Now, we're going to rely on a waveparticle [inaudible] to deduce the form ofthe main equation of quantum physics.The Schrodinger equation.But before going to this equation.I first would like to ask the followingquestion.Which picture is more fundamental?The particle-based picture or thewave-based picture?So here, I have simple example of what weusually mean by a particle in this, inthis example.Here is the baseball and so this is anobject which has a well-defined velocityand position at any given time.We can easily identify its position justby looking at it, and measure thevelocity.In contrast when we are talking aboutwaves, it doesn't really make sense to askthe question where is it located.So here, for instance, I have an exampleof wave which, is generated using a ropeand well, we cannot define a position ofthe wave.Okay?Part of what we can do, we can define itsphase velocity and the wavelength and thecorresponding wave vectors.So which is so wavelength lambda is equalto 2pi divided by k, k is the wavelength,right, omega is the frequency.So, the velocity of the wave is thecoefficient between the frequency and theefficiency of proportionality between thefrequency and the wavelength.Now there are two common expressions whichare two types of common expressions that Iuse to describe a simple sinusoidal wavethat we will see pretty often.So one is just using the sine or cosinefunction such as here.So in this example u is the vertical is,displacement, which is a function of x, aposition in time and s in this case, inthis horizontal direction.So another way to describe a wave is usingthis exponential function of an imaginaryline constant, which i here is the squareroot of minus 1.And they are equivalent to one another.One most commonly we're actually going touse the latter expression.Now, going back to the main question,what, what is more fundamental, particlesor wave?I would argue is that waves are in factmuch more fundamental and that we can'treally represent the wave in terms of aparticle, but we can surely decompose alocalized particle into waves.And this decomposition goes by the name ofFourier transform.So here I have an example of a genericFourier transform, which is a way,basically, to represent an arbitraryreasonable function of a fence, in termsof these exponentials e to the power ofikx.We should respond to this way ofresponding the wave at any given time.And the responding coefficients here whichmultiply these exponential areboth[INAUDIBLE] of the, of the function fof x.In particular, we can consider a functionf of x, which has a sharp a peak, see?With a narrow spread out, in let's saydelta x.And we can we can consider this functionas a mathematical description of an entitywhich is a particle-like entity.And, by decomposing it into if we're goingto transform, we represent it, it's alinear combination of waves.Now, we're going to move a little closerto the main subject of today's lecture,and, and remind ourselves the classicaldescription of electromagnetic waves orclassical description of light.So this electromagnetic waves follow fromthe Maxwell's equations, which have beenknown for a long time.And, here are those equations, Maxwell'sequations.And so, the first two equations hereessentially represent the Gauss's law.So, the first one tells that electriccharges, electric density[UNKNOWN] givesrise to an electric field.The second one tells us that there are nomagnetic charges, no monopoles.So the last week, we has, or essentially,the Faraday's law which tells that thechanging magnetic fields in time createsan electric field and vice versa.And also there is an Ampere's law thatcars, electric cars give rise to magneticfield.Now, even if there is no[INAUDIBLE] theproblem, that is even if there is thereare no charges present so, the right-handside of this equation is zero and thecurrent is zero.That is, if were in vacuum, even in thiscase a material solution exists to thisMaxwell's equation, this solution isexactly what electromagnetic waves are.In order to derive the wave equation, whatwe can do, we can apply the curl the curloperator to the both sides of this thirdequation.And the result of this would be a doublecurl of the electric field and theelectron side is equal to minus 1 over c,d over dt, curl of B in the right-handside.Now, this left-hand side can be simplifiedby using the following expression, sodouble curl is equal to gradient of thedivergence of the electric field minusLaplacian of the electric field.So the Laplacian here is of course, isjust the second derivative with respect tox, second derivative with respect to y,second derivative with respect to z.Since we have no charges in the problem,since we're operate in vacuum, so this guyis equal to zero, and so the, theleft-hand side is simply equal to thisLaplacian of e.Sometimes, it identically can be writtenas, as so.This triangle basically means Laplacian.Now the, the right-hand side can besimplified using the fourth equation, thefourth Maxwell's equations by writing itas minus 1 over c squared, secondderivative of the electric field, withrespect to time.So, so therefore, we have no closedequation, which is basically the waveequation that governs the behavior of theelectric field in vacuum.So let me write it down.So Laplacian minus 1 over c squared, d2over dt squared acting on the electricfield, E is equal to 0.Now, let's us simplify the problem alittle bit and let's assume that the wave,the solution to this equation depends onlyon one spatial coordinate.Let's call it x.So in which case the equation is going tobecome d2 over dx squared minus 1 over csquared, c of course is the speed oflight, secondary with respect to time.And here, I have a function, so thefunction itself is a variable, but itdepends on the one spatial coordinate intime.So, one can check that solution to thisequation can be written as E of x andt[UNKNOWN] times an exponential kxminus[UNKNOWN].So, if we differentiate this exponentialtwice, we're going to just pull out thewave [unknown] key.So it will appear here minus k squared andhere will have the frequency omega squaredover c squared and this whole thing mustbe equal to 0 in order to satisfy thisequation.So what we get is the result of thissolution we will get first of all thefunctional form of the so called planewave And a relation between Omega and K.So, which is the final result.So finally I would like to use the waveparticle duality for electrons now, inorder to guess, if you want, or to derivethe, the wave equation that governs theirquantum wave property.So here, we're relying on uh,uh, onexperiments of the type performed by,let's see, [inaudible] a fraction where weclearly saw that the data can beunderstood if we assume the collectionbehaves, behaves as waves.And, so, let's assume that this is indeedso.Let's assume that the free electron isdescribed by some wave function psi.So, the precise physical meaning of thisfunction will be discussed later,[unknown] throughout the course, but atthis stage, let's just assume that this isso.So, jst like in the case of protons, wehave an electric, an electric field whichhave the form of the plane waves.Here, we have some electron wave function,which has the form of the plane wave withthe sum momentum p and energy epsilon.But unlike protons, where energy scaleswith momentum linearly, so we know thatfor non-relativistic electrons the kineticenergy is basically mv squared over 2 or psquared over 2m.So what we, what we have to demand is thatwhatever equation governs the quantumproperties of electrons, it must give usthis functional form to describe a freeelectron, and this spectrum as aconstraint on this solution.And so, basically, we can construct suchan equation by hand that gives that hasthese properties.And this is what we can do, is we canwrite this equation by acknowledging tothe wave equation before, that gave us thelinear spectrum.Now, here if we plug in this guy into thisequation, so the first dividend is goingto give us minus h squared over 2m, andhere is the mass, of course.The second derivative is going to give usminus 1 over h bar squared p squared, andthe second term here is going to give usminus ih bar, d over dt is minus ih barepsilon.And we will have the same plane withmultiplying both terms, so we must demandthat this but these guys are equal tozero.Okay, so a lot of things cancel out here.So we have p squared over 2m here.And we have minus, minus, so basicallyminus epsilon here.And indeed we're, we reproduce thisdesired spectrum.Now this construction of course isnothing, but yes, we can, it's a veryconvenient, and yes, we can generalizethis equation just by writing it in aslightly different form.So let me do so, so what we're going todo, we're going to put the time derivativein the left-hand side.So basically this time derivative is goingto be in the left-hand side, ih bar, dover dt psi and this guy, we're going tointerpret it as an energy function.So after all, p squared over 2m is just anenergy of a free electron, but what we'regoing to do, we're going to put someenergy function, otherwise known asHamiltonian.So this guy is very important object, is aHamiltonian.Ad we're going to postulate that thisHamiltonian is essentially kinetic energy,p squared over 2m plus some potentialenergy, V of r.So our p here is an operator, minus ih bargradient [inaudible] whatever it is thatwe have here, so this wave function.So and this is the economical Schrodingerequation, that basically contains most, ifnot all [inaudible] physics.So of course the way we derive it inquotes is not really proper derivation.It was more like a guess, and in somesense, this is what Schrodinger give well,almost hundred years ago now.And the reason we actually believe thatthis equation is a true equation thatdescribes quantum physics, is because,because when we use it to actually solveproblems where apart from just freeparticles, where we both have anon-trivial, let's say, potentiallandscape, let's say in the case of anatom or in, in the case of scattering ofparticles.So this equation produces theoreticalresults, which are perfectly consistentwith experimental observations.So there is no question now, nowadays,that this equation indeed describesquantum reality.But the way people early in the early daysgot to this equation was very, verynon-trivial.So they had to edit a lot of piecestogether to guess what is the underlyingmathematical structure that governs thequantum behavior.