For more information, please see full course syllabus of Physical Chemistry

For more information, please see full course syllabus of Physical Chemistry

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### Schrӧdinger Equation & Operators

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro
- Schrӧdinger Equation & Operators
- Relation Between a Photon's Momentum & Its Wavelength
- Louis de Broglie: Wavelength for Matter
- Schrӧdinger Equation
- Definition of Ψ(x)
- Quantum Mechanics
- Operators
- Example I
- Example II
- Example III
- Example IV
- Example V
- Example VI
- Operators Can Be Linear or Non Linear
- Example VII
- Example VIII
- Example IX

- Intro 0:00
- Schrӧdinger Equation & Operators 0:16
- Relation Between a Photon's Momentum & Its Wavelength
- Louis de Broglie: Wavelength for Matter
- Schrӧdinger Equation
- Definition of Ψ(x)
- Quantum Mechanics
- Operators
- Example I 10:10
- Example II 11:53
- Example III 14:24
- Example IV 17:35
- Example V 19:59
- Example VI 22:39
- Operators Can Be Linear or Non Linear 27:58
- Operators Can Be Linear or Non Linear
- Example VII 32:47
- Example VIII 36:55
- Example IX 39:29

### Physical Chemistry Online Course

### Transcription: Schrӧdinger Equation & Operators

*Hello and welcome back to www.educator.com and welcome back to Physical Chemistry.*0000

*Today we are going to begin our discussion of Quantum Mechanics.*0004

*Let us just jump right on in.*0009

*We are going to be discussing the Schrӧdinger equation and something called operators.*0012

*Einstein has demonstrated that the relation between a photon’s of momentum and its wavelength is this right here.*0019

*A wavelength of a photon is able to planks constant divided by the photons momentum.*0027

*A photon is a particle of light and this is planks constant.*0033

*Louis de Broglie argued that matter also obeys this relation.*0040

*That a particle of mass M and velocity V will have a wavelength of this.*0046

*Momentum is just mass × velocity.*0055

*When we are talking about a specific particle with a definite mass and a definite velocity, it is the same relation.*0057

*This is the Broglie relation.*0064

*The Broglie waves have been experimentally confirmed.*0068

*In other words, particles do exhibit waves like behavior.*0075

*If matter then behaves like waves then theoretically at least, there should be some wave equation that describes the particles behavior.*0082

*Notice that I put these enclosed.*0091

*There is an equation, this is the Schrӧdinger equation and it looks like this.*0094

*I have written 2 versions of it.*0098

*There are actually several different ways that you can write this.*0100

*This is the important thing right here, let me go ahead and do this in red.*0104

*This red thing that you see right here.*0107

*This is the relationship that exists among between the different elements of this way function.*0114

*You have got to see what is the function that we are looking for and we have this different equation.*0124

*It just says that if I take the second derivative of this function, if I multiply it by some variation of planks constant divide by twice its mass, negate it.*0129

*If I add to that the function itself multiplied by the potential energy, I end up getting the function multiplied by the total energy of the system.*0141

*This is a different way of writing it and what I have done is basically taken this function and I have put it out here.*0151

*It will make more sense a little bit later in the lesson when I talk about this thing called operators.*0158

*Now the solutions of this differential equation, these right here, this particular function that we are looking for,*0163

*they describe how a particle of mass M moose in its particular potential field.*0172

*It is this Schrӧdinger equation that we are interested in,*0178

*In any given particular system that we are dealing with, we are going to come up with a Schrӧdinger equation for it.*0180

*We are going to solve the equation and then we are going to get the Z,*0185

*these different functions that describe how the particle is behaving at a given time, at a given speed, at a given whatever.*0190

*That is the whole idea.*0199

*What we want, the whole idea of Quantum Mechanics is to find this wave function from*0201

*the wave equation that we write down from the given set of data.*0208

*Z sub x are called the wave functions of the particle and they will end up telling us everything*0214

*we want to know about how the particle is behaving, that is the whole idea here.*0218

*This wave function contains all the information about the particle.*0224

*Whatever I need to know about it, its position, its momentum, its energy, is angle, whatever it is.*0227

*It is part of this function and I extract information from this function.*0234

*Z sub x is a measure of the amplitude of the matter wave.*0240

*Again, we are looking at matter as if it is a wave.*0244

*As if it is displaying wavelike properties.*0247

*Since that is the case, there is a wave function that describes its behavior.*0249

*Z sub x is a measure of the amplitude of that matter waves.*0255

*We are saying more about this later.*0258

*If you are interested in seeing how the Schrӧdinger equation can be obtained from the classical wave equation,*0261

*that is an argument for the plausibility of the Schrӧdinger equation or why the wave equation is actually called an amplitude, please see appendix 1.*0267

*These appendices that I'm going to be doing throughout this course, they are extra information.*0276

*They are not necessary, as far as the continuity of the course is concerned.*0282

*It is not like you have to necessarily watch them or do anything with them to continue on with the course.*0287

*They are just extra information for those of you that are interested in going a little deeper,*0293

*whether it is deeper conceptually, whether it is deeper mathematically, and things like that.*0297

*Quantum Mechanics is entirely mathematical.*0304

*At this level, my best advice is to accept and perform the mathematics without worrying too much about what the individual concepts mean.*0307

*When I talk about the mathematics, particular technique that we may be using whether it is differentiation, integration, something else,*0316

*we are going to be introducing some new mathematics that many of you may not have seen before.*0325

*It is not that I’m not going to explain what this physical significance is,*0329

*but in a lot of ways understanding in Quantum Mechanics is an emerging process.*0334

*Like it is in most sciences, in all sciences but it is a lot more so with Quantum Mechanics that it is with classical sciences that you are accustomed to.*0339

*It is really just a different way of thinking.*0347

*Quantum mechanics has this reputation of being very esoteric and really hard to wrap your mind around.*0350

*That is actually not true at all.*0356

*What you have to do is pull yourself away from trying to wrap your mind around it conceptually and*0359

*just developing a certain mathematical facility, just doing the math as is.*0365

*As you do it, as you become more comfortable with it, it will start to make sense why the math is actually taking the form that is taking.*0370

*For those of you that go on into higher science and particularly those of you*0378

*that want to take other courses in mathematics like Fourier series, a theoretical algebra, things like that.*0382

*All of this will actually come together.*0391

*For right now, we want you to develop a good mathematical facility with what is going on.*0393

*Do not worry too much if it does not entirely make sense to you.*0399

*Treating it that way, it is going to be a lot easier than you will expect it, I promise.*0402

*Let us go ahead and see what we can do.*0408

*Let us talk about operators.*0413

*I’m going to go ahead and rewrite the equation again.*0415

*Let me go ahead and write it in blue.*0417

*We have -H ̅² / 2 M.*0420

*We have D² / DX² MC.*0426

*I will go ahead and write that + the potential energy V × this C function = total energy of the system × this function.*0430

*Again, it is this C that we are looking for, that is what we want to find.*0444

*When you are doing algebra, you have something like 3x + 6 = 9 then solve for x.*0448

*A differential equation is the same sort of thing.*0453

*Now, instead of solving for a number x = 5, we want to get an actual function.*0456

*We are looking for a function, it is just another variable.*0461

*In this equation, this is the variable, this is your x.*0464

*Except x happens to be a function.*0469

*Let us go ahead and talk about operators.*0472

*An operator is a symbol that tells you to perform a task.*0475

*That tells you to perform not just a task, it could be one or more operations on a function, thus, producing a new function.*0497

*In fact, what we are really doing is we are giving a name to something that you have been doing for years and years.*0529

*We are producing a new function.*0535

*For example, when you take the derivative of a function you get a new function back.*0538

*The derivative of x² is 2x, the differential DDX is an operator, it is the differential operator.*0541

*That is what you are doing, we are just giving a name to it.*0549

*In other words, we start with some function f of x and we operate on it.*0554

*Let us do A with a little caret symbol over and we spit out a new function.*0560

*The symbol for the operator is symbolized by a capital letter with a caret over it.*0565

*It is symbolized by a capital letter with a caret symbol.*0575

*You have to define what the operator is.*0591

*We might say A is due this, B is due this, and we will see a little bit of that in just a moment.*0594

*Operators and operations are best described just by doing examples.*0602

*I’m just going to launch into the examples rather than try to explain it and it will make complete sense.*0605

*They are actually very easy to deal with.*0609

*Example 1, we will let A, this operator A equal to D² Dx².*0612

*In other words, the operator A means take the second derivative of some function.*0626

*Now what we want you to do is to find A of the sin of X.*0633

*Also written as A sin X.*0639

*You do not necessarily need to put parentheses around the function that you are operating on.*0642

*This says perform the operation A on the function sin X.*0647

*Well nice and simple.*0652

*You already know this, you have been doing this for ages.*0654

*Sometimes I will write the parentheses, sometimes I will not.*0656

*A = D² DX² of sin X.*0661

*I like to do things pictorially, so sin X when I take a derivative of sin of 5x, I end up with Φ cos 5x.*0670

*That is the first derivative.*0680

*The operators take the second derivative also.*0681

*When I take another derivative of that, I end up with -25 × the sin of 5x.*0685

*Operator, here is my definition of the operator.*0698

*The symbol A means do this and I have a function that I'm going to do that to, and I do it.*0700

*I end up with a function, it is that simple.*0706

*You have been doing it all along, you are just given it.*0709

*The only difference is that some of our operators tend to go a little bit more complex.*0717

*But you can handle it very easily.*0720

*Example 2, we will the operator B, we will define it as this D² DX² + multiplication by this thing called V, whatever V is.*0723

*V is a function of X.*0739

*We want you to find B of sin of 5X, the same function.*0742

*But now we want you to perform a different operation on it.*0748

*The operator is defined by this.*0752

*Although operators are only symbols, they can be treated just as though they were regular polynomials.*0757

*This whole thing is the operator.*0764

*I can just treat this, this way.*0766

*Here is what happens, B of the sin 5X = this is the operator, it was going to be D²/ DX² +*0769

*this function V of X × I will put sin of X here to perform this operation on sin X.*0781

*I can just treat this even though it is a symbol, operators are just like polynomial.*0789

*You can distribute them.*0796

*This says take the second derivative of the sin 5X and then add to it this V of X × the sin of 5X.*0799

*It is this and operate this way.*0814

*The operations, you can distribute the operations the same way you would distribute any number.*0818

*That is what makes these operators very powerful.*0824

*The second derivative of the sin 5X.*0829

*We already found that this before.*0831

*This first part is going to be -25 sin 5X and here we have whatever V happens to be × the sin 5X.*0834

*That is our new function from this operator, it is that simple.*0846

*You just do exactly what it says and you treat the operator whether it is 2 things, a binomial operator, a trinomial operator,*0851

*a quadranomial operator, you just distribute the way you do anything else.*0857

*Let us see example number 3, I hope we are not elaborating the point too much but I think it is always good to see a lot of examples.*0866

*Example 3, this time we will call the operator C.*0874

*It is equal to - i × H, I will do H bad DDX.*0877

*By the way, H ̅ is just planks constant divided by 2 π.*0880

*It is just some shorthand notation for it and we will see it again.*0895

*It is just a constant, that is all it is.*0898

*This says if I perform C on a given function, I’m going to take the first derivative of the function then I’m going to multiply it by H and multiply by -1.*0901

*Clearly, operator can be complex, imaginary as well as real.*0913

*It just do something to a function and get a new function.*0919

*This time, we want you to find this of the function e ⁺INX, where n is just some number.*0924

*The operator of A ⁺INX = this – I × H ̅ DDX of e ⁺INX.*0937

* When I go ahead and take the derivative of this, I'm going to get.*0954

*This is going to be - i × this H ̅, the derivative of e ⁺INX is IN e ⁺INX derivative of the exponential function in e ⁺INX.*0959

*We have - I² H ̅ and A ⁺INX.*0979

*I² = -1, - -1 that becomes +1.*0989

*You are left with just H ̅ Ne ⁺INX.*0993

*If you have noticed with the previous samples or example number 1,*1005

*notice, the original function was e ⁺INX.*1011

*Operating on that gave me back something × the original function.*1016

*This is going to be very important in a little while.*1025

*I probably noticed it with the first example, it is another example.*1027

*You may notice it in a few more examples before we actually talk about other things.*1031

*I just want to bring that your attention.*1037

*Interesting enough, sometimes the operator will change and become a bit of a completely different function.*1039

*Sometimes what the operator only does is multiply the original function by some constant, that is very important.*1044

*Let us go back to blue here for our examples.*1053

*Let us do example 4.*1057

*Example 4, this time our operator D = DDX.*1063

*This is the partial differential operator.*1070

*For those of you who have not done partial differentiation, there is actually nothing to learn.*1072

*If you have a function of 2 variables, let us say X² Y.*1078

*All you are doing when you are taking the partial, just take the derivative only with respect to X.*1083

*It means hold every other variable constant, that is all you are doing.*1087

*You already know what to do here.*1091

*Find D of XY² Z³.*1097

*In this particular case, we have a function of 3 variables X, Y, and Z.*1104

*It happens to be XY² Z³.*1109

*This operator is asking you to take the partial derivative of this whole function with respect to X.*1112

*All that means is that Y² is a constant, Z² is a constant.*1120

*They do not exist, you just leave them alone.*1123

*Let us see what we have got.*1128

*I’m sorry this is DDZ not DDX.*1131

*Sorry about that.*1137

*We are going to hold X constant, we are going to hold Y² constant, and we are going to differentiate just the Z³.*1137

*D of XY² Z³ = DDZ of XY² Z³.*1145

*This is a constant so it stays XY² and the derivative with respect to Z is 3Z².*1161

*I will just write it like this.*1169

*If I want to put a number in front, I can, not a problem.*1170

*You can write it anyway you want.*1174

*You can leave like this or you can write it this way.*1176

*There you go, that is it.*1178

*In this particular case, we start with a function and we end up with a different function.*1181

*This particular operator does not just multiply the original function by a constant where is the one before did.*1185

*Sometimes it does, sometimes it does not.*1193

*Again, that is going to be very important the differentiation between the two.*1195

*Let us go ahead and do another example.*1200

*This is going to be example 5, and this time we will go ahead and call our operator L.*1206

*The operators is going to be D² DX² + 2 DDX -3.*1213

*All this operator says is that if you are given some function, take the second derivative, add to it 2 ×*1230

*the first derivative of it and then subtract the number 3.*1238

*That is it, it just as a symbol, it is an operator.*1241

*It is saying do this.*1244

*We have got CL and this time we want you to do is see what is that we are going to find.*1246

*We want you to find L of X³.*1254

*L caret of X³ well that is equal to D² DX² + 2DDX - 3 of this function X³.*1264

*We just distribute, this one, this one, and this one had.*1281

*Adding and subtracting, very simple.*1285

*We get D² DX² of X³ + 2 × DDX of X³ – 3.*1288

*We have got the second derivative.*1312

*We have got 3X² and we got 2 × 3X so we are going to end up with 6X over here.*1315

*And this one we are going to have the derivative of the 3X³ is going to be 3X².*1322

*It is going to be 2 × 3X² this is going to be - 3X³.*1332

*We end up with 6X + 6X² -3X³.*1341

*Nothing strange, nice and normal.*1352

*Let us go ahead and do one last example.*1358

*This time we are going to perform operators sequentially.*1365

*Here we will let the operator A =- I H ̅ DDX.*1372

*We will let the operator B = X³.*1383

*When you see an operator equal to some function, that means multiply the function that you get by this.*1388

*In other words, when I'm operating B for example, if I do B of X² it is going to be X³ × X².*1393

*When some operator is just some function, it means multiply the function that you are supposed to operate on by this.*1403

*It is just multiply by, that is all it is.*1409

*I will write that here.*1413

*Sometimes you just need to do that, you need to multiply by X³.*1415

*Our task in this example is to find A caret, B caret of sin X.*1422

*We also want you to find B caret A caret of sin X.*1433

*It will be both.*1439

*Operators and sequence, when they are written like this, you start from the right most operator and work with your left.*1441

*In this particular case, we will do this first one.*1449

*A caret B caret of sin X.*1452

*Let us make B a little bit more clear here, sorry about that.*1458

*A caret B caret of sin X that is equal to A caret of B caret sin X.*1461

*I’m going to perform B first and I’m going to perform A on what it is that I got.*1470

*This is going to be A caret, now B sin X, B sin X was multiplied by X³ so I’m going to get X³ sin X.*1477

*A, perform the operation A on X³ sin X that is going to equal – I H ̅ DDX of X³ sin X.*1488

*We have a product rule here.*1504

*X³ and sin X are both functions of X, we will go ahead and leave that one out.*1505

*We end up with is - I H ̅ this × the derivative of that is s going to be X³ × the cos X + that × the derivative of this.*1510

*It was going to be 3X² sin X, there you go.*1522

*This is A caret B caret, we perform the B first then we performed A.*1526

*Let us go ahead and do the other one.*1542

*Let us do B caret A caret of sin X.*1544

*That is equal to B caret, we will do A first.*1549

*A caret of sin X is going to equal B caret of - I × H ̅ DDX of sin X = B caret,*1554

*The derivative of sin X is cos X, we have – I H ̅.*1569

*This is going to be cos X and then B means multiply by X³ so we end up with –I H ̅ X³ × the cos X.*1576

*This one, we perform the operation A first and then we apply the operator B.*1594

*Notice, in general, in this particular case AB does not equal BA.*1603

*Operators do not commute.*1611

*In other words, you know the 2 × 4 is 4 × 2, that is the property of the real number system, that is commutability.*1613

*Operators do not commute in general.*1620

*This like matrix multiplication, they do not commute in general.*1622

*In general, AB performed on some function F does not equal BA performed on some function F.*1626

*Operators do not commute, in other words.*1641

*Operators do not commute, in general that has profound consequences for quantum mechanics.*1647

*There are going to be times when the operators do commute, that has profound consequences for quantum mechanics, not commute in general.*1659

*We will be seeing this again.*1666

*In general, operators do not commute.*1668

*Let us go ahead and talk about our next topic here.*1675

*Back to operators, we have defined what operators are and done some examples, now operators can be linear or nonlinear.*1679

*Now we are going to give a very specific mathematical definition of what linear is.*1687

*Those of you who studied linear algebra, you already know this definition or you have seen it.*1692

*Those of you who have not done linear algebra, this is going to be they real mathematical definition of what linear means.*1696

*Linear does not just mean that the exponent on a variable is 1.*1703

*You have treated it like that for years now, ever since middle school*1707

*but now we are going to give you what the mathematical definition is, the criterion for linearity.*1710

*Operators can be linear or nonlinear.*1717

*We deal only with linear operators.*1733

*In quantum mechanics, we are only concerned with linear operators.*1736

*We deal only with linear operators which is very convenient because non linear operators are quite difficult.*1740

*Here is the definition of linear.*1754

*Here is what it say, they are 2 things that you have to check when you are given some operator to check whether is linear.*1763

*The definition is A of F + G.*1771

*I’m not going to use the X, these are functions of X or functions of Y, or function of whatever.*1780

*I’m not going to put the variable.*1789

*I’m just going to put F+ G.*1790

*It is equal to A of F + A of G.*1793

*What this says is the following.*1799

*We know that operators are things that you do to functions, you operate on a function.*1802

*A linear operator has to satisfy this, it says that if I’m given a function F and I’m given a function G,*1808

*if I add those two functions first and then operate on what I get when I add them,*1815

*that I will get the same thing if I operate on F separately, operate on G separately and then add them.*1821

*That is what linear means, it means I can switch the order of addition and operation.*1828

*Add first then operate, or operate first then add, that is what linear means.*1832

*That was the first thing you has to satisfy.*1841

*The second thing that you saw was the following.*1844

*A of CF = CA of F.*1847

*If I'm given some function and if I multiply the function by some constant C and operate it, I should get,*1853

*If the operator is linear, it means I can go ahead and take the function, operate on it first and then multiply by the constant.*1861

*Here linearity implies that I can switch the order of operation and multiplication by a constant.*1868

*Also, I can switch the order of addition of two functions and operation or operation than addition.*1874

*These two things have to be satisfied when for an operator to be called linear.*1883

*When you are presented with an operator, in order to check linearity you have to check these two things.*1890

*Let us go ahead and write that down.*1900

*Confirm linearity we have to verify that for a given operator, then 1 and 2 are satisfied for a given operator.*1902

*Let us go ahead and do some examples.*1961

*This is the only way this is going to make sense.*1962

*Determine whether the operator defined by A of F = S² is linear or nonlinear.*1969

*This is a different way of defining it.*1975

*Notice, in the previous examples I gave you the operator and I set it this.*1978

*Here it actually specifies it explicitly.*1981

*A of F is the same is just S².*1983

*Operating on F means just taking a function S and squaring it, that is what the operation is.*1986

*The operation square, that is what it is.*1992

*We have to show whether this is linear or not.*1996

*Here is what we have to verify, the definition of linearity.*1999

*Let me go ahead and work in red here for these examples.*2002

*I have to show that A of F + G= A of F + A of G.*2006

*I’m given two functions F and G.*2021

*I’m going to add them and then operate them and square it.*2023

*What I’m going to do is square the square of G and I’m going to add them.*2026

*I’m going to see if the left side and the right side are the same.*2029

*If they are, it is linear, it is a linear operator.*2031

*If not, it is not a linear operator.*2034

*It is that simple.*2036

*Let us go ahead and do, I will go ahead and write the second one too.*2038

*I have to show that A of C of F = C A of F.*2041

*In other words, I'm going to take F and I’m going to square it and I’m going to multiply by a constant.*2047

*And then I'm going to take F and multiply by a constant and I will square it.*2054

*If those two ends up being the same, it is linear.*2056

*If they end up not being the same, it is non linear.*2058

*Both have to be satisfied.*2060

*One might, the other might not, that does not count.*2062

*Both have to be satisfied.*2065

*Let us go ahead and check number1.*2068

*Let us go ahead and do A of F + G.*2072

*A of F + G, A of F is squaring.*2078

*If I take F + G and I square it, that is going to equal F + G².*2083

*F + G² I just multiply that out, that is equal to S² + 2 FG + G².*2090

*This is my left side, this side right here.*2101

*Now the question is, does it equal A of F + A of G?*2105

*A of F = F².*2118

*A of G = G².*2121

*Does F² + 2 FG + G²= F² + G²?*2125

*No, it does not.*2129

*This is not a linear operator.*2131

*It is that simple, you just have to perform the operations on the left, operations on the right, and see if they are equal.*2133

*At this point I can stop, number 1 is dissatisfied.*2139

*Therefore, it is not linear.*2145

*I do not have to check 2.*2148

*However, if you want to go ahead and check 2, that is not a bad idea.*2148

*A of CF that is the left side.*2154

*A of CF or A of whatever = the whatever².*2158

*This is equal to CF² which is equal to C² F².*2162

*Our question is if this equal to C of A of F?*2169

*A of F is F², this is C of F².*2178

*Does C of F² = CF²?*2182

*It does not, not linear.*2184

*You do not have to do 1 before 2, you can do 2 before 1.*2188

*It does not matter.*2191

*This is not a linear operator.*2191

*The operation that tells you to square something, whatever it is that you are given is not a linear operator.*2195

*But you knew that already.*2202

*You knew that from the fact that back in high school and calculus, it is not linear, it is quadratic.*2204

*Quadratic functions are not linear.*2210

*Let us go ahead and do this one.*2215

*Determine when the operator defined by A of F = D² DX².*2218

*F is the operation of taking the second derivative of something a linear operator.*2225

*Let us find out.*2230

*I think I may go to black, I’m sorry.*2232

*Again, let me go back to black, sorry.*2236

*We need to show that A of F + G = A of F + A of G.*2238

*Add first then operate, operate on each then add.*2247

*They have to be equal to each other.*2250

*Let us go ahead and do the left side first.*2254

*A of F + G = D² DX² of F + G.*2256

*I know that when I do differentiation, I know that the differential operator is linear.*2266

*I know it from calculus, we do not call it an operator but the process of differentiation is linear.*2271

*Therefore, this is going to equal D² F DX² or if you like F double prime + D² G DX² or G double prime.*2277

*Let us do A of F + A of G.*2291

*This is going to equal D² D A² of F is this + A of G is going to be D² G DX².*2298

*They are equal.*2318

*Let us go ahead and do 2.*2320

*We operate A of CF, that is going to equal the second derivative of this thing C of F.*2324

*I know I can pull constants out, that is equal to C D² F DX² or CF double prime.*2331

*And if I have C of A of F that is equal to C × D² F DX².*2342

*This and this are equal.*2354

*Yes, this is a linear operator.*2356

*You just have to check 1 and check 2.*2361

*Let us see the next one, determine whether the operator defined by A of F = LN of F is linear or nonlinear.*2368

*I’m given some function and I take the log of that function, that is my operator, taking the log of whatever is that I’m given.*2377

*Is this linear or nonlinear?*2384

*I think you are accustomed to this, it is good to write out what the criterion is.*2387

*We need to show that A of F + G, the definition in other words = A of F + A of G and we need to show that A of C of F = C × A of F.*2394

*Let us go ahead and do A of F + G = log of F + G.*2412

*A of F + A of G = the log of F + log of G.*2423

*This and this do not equal each other.*2439

*I will just go ahead, at this point we can stop, it is nonlinear.*2445

*However, let us go ahead and do the other one.*2448

*We have A of C of F = the log of C of F.*2451

*I will go ahead and put our little caret there.*2462

*C of A of F, I can probably do a little bit more with this one.*2464

*The log of something × something, let us go ahead and expand it.*2477

*This is going to be equal to the log of C + the log of F.*2480

*We will go ahead and leave that one.*2487

*The C of F of A = C × the log of F.*2490

*This and this, they are not equal.*2497

*This is not a linear operator.*2501

*We have introduced the Schrӧdinger equation.*2512

*We have introduced the notion of operators which is profoundly important in Quantum Mechanics.*2514

*We will go ahead and close this lesson off like this.*2520

*Thank you so much for joining us here at www.educator.com.*2522

*We will see you next time.*2524

0 answers

Post by dulari hewakuruppu on April 9, 2015

I cannot find the link to download the lecture slides ..could you kindly help me locate it

1 answer

Last reply by: Professor Hovasapian

Thu Mar 12, 2015 4:19 AM

Post by James Lynch on March 9, 2015

Where do I find the appendix referred to in this lesson?

1 answer

Last reply by: Professor Hovasapian

Sun Feb 22, 2015 7:48 PM

Post by David LÃ¶fqvist on February 22, 2015

Example 1. You went from asking for Ã‚Sin(x) to solving for Ã‚Sin(5x)?