Write the schrodinger equation for simple harmonic oscillator and its solution
Answers
We will start with a harmonic oscillator to start a common numerical method to solve a one-dimensional, time-independent hornodinger equation. The analytical solution of the harmonic oscillator will be derived and described earlier. A specific integration algorithm (Numerov) will be used. The expansion of numerical method does not offer any special difficulty for other, more common types of probabilities. For a particle of mass in under a potential V (X), a one-dimensional, time-dependent, Shreidinger equation is given by: - ¯
-(h^2 d^(2 ) Ψ)/(2m〖dx〗^2 ) + V(x)Ψ(x) = EΨ(x),----------------------(1)
Where ψ (x) is the wave function, in the normal compound, and the wh Planck is divided by static H 2π. In the following we are focusing on different spectrum: the set of separate energy values for which there is generalized solution in eq (1.1), which are localized in space. 1.1 Harmonic Oscillator Harmonic Oscillator is a fundamental problem in quantum mechanics, along with classical mobility. It represents the simplest model system in which attractive forces are present and all types of vibration are an important paradigm for events. For example, in the context of an independent harmonic oscillator that goes as a normal component mode, vibrations can be described around the synchronous positions of a system of interacting particles, through a proper coordination change. The same is in Quantum Mechanics. Study quantum of the quantum oscillator allows a deeper understanding of volume and its effects and wave functions of bound states. In this chapter we will first recall the main consequences of the principle of harmonic oscillator, then we will show how to set up a computer code which allows the Shrodinger equation to solve the numerical solution for the harmonic oscillator. As a result, the code can be easily modified and customized (simple classified), no interaction capability. This will allow problems to be studied that, unlike the harmonic oscillator, there is not a simple analytical solution.