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Lecture Comments (5)

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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

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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?

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Last reply by: Professor Hovasapian
Sun Feb 22, 2015 7:48 PM

Post by David Löfqvist on February 22, 2015

Example 1. You went from asking for ÂSin(x) to solving for ÂSin(5x)?

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 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

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 school1707

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