The sc hr o ding er w av e equati on macquarie university. Depending on the medium and type of wave, the velocity v v v can mean many different things, e. Deriving time dependent schrodinger equation from wave. The schrodinger equation has two forms, one in which time explicitly appears, and so describes how the wave function of a particle will evolve in time. The wave equation is an important secondorder linear partial differential equation for the description of wavesas they occur in classical physicssuch as mechanical waves e. Wave equation the purpose of these lectures is to give a basic introduction to the study of linear wave equation. The 3dimensional wave equation is a linear, homogeneous partial differential equation with constant coefficients. We saw that a pure sinusoidal wave can by represented by. This will result in a linearly polarized plane wave travelling in the x direction at the speed of light c.
The schrodinger wave equation for the hydrogen atom. Note that fn nf1, in other words, the frequencies of higher harmonics are just integer multiples of the fundamental frequency f1. Although we will not discuss it, plane waves can be used as a basis for. Simple derivation of electromagnetic waves from maxwell.
On the derivation of the timedependent equation of schro. We illustrate a simple derivation of the schrodinger equation, which requires only knowledge of the electromagnetic wave equation and the basics of einsteins special theory of relativity. One example is to consider acoustic radiation with spherical symmetry about a point y fy ig, which without loss of generality can be taken as the origin of coordinates. We shall first postulate the wave function for the simplest conceivable system. In this frame, the velocity u0 is zero, so the gas is not moving. Only in this frame can we get the wave equation because the wave equation has two equal sound speeds cmoving. J n is an even function if nis an even number, and is an odd function if nis an odd number. We now turn to the 3dimensional version of the wave equation, which can be used to describe a variety of wavelike phenomena, e. You could write out the series for j 0 as j 0x 1 x2 2 2 x4 2 4 x6 22426 which looks a little like the series for cosx. Derivation of the wave equation in these notes we apply newtons law to an elastic string, concluding that small amplitude transverse vibrations of the string obey the wave equation. Deriving time dependent schrodinger equation from wave mechanics, schrodinger time independent nilesh p. Solution of the wave equation by separation of variables ubc math. Bardapurkar 32 introduction quantum mechanics is an essential part of undergraduate syllabus in physics as well as in chemistry. In order to do this, we s tart with an equation to describe a plane wave moving through.
For physical examples of nonspherical wave solutions to the 3d wave equation that do possess angular dependence, see dipole radiation. The wave equation governs a wide range of phenomena, including gravitational waves, light waves, sound waves, and even the oscillations of strings in string theory. Newtons laws, the schrodinger equation does not give the trajectory of a particle, but rather the wave function of the quantum system, which carries information about the wave nature of the particle, which allows us to only discuss the probability of nding the particle in di erent regions of space at a given moment in time. The schroedinger equation can not be derived from classical mechanics. Wave equation in 1d part 1 derivation of the 1d wave equation vibrations of an elastic string solution by separation of variables three steps to a solution several worked examples travelling waves more on this in a later lecture dalemberts insightful solution to the 1d wave equation.
But classical mechanics can be rederived from the schroedinger equation in some limit. In 10 it was shown the existence of the wave operator for equation 1. Hence it seems appropriate to include the following. There are many examples of wave equations in the physical sciences, char. Simple derivation of electromagnetic waves from maxwells. The mathematics of pdes and the wave equation michael p. We now extend the wave equation to threedimensional space and look at some basic solutions to the 3d wave equation, which are known as plane waves. Photonic communications engineering opti 500b, lecture 11 415 then.
University of calgary seismic imaging summer school august 711, 2006, calgary abstract abstract. In particular, we examine questions about existence and. Rost2 received december 6, 2000 few have done more than martin gutzwiller to clarify the connection between classical timedependent motion and the timeindependent states of quantum systems. The wave equation is a secondorder linear hyperbolic pde that describesthe propagation of a variety of waves, such as sound or water waves. Obtaining the schrodinger wave equation let us now construct our wave equation by reverse engineering, i. These orbital designations are derived from corresponding spectroscopic characteristics of lines involving them. For nonrelativistic quantum physics the basic equation to be solved is the schr odinger equation. This equation is also known as the wave equation o f sound or string vibration 18, 22. The principal quantum number is named first, followed by the letter s, p, d, or f as appropriate. Chapter 4 the wave equation another classical example of a hyperbolic pde is a wave equation. Some notes on understanding the schrodinger wave equation.
Most of the material in this chapter is taken from thornton and rex, chapter 6. Chapter 4 schroedinger equation mit opencourseware. The observable directly linked to the wave function is the probability to nd the particle at position rat time t, namely, j. Let ux, t denote the vertical displacement of a string from the x axis at. We can separate equation into three separate differential equations, each depending on one coordinate. There are no boundary conditions required here, although to find a unique solution some kind of side condition is required. In general, the wave function behaves like a, wave, and so the equation is, often referred to as time dependent schrodinger wave equation. It arises in fields like acoustics, electromagnetics, and fluid dynamics. Erbil a ege university, science faculty, physics department bornova izmir 35100, turkey we found a simple procedure for the solution of the timeindependent schrodinger equation in one dimension without making any approximation.
The wave equation is an important secondorder linear partial differential equation for the. The string has length its left and right hand ends are held. Chemical bonding includes two or more atoms coming mutually due to the force of interaction between them. This is a short mathematical exercise to get to the root of what the equation is really describing, and how to go through the math step by step to derive information about the simplest of the atoms, hydrogen. Notes on the algebraic structure of wave equations mit math. The mathematics of pdes and the wave equation mathtube. Solution of the wave equation by separation of variables the problem let ux,t denote the vertical displacement of a string from the x axis at position x and time t. Solution of the wave equation by separation of variables. D for a wave equation is that it is antihermitian, as opposed to a.