Quantum mechanics has about a dozen exactly solvable potentials. Normally, the time-independent Schrödinger equation for them is solved by using a generalized series solution for the bound states (using the Fröbenius method) and then an analytic continuation for the continuum states (if present). In this work, we present an alternative way to solve these problems, based on the Laplace method. This technique uses a similar procedure for the bound states and for the continuum states. It was originally used by Schrödinger when he solved the wave functions of hydrogen. Dirac advocated using this method too. We discuss why it is a powerful approach to solve all problems whose wave functions are represented in terms of confluent hypergeometric functions, especially for the continuum solutions, which can be determined by an easy-to-program contour integral.
Quantum mechanics has about a dozen exactly solvable potentials. Normally, the time-independent Schrödinger equation for them is solved by using a generalized series solution for the bound states (using the Fröbenius method) and then an analytic continuation for the continuum states (if present). In this work, we present an alternative way to solve these problems, based on the Laplace method. This technique uses a similar procedure for the bound states and for the continuum states. It was originally used by Schrödinger when he solved the wave functions of hydrogen. Dirac advocated using this method too. We discuss why it is a powerful approach to solve all problems whose wave functions are represented in terms of confluent hypergeometric functions, especially for the continuum solutions, which can be determined by an easy-to-program contour integral.
Abstract In 1926, Erwin Schrödinger wrote a series of papers that invented wave mechanics and set the foundation for much of the single-particle quantum mechanics that we teach today. In his first paper, he solved the Schrödinger equation using the Laplace method, which is a technique that is quite powerful, but rarely taught. This is unfortunate, because it opens the door to examining quantum mechanics from a complex-analysis perspective. Gaining this experience with complex analysis is a useful notion to consider when teaching quantum mechanics, as these techniques can be widely used outside of quantum mechanics, unlike the standard Frobenius summation method, which is normally taught, but rarely used elsewhere. The Laplace method strategy is subtle and no one has carefully gone through the arguments that Schrödinger did in this first paper, instead it is often just stated that the solution was adopted from Schlesinger’s famous differential equation textbook. In this work, we show how the Laplace method can be used to solve for the quantum–mechanical energy eigenfunctions of the hydrogen atom, following Schrödinger’s original solution, with all the necessary details, and illustrate how it can be taught in advanced instruction; it does require familiarity with intermediate-level complex analysis, which we also briefly review.
A simple procedure has been found for the general solution of the time-independent Schrödinger Equation (SE) with the help of quantization of potential area in one dimension without making any approximation.Energy values are not dependent on wave functions. So, to find the energy values, it is enough to find the classic turning points of the potential function. Two different solutions were obtained, namely, symmetric and anti symmetric in bound states. These normalized wave functions are always periodic. It is enough to take the integral of the square root of the potential energy function to find the normalized wave functions. If these calculations cannot be made analytically, they should then be performed by numerical methods. The SE has been solved for a particle in many one-dimension and the spherical symmetric central potential well, the relativistic theory of Dirac as examples.
A simple procedure has been found for the general solution of the time-independent Schrödinger equation (SE) with the help of quantization of potential area in one dimension without making any approximation. Energy values are not dependent on wave functions. So, to find the energy values, it is enough to find the classic turning points of the potential function. Two different solutions were obtained, namely, symmetric and antisymmetric in bound states. These normalized wave functions are always periodic. It is enough to take the integral of the square root of the potential energy function to find the normalized wave functions. If these calculations cannot be made analytically, they should then be performed by numerical methods. The SE has been solved for a particle in many one-dimension and the spherical symmetric central potential wells as examples. Their energies and normalized wave functions were found as examples. These solutions were also applied to the theories of scattering and alpha decay. The results obtained with the experimental values were compared with the calculated values. One was seen to be very fit.
Abstract The Morse potential is an important problem to examine due to its applications in describing vibrations and bond breaking in molecules. It also shares some properties with the simpler harmonic oscillator, at the same time displaying differences, allowing for an interesting contrast to its well-studied counterpart. The solution of the Morse potential is not usually taught in a quantum mechanics class, since using differential equations makes it very tedious. Here, we illustrate how to solve the Morse potential using the Schrödinger factorization method. This operator method is a powerful tool to find the energy eigenvalues, eigenstates, and wavefunctions without using differential equations in position space, allowing us to solve more problems without requiring a discussion of hypergeometric or confluent hypergeometric functions.
Incorporating relativity into quantum mechanics presents challenges, one of which is the inconsistent possibility of multiple particles within a single-particle theory. This chapter describes the traditional, semi-analytic bound state in a 1-D box problem that requires a trial-and-error search algorithm. It examines techniques for finding the bound states for an arbitrary potential as a numerical eigenvalue problem and employs similar techniques for the relativistic hydrogen states of the Klein-Gordon equation. The chapter also employs a Monte Carlo algorithm to simulate the spontaneous decay of atoms or nuclei. It presents the numerical solution of the time-independent Schrodinger equation and also examines partial wave expansions, which leads to methods of computing the spherical harmonics and the associated Legendre functions, and ultimately the 3-D hydrogenic wave functions. The chapter deals with the Feynman path integral formulation of quantum mechanics, and its solution via the Metropolis algorithm.
Chapter Preview: We begin at the beginning with Plank's black-body radiation spectrum applied to the cosmic background. We move on to bound states; first finding them semi-analytically with a trial-and-error search, and then finding them numerically as an eigenvalue problem. We include relativistic effects in hydrogen bound states by solving the Klein-Gordon equation. We then examine the numerical solution of the time- independent Schrödinger equation, and various types of bound state wave functions. We move on to wave packet solutions of the time- dependent Schrödinger equation. Then we study scattering from: a 3-D square well, the Coulomb potential, three stationary disks, and on a billiard table (see also classical billiards in § 3.8 ). We next look at quantum mechanics in momentum space: integral equations, bound states, the delta-shell potential, hyperfine structure of hydrogen, SU3 symmetry of quarks. Next to last we examine several examples of coherent states. We end the chapter with the Feynman path integral formulation of quantum mechanics, and its solution via the Metropolis algorithm. If not enough, Chapter 8 discussion of quantum entanglement fits in well here too.