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 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 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 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.
An alternative method for obtaining analytical solutions of the time-independent Schrödinger equation (TISE) is developed and applied to 1-dimensional model quantum systems of interest in atomic and molecular physics. The method is based on proposing a wave function that is a product of two functions, one of them an integrating factor. The differential equation resulting from operating the TISE on the product wave function is compared with a differential equation with a known solution. This comparison produces a system of differential equations for the integrating factor. This system of equations is easily solved for model quantum systems such as the harmonic oscillator, the hydrogen atom and the Morse oscillator. The method presented simplifies some approaches shown in textbooks, based on asymptotic analyses of the TISE, and power series methods with recurrence relations. In addition, the method presented here facilitates the understanding of the relationship between the ordinary differential equations of the mathematical physics and the TISE of physical models as the harmonic oscillator, the Hydrogen atom, and the Morse oscillator.
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).
In this chapter, the authors solve the Schrodinger equation for a harmonic oscillator using two methods, analytical method and algebraic method. They discuss the quantum properties of the harmonic oscillator. A harmonic oscillator is a system that exhibits simple harmonic motion or periodic motion, a motion that repeats itself after equal intervals of time. Many systems such as spring, simple pendulum, vibrating string and molecular vibration can be approximated as simple harmonic oscillators. The authors consider a simple harmonic oscillator, a spring whose spring constant is "k" attached to a mass "m" as a simple harmonic oscillator. They describe the Schrodinger equation in spherical coordinates. The authors also solve the three-dimensional Schrodinger equation for the hydrogen atom. For the hydrogen atom, the potential energy function is dependent only on radial distance and is independent of angular variables. The hydrogen atom consists of a positively charged proton surrounded by an electron that orbits around it.