boson particle sources and their use in variational calculus.
The sixth chapter begins with the equation of wave motion in one dimension and arrived at the three-dimensional Laplacian operator. After this, time-independent and time-dependent Schrödinger’s equations have been derived in a simple way. It also deals with the interpretation of wave function and mentions the conditions of acceptable wave functions taking into consideration the graphical presentation. This chapter also gives idea about the condition of normalisation and orthogonality and determination of normalisation constant. The condition of orthogonality has also been discussed in detail, which gives an important point regarding how two wave functions become orthogonal. It also speaks about orthonormality. The ideas of Eigen function and Eigen value have also been introduced, and degeneracy has been defined. This chapter also incorporates the transformation of Laplacian operator into spherical polar coordinates in a graspable manner. The Ehrenfest theorem and its proof have also been included. The chapter also provides the idea of matrix representation of wave function, matrix representation of operator, properties of matrix elements, and matrix form of time-dependent Schrödinger equations. At the end of the chapter, adequate references, solved problems, and questions on concepts have also been cited.
Abstract A new approach to Quantum Scattering using the Theory of Sources and Absorbers is developed in these chapters. Existing theories of quantum scattering usually calculate the scattered wavefunction and the scattering cross-section by obtaining the solution to Schrödinger’s equation and ignoring the possibility of representing the problem in terms of the principle of stationary action. The action integral approach has several advantages, including quadratic convergence to the accurate results in the variational approach; also, the symmetry of the Lagrangian is readily manifest. Furthermore, the “near-field” wavefunctions and evanescent wave contributions are invariably ignored. Computational methods using discretization of space require the equivalent of “radiation boundary conditions” at large distances, and these are not cleanly implementable.
The generalization of the variation of the action-integral operator introduced by Schwinger in the derivation of the principle of stationary action enables one to use this principle to obtain a description of the quantum mechanics of an open system. It is shown that augmenting the Lagrange-function operator by the divergence of the gradient of the density operator, a process which leaves the equations of motion unaltered, leads to a class of generators whose associated infinitesimal transformations yield variations of the action-integral operator for an open system, similar in form and content to those obtained for the total, isolated system. The augmented Lagrange-function operator and the associated action-integral operator are termed proper operators, since only their variation yields equations of motion for the observables of an open system, in agreement with the expressions obtained from the field equations. Modifying the generator in this manner is shown to be equivalent to requiring that the open system be bounded by a surface through which there is a local zero flux in the gradient vector field of the electron density. Only the observables of such properly defined open systems are described by the correct equations of motion. The physical significance of such proper open systems is discussed.
In (1), a method for finding the propagator of a generalized quantum oscillator directly from the time-dependent Schrodinger equation is presented. This method assumes the form Q(t)exp[i (B(t)xx + A(t)yy + C(t)xy) ] which is inserted into the Schrodinger equation. The equation is broken into purely imaginary and real parts. Time dependent coefficients of xx, xy and yy are found leading to four differential equations which may be solved to ultimately yield a full solution for the propagator. In (2), it is suggested that the propagator is given by Q(t) exp(i Integral (t=0 to Tf) Lc dt1)). In order to use this approach, a solution to the classical Lagrangian Lc is found and x written as: XA(Tf-t)+YB(t-0) where X and Y are the initial and final x positions (and are numbers). Thus, dx/dt may be computed, and the classical action integral (t=0 to Tf) Lc dt) found through integration. In this note, we consider the quantum oscillator and try to demonstrate the two approaches are the same. A key point is that -id/dt (partial) in the Schrodinger equation really represents -i d/dTf (partial).