Double slit experiment with electrons and photons, wave particle duality, heisenberg uncertainty. 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. A one dimensional mechanical equivalent of this equation is depicted in the gure below. Its physical interpretation is that fr,k,tdrdk is the number of electrons wavepackets at point r with wavenumber k in the phase space volume drdk. Overview, scale of quantum mechanics, boundary between classical and quantum phenomena. The wave function wf in the whole region is built with the use of requirements of continuity and niteness for 0x in the whole region.
We are now interested in the time independent schrodinger equation. Classical wave equations and solutions lecture chemistry. The electric and magnetic elds produced by an arbitrary charge density are complicated. Pdf classical solution of the wave equation researchgate. An example using the onedimensional wave equation to examine wave propagation in a bar is given in the following problem. Ditto from classical to wavemechanical dy namics 9 at first glance, the timedependent sc hrodinger equation 32 merely describes. A stress wave is induced on one end of the bar using an instrumented. Wave equations in classical physics are equations which describe waves. The wave equation 3 this is the desired wave equation, and it happens to be dispersionless. There are one way wave equations, and the general solution to the two way equation could be done by forming linear combinations of such solutions.
Lecture 4 postulates of quantum mechanics, operators and. Therefore, the general solution, 2, of the wave equation, is the sum of a rightmoving wave and a leftmoving wave. Aug 29, 2014 this video shows how to solve the classical wave equation using separation of variables. What are you expected to do when faced with one of the many differential equations in. But, in a very simple way, a wave is a something that travels in a given direction with a given velocity. Elementary solutions of the classical wave equation 1. Phase velocity is the speed of the crests of the wave. The mathematics of pdes and the wave equation mathtube. It is argued here that one may arrive at new energy methods using this classical wave equation as starting point. In many cases for example, in the classic wave equation, the equation describing the wave is linear. K wave like nature of matter is shown by performing the doubleslit experiment using water waves. Lecture notes on classical mechanics a work in progress.
On almost global existence for nonrelativistic wave. Wave equation, classical and schrodingers, defined in. In many realworld situations, the velocity of a wave. May 27, 2012 the difference between classical an quantum mechanics is not the wave function itself but the interpretation, e. Classical solution of the wave equation mn sergeenko the national. 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. Difference between classical wave function and quantum wave. The onedimensional wave equation the mathematical description of the onedimensional waves can be expressed as solutions to the wave equation. It arises in fields like acoustics, electromagnetics, and fluid dynamics. The pdes above are examples of the three most common types of linear equations. The ap proach used here has the advantage of applying to classical equations. Energy and power in waves 1 energy in a string the kinetic energy of a mass m with velocity v is 1 2.
The wave function is the ratio between the kinetic energy that the electron has when it is unobserved and the energy that it. The classical wave equation and separation of variables pdf 5. Differential equations department of mathematics, hkust. Alan doolittle postulates of quantum mechanics postulate 1 the wave function. It arises in fields like acoustics, electromagnetics, and fluid dynamics historically, the problem of a vibrating string such as that of a musical. We turn our attention again to the classical wave equation.
The wave equation is a secondorder linear hyperbolic pde that describesthe propagation of a variety of waves, such as sound or water waves. Finding the solution to 1d wave equation three cases of k a. A simple wave will look the same as it moves along think of waves on the deep ocean not on the beach. Plancks constant, interference, fermats principle of least time, debroglie wavelength. Derivation of the schrodinger equation from classical physics. The simple harmonic oscillator recall our rule for setting up the quantum mechanical problem. Dispersion relationships the dispersion relationship is the law which relates the angular frequency.
But avoid asking for help, clarification, or responding to other answers. Lecture notes quantum physics i physics mit opencourseware. Thus, the relation between the frequency of the wave. Schrodingers equation 2 the simple harmonic oscillator example. The wave function in quantum mechanics kiyoung kim department of physics, university of utah, slc, ut 84112 usa abstract through a new interpretation of special theory of relativity and with a model given for physical space, we can nd a way to understand the basic principles of quantum mechanics consistently from classical theory. In chapter 5 some elements of the theory of linear integral equations are given. Erwin schrodinger formulated a version of quantum mechanics that was based on waves. Pdf the classical limit of wave quantum mechanics is analyzed.
What equation of motion has such a formula among its. This wave will be moving with a phase velocity given by vphase. Download fulltext pdf download fulltext pdf download fulltext pdf classical solution of the wave equation article pdf available in international journal of modern physics a 1817. Now we want to do the same here for waves as we did for the ball and spring. According to shpenkov, the classical wave equation is able to derive a periodic table of elements which is close to mendeleyevs periodic table, and also other phenomena related to the structure of molecules. Wesley weiherdammstrasse 24 78176 blumberg, germany from the extensive observations and the ideas of newton and from classical physical optics the velocity of a quantum particle is given by w s e, where s is the poynting vector and e the wave energy density. Another physical setting for derivation of the wave equation in one space. Schrodinger equation in 3d and angular momentum, the angular momentum operator, eigenstates of angular momentum, the radial wave equation pdf hydrogen atom, hydrogen atom spectrum pdf need help getting started. Depending on the medium and type of wave, the velocity v v v can mean many different things, e. Schrodingers equation 2 the simple harmonic oscillator. He wrote down a wave equation the socalled schrodinger equation that governs how the waves evolve in space and time. The crucial difference from ns is that a dispersive term replaces the dissipative term of classical. The experiment is set up in the same way, except that gun shooting bullets is replaced by a wave source and the entire system is placed in a shallow pool of water.
That means that the net amplitude caused by two or more waves traversing the same space is the sum of the amplitudes which would have been produced by the individual waves separately. It is shown that the basic requirements of continuity and finiteness to the. We derive the wave equation which takes the same form as all the other wave equations weve seen so far, and then look at the properties of the waves. A homogeneous, elastic, freely supported, steel bar has a length of 8. Chapter 1 elementary solutions of the classical wave equation.
The solutions of the one wave equations will be discussed in the next section, using characteristic lines ct. Bardapurkar3 1department of physics, badrinarayan barwale mahavidyalaya, jal na, maharashtra, india. The wave equation in one dimension later, we will derive the wave equation from maxwells equations. Chapter 4 the wave equation another classical example of a hyperbolic pde is a wave equation. What links here related changes upload file special pages permanent link. The general solution 0x of the wave equation and the quantization condition is written in terms of the classical action. If integrated in all space over k, we would get the.
This equation determines the properties of most wave phenomena, not only light waves. Most of you have seen the derivation of the 1d wave equation from newtons and. Here it is, in its onedimensional form for scalar i. Murray, born in scotland 1931, professor emeritus of the university of oxford, known for his work and his books on mathematical biology. The wave equation outline of mechanical waves longitudinal and transverse waves waves in a string, sound waves the wave equation description of waves using functions of two variables travelling waves the wave equation 0 y v y 1 2 2 2 2 2 x t waves in a string. The classical wave equation is a second order linear partial differential equation given by. The wave equation is an important secondorder linear partial differential equation for the. 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.
Chapter 1 transport properties university of virginia. Laplaces equation is elliptic, the heat equation is parabolic and the wave equation is hyperbolic, although general classi. This time to look at solutions which are moving with a constant speed v, say for instance in the xdirection. We show how the second order wave equation can be decomposed into two. Wave nature of the electron and the internal structure of an atom pdf lecture 2 supplement. Thanks for contributing an answer to physics stack exchange. Lecture notes on classical mechanics a work in progress daniel arovas department of physics university of california, san diego may 8, 20. Chapter 1 elementary solutions of the classical wave. The experiment is performed by a motor, which gently moves the wave source up and down. It is invariant under rotations of x, y and z in fact, it is invariant under a larger group of symmetries, lorentz transformations. 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. The dispersion relation can then also be written more compactly as. As we stated in the last chapter, we need a true wavelike representation for the electron enter the schrodinger diffeq and heisenberg matrix derivations. Chapter 2 lorentz contraction from the classical wave.
Given that general relativity and quantum mechanics are much harder theories to apply, it is no wonder that scientists revert to classical mechanics whenever possible. Weve found a formula for the shape and motion of a wave that is oscillatory both in space and in time. I understand that you might not have gotten to calculus and differential equations. In chapters 3 and 4 the structural properties of the solutions of hyperbolic and parabolic partial differential equations are studied and much attention is paid to basic problems of the theory of wave equation and heat conduction equation. The equations of motion of classical physics 19 the current density is given by j. In this section well give a derivation of the schrodinger equation. It may not be surprising that not all possible waves will satisfy the wave equation for a specific system since waves solutions must satisfy both the initial conditions and the boundary conditions. Now recall that if we have a wave going from a medium with impedance z1 into a medium with impedance z2, the amplitude of the transmitted and re. Previous proofs utilized, in addition, either lorentz invariance 6 or direct estimation of the fundamental solution of the linear wave equation 5. An equation might tell us, for any one moment, how to calculate the force of the wave when it hits an unwary child, based on the height of the wave amplitude. If you want to learn differential equations, have a look at differential equations for engineers if your interests are matrices and elementary linear algebra, try matrix algebra for engineers if you want to learn vector calculus also known as multivariable calculus, or calculus three, you can sign up for vector calculus for engineers. Classical mechanics is about movement and change, while the strong emphasis on stationary states in traditional quantum courses makes the quantum world seem static and irreconcilably di. When this is true, the superposition principle can be applied.
Then i will discuss george shpenkovs interpretation of classical wave. The actual frequency of the wave in our new reference frame is still given by the planckeinstein equation. These questions are studied in regularity theory for pdes. When the elasticity k is constant, this reduces to usual two term wave equation u tt c2u xx where the velocity c p k.
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