# Modern Physics with Modern Computational Methods - John

läsåret 01-02

Among one of the widely used is perturbation method (Marion, 1970; solved the Dufﬁng-harmonic oscillator by expanding the term x3 1þx2 into a polynomial form x€þx 3 x5 þ¼ 0. The forced oscillator chosen here is a simple oscillator which is subject to damping and is driven by a periodic force that is simple harmonic in nature. Its equation of motion is given by x¨+ µ ˙+ ω 0 2 x = F 0 sin(ωt) (3) where ω 0 is frequency of the simple harmonic oscillator, µ is the damping force per unit velocity per unit mass, F 0 is the Equation Solving; The Physics of the Damped Harmonic Oscillator; On this page; Contents; 1. Derive Equation of Motion; 2. Solve the Equation of Motion where F = 0; 3. Underdamped Case (ζ<1) 4.

In this example, you will simulate an harmonic oscillator and compare the numerical solution to the closed form one. Schrödinger’s Equation – 2 The Simple Harmonic Oscillator Example: The simple harmonic oscillator Recall our rule for setting up the quantum mechanical problem: “take the classical potential energy function and insert it into the Schrödinger equation.” We are now interested in the time independent Schrödinger equation. This algorithm reduces the solution of Duffing-harmonic oscillator differential equation to the solution of a system of algebraic equations in matrix form. The merit of this method is that the system of equations obtained for the solution does not need to consider collocation points; this means that the system of equations is obtained directly. Simple Harmonic Oscillator #1 - Differential Equation Now if you know about solving differential equations, we can actually find the particular function x(t) that satisfies that equation.

After substituting Equations 5.6.6 and 5.6.8 into Equation 5.6.5, the differential equation for the harmonic oscillator becomes d2ψv(x) dx2 + (2μβ2Ev ℏ2 − x2)ψv(x) = 0 Exercise 5.6.1 Make the substitutions given in Equations 5.6.6 and 5.6.8 into Equation 5.6.5 to get Equation 5.6.9. The Equation for the Quantum Harmonic Oscillator is a second order differential equation that can be solved using a power series.