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arXiv:2008.01831 · Journal of Mathematical Physics

Unitary, continuum, stationary perturbation theory for the radial Schrödinger equation

Scott E. Hoffmann

Published Mar 1, 2021• 1 citation
arXivHTMLPDF

Abstract

In nonrelativistic quantum mechanics, we require that the free and interacting Hamiltonians be related by a unitary transformation, as has been done by other authors. We then derive a stationary perturbation theory for the radial Schrödinger equation for scattering from a spherically symmetric potential. A resulting advantage over the more commonly used Green function method is that the expression for the interacting state vector is normalized to each order in the coupling constant, unlike, in general, the result of the Green function method. Other authors have applied the unitary transformation concept to time-dependent perturbation theory to give unitarity of the time evolution operator to each order in perturbation theory, with results that show improvement over the standard perturbation theory. In this paper, general formulas are obtained for the phase shifts at the first and second order in the coupling constant. We test the method on a simple system with a known exact solution and find complete agreement between our first- and second-order contributions to the s-wave phase shifts and the corresponding expansion to the second order of the exact solution.

DOI
10.1063/5.0023630

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In nonrelativistic quantum mechanics, we require that the free and interacting Hamiltonians be related by a unitary transformation, as has been done by other authors. We then derive a stationary perturbation theory for the radial Schrödinger equation for scattering from a spherically symmetric potential. A resulting advantage over the more commonly used Green function method is that the expression for the interacting state vector is normalized to each order in the coupling constant, unlike, in general, the result of the Green function method. Other authors have applied the unitary transformation concept to time-dependent perturbation theory to give unitarity of the time evolution operator to each order in perturbation theory, with results that show improvement over the standard perturbation theory. In this paper, general formulas are obtained for the phase shifts at the first and second order in the coupling constant. We test the method on a simple system with a known exact solution and find complete agreement between our first- and second-order contributions to the s-wave phase shifts and the corresponding expansion to the second order of the exact solution.
Rayleigh-Schrödinger time-independent perturbation theory is a key method in quantum mechanics for solving problems where the Hamiltonian of a quantum system cannot be exactly solved. By expanding the solutions in series, this method allows one to obtain approximations for quantum states and eigen-energies, especially for systems where the interactions or perturbing forces are small compared to the principal part of the Hamiltonian. This is crucial for understanding complex phenomena, predicting the behaviours of atomic and molecular systems, and calculating corrections to basic theoretical models, thereby improving the accuracy of quantum predictions. In this work, we will show that using the unitary operators, we arrive at the equations giving the different order of corrections of both energy as well as wave functions of the time-independent perturbation theory. Our approach closely mirrors the famous formalism of Rayleigh-Schrodinger perturbation theory (RSPT). The use of unitary operators simplifies the derivation of higher-order corrections, making the process more accurate and accessible to researchers. This technique harnesses the potential of unitary operators within a perturbation series framework. By combining these two powerful techniques, we offer a comprehensive framework for addressing normalization challenges in quantum mechanical calculations. This methodology not only facilitates a deeper understanding of perturbation theory but also serves as a pedagogical stepping stone, encouraging postgraduate researchers to apply their foundational knowledge and acquired skills in practical problem-solving situations with confidence. This is therefore a second proof or a new method to derive the fundamental equations of Rayleigh Schrodinger perturbation Theory RSPT using the unitary transformations.
The author discusses a method for solving the Schrodinger equation with spherically symmetric potentials. The method is based on the use of a certain integral transform for the radial wavefunction. In the representation defined by this transform the Schrodinger equation becomes an integral equation of Volterra type. In a number of cases this equation can be solved exactly without any use of perturbation theory. The resulting solution is represented by a locally finite sum of functions, which can be easily calculated since they are defined by integrals involving simple algebraic functions. For this reason, the method is well suited for practical calculations. The author derives exact solutions for the Yukawa and exponential potentials. As an application the author calculates the bound state spectrum and the scattering cross-sections.
Solutions of the Schrödinger equation with an exact time dependence are derived as eigenfunctions of dynamical invariants which are constructed from time-independent operators using time-dependent unitary transformations. Exact solutions and a closed form expression for the corresponding time evolution operator are found for a wide range of time-dependent Hamiltonians in d dimensions, including non-Hermitean -symmetric Hamiltonians. Hamiltonians are constructed using time-dependent unitary spatial transformations comprising dilatations, translations and rotations and solutions are found in several forms: as eigenfunctions of a quadratic invariant, as coherent state eigenfunctions of boson operators, as plane wave solutions from which the general solution is obtained as an integral transform by means of the Fourier transform, and as distributional solutions for which the initial wavefunction is the Dirac δ-function. For the isotropic harmonic oscillator in d dimensions radial solutions are found which extend known results for d = 1, including Barut–Girardello and Perelomov coherent states (i.e., vector coherent states), which are shown to be related to eigenfunctions of the quadratic invariant by the ζ-transformation. This transformation, which leaves the Ermakov equation invariant, implements SU(1, 1) transformations on linear dynamical invariants. coherent states are derived also for the time-dependent linear potential. Exact solutions are found for Hamiltonians with electromagnetic interactions in which the time-dependent magnetic and electric fields are not necessarily spatially uniform. As an example, it is shown how to find exact solutions of the time-dependent Schrödinger equation for the Dirac magnetic monopole in the presence of time-dependent magnetic and electric fields of a specified form.
We obtain a complete series solution of stationary Schrödinger's equation in the general quantum systems. It is exact in the sense that any approximation means is not used, or that the whole corrections or contributions from all order perturbations are involved if the perturbation concept is introduced. Furthermore, the useful forms of our exact solution and a new expression the complete Green operator are given out. As a universal and analytical solution, it is helpful for the theoretical derivations and practical calculations in quantum theory.
Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.
We present a new derivation of Time-Independent Perturbation Theory (PT) that is shorter and more general than the usual ones presented in Quantum Mechanics (QM) texts. Expressing the task of PT in terms of Green's functions, the key to our approach is to write the bound-state equation for the perturbed wave function directly in terms of the unperturbed Green's function. Not only does this greatly simplify the derivation of PT expressions for energies and wave functions, but it does so without using the orthogonality or completeness properties of unperturbed Hamiltonian eigenstates, which form the basis of standard derivations of PT. As a result, our approach does not require the potentials to be energy independent as is required in standard derivations, nor does it require the inverse free Green's operator G0−1(E) to be a linear function of energy, as is the case in QM. Consequently, our approach is applicable not only to QM but also to various extensions of QM, including Quantum Field Theory. For students, our approach provides an easy-to-understand introduction to the use of Green's functions in QM while at the same time presenting a description of PT that is shorter and more fundamental than the standard ones currently advocated.