1. INTRODUCTION In this chapter we shall apply the method of invariant imbedding to two types of problems that occur often in quantum mechanical studies. The first general class is frequently referred to as the “phase shift” problem, while the second is a particular form of the eigenvalue problem. In this latter case the method of the previous chapter is, within certain technical limits, applicable. However, the way in which we shall now make use of the imbedding scheme gives rise to a completely different characterization of the eigenvalues than the one we have been discussing. It seems particularly desirable to include a presentation of this method because the results illustrate so strikingly the way in which different kinds of imbeddings can yield completely different formulas. Although the phase shift problem is of less interest in itself, since our results can be derived in a rather more elementary way, we include it since once again, it demonstrates the versality of the method (see [1,2]).
INTRODUCTION In this chapter we shall apply the method of invariant imbedding to two types of problems that occur often in quantum mechanical studies. The first general class is frequently referred to as the “phase shift” problem, while the second is a particular form of the eigenvalue problem. In this latter case the method of the previous chapter is, within certain technical limits, applicable. However, the way in which we shall now make use of the imbedding scheme gives rise to a completely different characterization of the eigenvalues than the one we have been discussing. It seems particularly desirable to include a presentation of this method because the results illustrate so strikingly the way in which different kinds of imbeddings can yield completely different formulas. Although the phase shift problem is of less interest in itself, since our results can be derived in a rather more elementary way, we include it since once again, it demonstrates the versality of the method (see [1,2]).
In the previous chapter, the postulates helped us to use the wave function to find the expected results of experimental measurements by letting a dynamical quantity F (x, p, t) → F (xˆ, pˆ, t) and integrating over space. In the examples, however, the wave functions were typically given and not derived from first principles. We now endeavor to find the appropriate wave functions for a variety of physical systems in one dimension by solving the Schro¨dinger equation where for time-independent cases, the total energy is an eigenvalue. In subsequent chapters, we will extend the method to three dimensions and develop the appropriate operators, eigenvalues, and eigenfunctions for angular momentum. We will also develop methods for examining time-dependent systems. The latter chapters address special systems of interest that may be referred to as applications of quantum mechanics.
In the previous chapters, we have shown how to solve the Schrodinger equation for several systems as a particle in an infinite and finite potential well, scattering, harmonic oscillator, and hydrogen atom. For all these systems, we were able to solve Schrodinger's equation exactly. However, there are complicated potentials for which Schrodinger's equation cannot be solved exactly. For such situations, we resort to approximation methods for finding approximate solutions to the Schrodinger equation. In this chapter, we discuss several approximate methods for solving Schrodinger's equation.
Addendum: Dec. 15, 2020. The title of the first reference is: Phase Shift of exp[i delta(x)] for Bound Quantum Systems In classical mechanics, interactions follow Newton’s second law which is equivalent to a conservation of energy equation with a potential V(x) for conservative forces. The Schrodinger equation, we argue, allows for two kinds of interactions. The first is like the classical situation and involves average kinetic energy as part of a classical conservation equation with V(x). This is also associated with a (p rms) i.e. root mean square momentum, which is identical to classical momentum. There exists, however, also = Sum over p p P(p/x) with P(p/x)=a(p)exp(ipx)/W(x) ( W(x) is the wavefunction). This momentum is responsible for dW/dx and in turn the change in spatial density W(x)W(x) i.e. quantum humps. , like prms follows from interactions of the particle with V(x) in a bound system and possibly other reasons, but does not follow Newton’s law. In this note, we consider the more general case of which appears in the Aharonov-Bohm problem. It seems one may form a generalized = A(x) + i.e. calculate various pieces of which govern the spatial density, i.e. interference, without necessarily contributing to KEave(x) in KEave(x)+V(x)=E ((1)). In particular, A(x), the magnetic vector potential behaves like a portion of and is involved in interference, but plays no role in ((1)). We suggest two different kinds of interactions may occur in a quantum system. The first is the usual Newton type interaction applied to averages, but a second involves interactions that create which does not behave according to Newtonian mechanics. For a bound state problem with V(x), both types of interactions exist, thus one has two “momenta”, p rms and , with only the latter obeying Newtonian mechanics. For the time-independent Schrodinger equation, Newtonian conservation of energy applies only to average values such as KEave(x) and V(x) which is itself an average, we argue. Given that there exists microscopic motion and a distribution a(p)exp(ipx), these may give rise to non-Newtonian behaviour and interactions other than the usual -dV/dx in ((1)).
Abstract The generalized Hellmann–Feynman and curvature theorems are derived for the generalized Sturm–Liouville‐type eigenvalue equation which is widely encountered in many branches of physics. In classical mechanics, the Sturm–Liouville eigenvalue equation primarily arises from applying the theory of small oscillations to vibration problems while, in quantum mechanics, the most important example is the time‐independent Schrödinger equation. The generalized theorems, respectively, provide simple useful expressions for the first and second λ derivatives of the eigenvalue where λ is an arbitrary real parameter appearing in the eigenvalue equation. These results, which apply to both classical and quantal systems, include as special cases the well‐known quantum mechanical Hellmann–Feynman and curvature theorems for the time‐independent Schrödinger equation, thus unifying the formalism. This connection between classical and quantum mechanics is of interest per se, and also because classical eigenvalue equations find application in the treatment of some quantum chemical problems. Two derivations of both generalizations are presented, the first proof applying to exact and Rayleigh–Ritz (linear variational) solutions of the eigenvalue problem, and the second, more general one, to arbitrary optimal variational as well as exact solutions. Necessary and sufficient conditions for the validity of the results are discussed. To demonstrate the insight they afford, several applications of the theorems in conjunction with perturbation theory are described.
This study delves into the realm of Schrödinger equations and wave mechanics, aiming to see if they could provide a foundation comparable in generality to that formulated in the preceding chapter, and to bridge the gap between classical and quantum physics. It begins by deriving a family of Schrödinger-like equations. The study introduces an intuitive approach for deriving an entire spectrum of Schrödinger-like equations, encompassing classical and novel variants, through the utilization of the complex wave function and its conjugate, along with the assumptionsA01, A08. Examining the concepts of the complex wave function ψ and the multi-vector wave function Ψ reveals their equal generality, suggesting the applicability of Schrödinger-like equations in the context of multi-vectors, including both ψ and the phase Φ . To explore the correspondence principle between quantum and classical mechanics, a well-studied example is dissected in Section 2.4, necessitating the development of a suitable wave function (2.50). The use of the real function [1, (1.50)] to form the complex test function (2.50) is a logical step, and its substitution into the Schrödinger equations aims to yield an identity. The substitution of (2.50) reveals a discrepancy between the left and right sides of the Schrödinger equations (2.25, 2.26), highlighting their inadequacy in describing the classical mechanics example accurately. This observation underscores the limitations of classical Schrödinger equations and their departure from the correspondence principle. Consequently, a more comprehensive set of Schrödinger-like equations (2.58-2.68) is derived, rendering the previously considered equations (2.25, 2.26) irrelevant. Substituting the test function (2.50) into equations (2.58-2.68) demonstrates their soundness and suitability for the example, establishing the correspondence principle between classical and quantum mechanics. Our analysis indicates that the classical set [1, (1.4-1.6)] exhibits a higher degree of generality than the quantum one (2.58-2.68), thereby deferring the definitive validation of this assertion to forthcoming research endeavours. These findings underscore the potential significance of our framework in bridging the gap between quantum and classical mechanics, promising further exploration and validation in future studies.