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A method to obtain the solution of the time- independent Schrodinger equation for the simple harmonic oscillator

Anirban Bose

Published Jan 2, 2026• 0 citations
Publisher

Abstract

A method has been introduced to derive the solution of the time-independent Schrodingerequation for the simple harmonic oscillator. A trial solution has been chosen as theproduct of the divergent part of the approximate asymptomatic solution of theSchrodinger equation and an unknown function. By inserting this trial solution into the same equation, an equation of that unknown function is obtained.This equation has a recursive property which has been exploited to obtain thesolutions of the equation. It is all about reconstruction of the equation with dif-ferent solutions of that unknown function by repetitive differentiation and thatultimately leads to the wave functions of the Schrodinger equation for harmonic

DOI
10.5281/zenodo.18133870

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A method to obtain the solution of the time- independent Schrodinger equation for the simple harmonic oscillator

Anirban Bose

A method has been introduced to derive the solution of the time-independent Schrodinger equation for the simple harmonic oscillator. A trial solution has been chosen as the product of the divergent part of the approximate asymptomatic solution of the Schrodinger equation and an unknown function. By inserting this trial solution into the same equation, an equation of that unknown function is obtained. This equation has a recursive property which has been exploited to obtain the solutions of the equation. It is all about reconstruction of the equation with different solutions of that unknown function by repetitive differentiation and that ultimately leads to the wave functions of the Schrodinger equation for harmonic oscillator.

•Jan 5, 2026•0 citations

A method to obtain the solution of the time- independent Schrodinger equation for the simple harmonic oscillator

Anirban Bose

A method has been introduced to derive the solution of the time-independent Schrodinger equation for the simple harmonic oscillator. A trial solution has been chosen as the product of the divergent part of the approximate asymptotic solution of the Schrodinger equation and an unknown function. Choosing the divergent exponential factor has an advantage. Because if the equation has to have a meaningful bound state solution, the other part of the product is compelled to converge faster than the diverging exponential, so that their product, which is the bound state solution, must converge at large distances. This requirement makes the other part an oscillatory function with (n+1) extrema when it has n nodes. Since its derivative vanishes at each of these extrema, its derivative necessarily possesses at least (n+1) nodes. Such a nodal structure is characteristic of the next higher bound state, motivating an investigation into whether the derivative of the remaining part is directly related to the wave function of the next higher energy eigen state. This investigation demonstrates that the wave functions of the harmonic oscillator can be generated by successively reconstructing the equations of the higher eigen states through repeated differentiation.

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Solution of Time-Independent Schrodinger Equation (TISE) by Using Finite Difference Approach

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This research investigates the numeric solution of the time-independent Schrödinger equation for the quantum harmonic oscillator by finite difference approach. The harmonic oscillator, described by a quadratic function potential, is a fundamental model in quantum mechanics due to its broad applications, ranging from molecular vibrations to quantum field theory. The time-independent Schrödinger equation is a second-order differential equation that typically poses challenges when solved analytically for complex potentials. The finite difference method become an attractive choice as it transforms the continuous differential equation into a system of linear equations that can be computationally solved through computer programming code. In this study, the spatial domain is discretized, and the second derivative is calculated by using central differences, transforming the TISE into a tridiagonal matrix representing Hamiltonian of system. By finding solutions to this matrix eigenvalue problem, wavefunctions and eigenvalues are obtained. The study results demonstrate that the finite difference approach effectively solves the TISE for the harmonic oscillator. The results obtained by using the finite difference method closely approximate the analytical results. The linear regression results show respectively that the gradient (β1), regression coefficient (β0) and coefficient of determination (R²) approach ideal values of 1, 0, and 1. The z-test results also show that the value of calculated z < critical z, indicating that the wavefunction and probability density, whether estimated by using finite difference approach or analytical methods, are equivalent with confidence level of 95 percent.

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- Approximation Methods I: Time-Independent Perturbation Theory

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The starting point in a quantum mechanical problem is the construction of a Schrödinger equation. We try to solve it exactly to obtain wave function. A Schrödinger equation is exactly solvable only for a certain number of simple potentials. In most of the cases the Schrödinger equation cannot be solved exactly. The solution of the Schrödinger equation for the real systems is usually complicated and often exact solutions are very difficult to find or solutions not exist. Examples include as anharmonic oscillator, hydrogen atom in an electric field, a plane rotator in the presence of an applied electric field and so on. Therefore, we need to rely on approximation methods. The simple and most interesting and powerful approximate method is the perturbation theory developed by Schrödinger in 1926 [1]. Earlier Lord Rayleigh [2] analysed harmonic vibrations of a string perturbed by small inhomogeneities. The work of Rayleigh was pointed out by Schrödinger. Consequently, the method is also called Rayleigh-Schrödinger perturbation theory. This method is powerful for determining the changes in the discrete energy values and the associated eigenfunctions of a system due to a weak perturbation, provided the eigenpairs of the undisturbed system are available. The approximations obtained from the perturbation theory are used to calculate energy level diagrams and to understand the splitting and assignments of atomic and molecular spectra [35]. The perturbation theory has been proven to be useful in several fields of theoretical physics and also in chemistry. It provides considerable insight into various phenomena. The main disadvantage of this method is that in certain problems the perturbation series converges slowly or diverges.

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The time evolution of a quantum harmonic oscillator with a series of sudden jumps of the mass or the frequency is determined in the form of a recursion relation. An approximating method is developed for determining the time evolution of harmonic oscillators with arbitrary derivable functions of the frequency or the mass. The approximate solution is shown to tend to the analytical one in the limiting case. As a demonstration of the approximating method, the solution of the problem of damped oscillation in the square of the oscillator frequency is presented.

Physical Review A•Jun 1, 1994•44 citations
A method has been introduced to derive the solution of the time-independent Schrodingerequation for the simple harmonic oscillator. A trial solution has been chosen as theproduct of the divergent part of the approximate asymptomatic solution of theSchrodinger equation and an unknown function. By inserting this trial solution into the same equation, an equation of that unknown function is obtained.This equation has a recursive property which has been exploited to obtain thesolutions of the equation. It is all about reconstruction of the equation with dif-ferent solutions of that unknown function by repetitive differentiation and thatultimately leads to the wave functions of the Schrodinger equation for harmonic
A method has been introduced to derive the solution of the time-independent Schrodinger equation for the simple harmonic oscillator. A trial solution has been chosen as the product of the divergent part of the approximate asymptomatic solution of the Schrodinger equation and an unknown function. By inserting this trial solution into the same equation, an equation of that unknown function is obtained. This equation has a recursive property which has been exploited to obtain the solutions of the equation. It is all about reconstruction of the equation with different solutions of that unknown function by repetitive differentiation and that ultimately leads to the wave functions of the Schrodinger equation for harmonic oscillator.
A method has been introduced to derive the solution of the time-independent Schrodinger equation for the simple harmonic oscillator. A trial solution has been chosen as the product of the divergent part of the approximate asymptotic solution of the Schrodinger equation and an unknown function. Choosing the divergent exponential factor has an advantage. Because if the equation has to have a meaningful bound state solution, the other part of the product is compelled to converge faster than the diverging exponential, so that their product, which is the bound state solution, must converge at large distances. This requirement makes the other part an oscillatory function with (n+1) extrema when it has n nodes. Since its derivative vanishes at each of these extrema, its derivative necessarily possesses at least (n+1) nodes. Such a nodal structure is characteristic of the next higher bound state, motivating an investigation into whether the derivative of the remaining part is directly related to the wave function of the next higher energy eigen state. This investigation demonstrates that the wave functions of the harmonic oscillator can be generated by successively reconstructing the equations of the higher eigen states through repeated differentiation.
This research investigates the numeric solution of the time-independent Schrödinger equation for the quantum harmonic oscillator by finite difference approach. The harmonic oscillator, described by a quadratic function potential, is a fundamental model in quantum mechanics due to its broad applications, ranging from molecular vibrations to quantum field theory. The time-independent Schrödinger equation is a second-order differential equation that typically poses challenges when solved analytically for complex potentials. The finite difference method become an attractive choice as it transforms the continuous differential equation into a system of linear equations that can be computationally solved through computer programming code. In this study, the spatial domain is discretized, and the second derivative is calculated by using central differences, transforming the TISE into a tridiagonal matrix representing Hamiltonian of system. By finding solutions to this matrix eigenvalue problem, wavefunctions and eigenvalues are obtained. The study results demonstrate that the finite difference approach effectively solves the TISE for the harmonic oscillator. The results obtained by using the finite difference method closely approximate the analytical results. The linear regression results show respectively that the gradient (β1), regression coefficient (β0) and coefficient of determination (R²) approach ideal values of 1, 0, and 1. The z-test results also show that the value of calculated z < critical z, indicating that the wavefunction and probability density, whether estimated by using finite difference approach or analytical methods, are equivalent with confidence level of 95 percent.
We found a simple procedure for the solution of the time - independent Schrodinger equation in one dimension without making any approximation. The wave functions are always periodic. Two difficulties may be encountered: one is to solve the equation E=U(x), where E and U(x) are the total and potential energies, respectively, and the other is to calculate the integral of the square root of U(x). If these calculations cannot be made analytically, it should then be performed by numerical methods. To find the energy and the wave function of the ground state, there is no need to calculate this integral, it is sufficient to find the classical turning points, that is to solve the equation E=U(x).
The starting point in a quantum mechanical problem is the construction of a Schrödinger equation. We try to solve it exactly to obtain wave function. A Schrödinger equation is exactly solvable only for a certain number of simple potentials. In most of the cases the Schrödinger equation cannot be solved exactly. The solution of the Schrödinger equation for the real systems is usually complicated and often exact solutions are very difficult to find or solutions not exist. Examples include as anharmonic oscillator, hydrogen atom in an electric field, a plane rotator in the presence of an applied electric field and so on. Therefore, we need to rely on approximation methods. The simple and most interesting and powerful approximate method is the perturbation theory developed by Schrödinger in 1926 [1]. Earlier Lord Rayleigh [2] analysed harmonic vibrations of a string perturbed by small inhomogeneities. The work of Rayleigh was pointed out by Schrödinger. Consequently, the method is also called Rayleigh-Schrödinger perturbation theory. This method is powerful for determining the changes in the discrete energy values and the associated eigenfunctions of a system due to a weak perturbation, provided the eigenpairs of the undisturbed system are available. The approximations obtained from the perturbation theory are used to calculate energy level diagrams and to understand the splitting and assignments of atomic and molecular spectra [35]. The perturbation theory has been proven to be useful in several fields of theoretical physics and also in chemistry. It provides considerable insight into various phenomena. The main disadvantage of this method is that in certain problems the perturbation series converges slowly or diverges.
The time evolution of a quantum harmonic oscillator with a series of sudden jumps of the mass or the frequency is determined in the form of a recursion relation. An approximating method is developed for determining the time evolution of harmonic oscillators with arbitrary derivable functions of the frequency or the mass. The approximate solution is shown to tend to the analytical one in the limiting case. As a demonstration of the approximating method, the solution of the problem of damped oscillation in the square of the oscillator frequency is presented.