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Lecture Comments (2)

1 answer

Last reply by: Professor Hovasapian
Mon Feb 16, 2015 2:39 PM

Post by Anhtuan Tran on February 14, 2015

Hi Professor Hovasapian,
Are you going to use the Dirac's Notation (or also known as Bra-ket notation) in this course. If so, where can I find it?
Thank you.

The Harmonic Oscillator III

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
  • The Harmonic Oscillator III 0:09
    • The Wave Functions Corresponding to the Energies
    • Normalization Constant
    • Hermite Polynomials
    • First Few Hermite Polynomials
    • First Few Wave-Functions
    • Plotting the Probability Density of the Wave-Functions
    • Probability Density for Large Values of r
    • Recall: Odd Function & Even Function
    • More on the Hermite Polynomials
    • Recall: If f(x) is Odd
    • Average Value of x
    • Average Value of Momentum

Transcription: The Harmonic Oscillator III

Hello, welcome back to and welcome back to Physical Chemistry,0000

Today, we are going to round out our discussion of the quantum mechanical harmonic oscillator.0004

Let us jump right on in.0008

We said that our energy levels for the quantum mechanical harmonic oscillator, our energy levels are as follows.0012

We found that E sub R = H ̅ ω × R + ½, where R is that harmonic oscillator μ quantum numbers.0024

R= 012 and so on, all the way through.0040

Now let us talk about the wave functions.0046

We saw the Schroeder equation, we are finding energies and we are finding wave functions.0049

The wave function represents what the particle is doing.0053

We do things to the wave function in order to extract information about the actual physical system that we are dealing with.0057

The wave functions ψ sub R corresponding to the energies.0064

For each R value you have the energy and you have the wave function ψ corresponding to the wave functions.0083

No, the wave functions corresponding to the energies.0092

They look like this.0103

Ψ sub R= N sub R H sub R Α ^½ × X E ⁻Α X²/ 2.0110

This is reasonably complex.0130

Let us see what each of these are.0131

Α here is going to equal to K × μ divided by H ̅ ^½.0135

We introduce some symbols just to make it look a little more clean.0146

N sub R, let us go ahead and do red.0155

N sub R is the normalization constant.0158

We want to normalize so the integral was equal to 0.0166

The integral =1, the normalization constant.0172

N sub R = 1/ 2 ⁺R × R factorial ^½ × Α/ π¹/4, there we go.0179

The H sub R they are called the Hermite polynomials.0202

H sub R Α ^½ X, they are called Hermite polynomials.0211

They are part and parcel of the solution of the particular Schrödinger differential equation.0228

H sub R is called the first degree hermite polynomial.0234

It is not going to be the end of the world if you say hermite, it is not a big deal.0252

Let me go ahead and lists the first few hermite polynomials, 0258

then I will go ahead and list the first few of the actual complete wave functions for the harmonic oscillator.0261

Let me do this in blue.0271

H sub R, this is in parenthesis Α ^½ X.0273

It is whatever is in the parenthesis, goes into the variables.0278

The first few hermite polynomials0293

We have H sub 0 of Z = 1.0309

H sub 1 of Z = 2 Z.0318

H sub 2 of Z = 4 Z² – 2.0325

H sub 3 of Z = 8 Z³ - 12 Z.0338

Let us go ahead and do H sub 4 Z that is equal to 16 Z⁴ -48 Z² + 12.0350

We said H sub R of Α¹/2 X.0365

If we did H of 2, this one right here.0373

4 Z² - 2 it becomes 4 × Α¹/2 X² -2.0382

Whatever is in parentheses goes into the variable.0392

It is just basic functional notation.0395

Let us go ahead and do the first few, these are just the hermite polynomials.0399

Let us do the first few wave functions, the complete wave functions.0407

When we put everything together, normalization constant, hermite polynomial, and the rest of it.0413

The first few wave functions we have, Z sub 0 of X = Α/ π¹/4 E ⁻Α X²/ 2.0418

Z sub 1 of X = 4 Α³/ π¹/4 × X × E ⁻Α X²/ 2.0438

As you can see, these tend to be very complicated very quickly.0455

But again, most of it is just that the constants that tend to be unwieldy.0458

The functions themselves are not that difficult to deal with.0463

Ψ sub 2 that is what we are, we are at number ψ sub 2.0467

It is going to end up being Α/4 π¹/4 × 2 Α X² -1 × E ⁻Α X²/ 2.0472

Let us go ahead and finish up with ψ sub 3 so we can move on to some other things.0489

It is going to equal Α³/ 9 π¹/4 × 2 Α X³ -3 X × E ^- Α X²/ 2.0493

This is just a representation of the first 4 wave functions.0514

Let us go ahead and actually plot these 4 wave functions to see what they look like.0518

Let us plot the probability density which is ψ².0524

Let us go ahead and do that. Let us do this on one page here.0532

Let me go ahead and just do this.0536

This is going to be the wave function, this is going to be ψ of R of X.0555

And over here we are going to do the probability density that is you remember, the modulus.0560

That is the ψ sub R² which is nothing more than ψ complex × ψ itself.0565

Since this is real, this is going to be to ψ² but this is the symbol, just in case.0572

This is going to be the 0, this is going to be the 0.0578

We have something like a high point there.0584

This is going to be to R =0 and here we are going to have the probability density of that.0596

This is the wave function, this is the probability density, this is where you are most likely to actually find the particle.0611

This says that when you are at the R= 0, when your energy level is E sub 0 = ½ H ̅ ω.0625

When you are at the first energy level, the chances are that 0637

you are more likely to find the particle near the equilibrium position than you are near the extremes, near the amplitudes.0641

We will go ahead and go to R = 1.0651

We are going to end up with something which is this way, this way.0657

Then we will go ahead and do the probability density for this.0663

We are going to end up with node there and there.0665

It is going to be like this, something like that.0670

This is the first energy level.0676

We will go ahead and do R2, R=0 R =1 R =2.0680

We have something like this.0687

That is the wave function and now we will go ahead and do the probability density.0700

This is going to end up with 0 here, and 0 here, a high point and high point.0704

This is going to be something like that.0711

This is the second energy level.0716

We will go ahead do that and that.0719

Let me see what we got.0726

We got 123, 123, we have a low point, a high point, low point, and a high point.0728

Of course we are going to have 1, 2, 3.0744

We are going to have a high point 01234, something like that.0750

The wave function as the wave function goes from as R goes from 0 to 1 to 2 to 3, the energy is rising.0758

This is just the wave function.0767

It is a probability density that gives us the most information.0769

At the lower energy levels, the 0, 1, 2, we tend to find it mostly in the center.0772

As the quantum number rises 12345, you notice that it is more evenly distributed.0779

It means that the particles or the particle is spending more time sort of evenly distributed.0785

This getting a little bit closer to the edges.0791

Notice, it is spending more time everywhere instead of spending most of its time towards the equilibrium position.0794

That is all that is happening here.0799

This is the wave function, this is the probability density, this gives us the probability of where you actually going to find the particle.0800

The places where it is 0, you are not going to find the particle here, that is what this is saying.0808

You are not going to find the particle there.0815

Those are nodes of the places where this is X.0816

Here is the 0, the equilibrium position.0821

As you go farther and farther away, that this is the oscillation point.0823

The amplitude this way - amplitude this way, this is the equilibrium position.0831

As you get much higher and higher and higher in energy, you are going to find 0835

the particle distributed more evenly between the amplitude and the - amplitude between here and here.0839

But there are places where you absolutely not find a particle, that is what this means.0848

Places where the probability density is 0, you will never find the particle there. 0852

Because we are talking about something that is quantize.0856

For large values R, as R gets really big and R goes on to infinity.0865

Where large values of R, the probability density looks like this.0877

We will end up with something looking like this.0900

The correspondence principle, you remember we talk about it once.0910

The correspondence principles says as quantum numbers increase, 0922

the quantum mechanical system starts to display classical mechanical behavior.0934

In this case, as the energy and displacement rise, 0963

the particle is more likely to be found at the extremes points, the turning points.0987

Because it is moving more slowly at the extreme points.1032

It is basically telling me that as the quantum number rises, as energy rises, 1048

as displacement start to increase, you are more likely to find the particle near the edges.1055

The particle is going like this, back and forth.1062

At the extremes, it is actually moving quite slowly because it slowing down at 0.1066

Its kinetic energy is virtually 0 so it is moving more slowly.1071

Because it is moving more slowly, you are more likely to find it at the extremes of the equilibrium position, the center.1077

You are more likely to find it because it is passing through the equilibrium so fast, the kinetic energy is maximized.1084

It is going this way and zipping through the center and coming here.1090

It is zipping through the center, it is slowing down going there.1095

This is what the classical harmonic oscillator would do.1100

It would spend more time at the extremes, less time in the center.1104

Again, as the quantum number increases, the correspondence principle says that 1109

the quantum mechanical behavior is going to start to look more like a classical mechanical harmonic oscillator.1114

It is going to spend more time at the extremes and less time at the equilibrium position.1119

In other words, it is going to be vibrating so much.1123

You are virtually not going to find the particle at its equilibrium position.1125

You are going to be finding it more often as its extreme points.1129

Let us go ahead and finish off with a discussion of the hermite polynomials.1137

Let us recall, an odd function is if you were to put -X in for X for the function, you end up getting negative of the original function back.1146

You put in -X into the function and what you get back is actually negative of the original.1167

That is an odd function.1173

An even function says that if you were to put -X into your function, you just get the original function back.1175

And odd function is symmetric about the origin.1186

An even function is symmetrical about the Y axis.1189

Something like this.1194

The cubic curve as an odd function, it is symmetric about the origin.1197

X² is symmetric about the Y axis.1202

The hermite polynomials, we are doing this discussion in order to help us with our math.1207

The hermite polynomials are even when R is even and they are odd when R is odd.1219

Recall also that if F of X is odd, then the integral from - infinity to infinity of F of X DX = 0 1238

because we are integrating below the X axis, above the X axis the integral cancel.1255

Ψ sub R= N sub R H sub R of Α¹/2 X E ⁻Α X²/ 2.1264

When R is even, ψ sub R is even.1285

When R is odd, the ψ sub R is odd because this is an even function.1295

Therefore, when the hermite polynomial is even, when R is even, the hermite polynomial is even which makes this whole thing even.1309

When R is odd, the hermite polynomial is odd which makes and odd function × an even function, it makes it odd.1317

In either case, ψ sub R² is even.1326

If ψ sub R is even, the square of it is even. If the ψ sub R is odd, the square of it is even.1344

The expectation value of X, the average value of X is equal to the integral – infinity to infinity.1357

Ψ sub R complex ψ sub R × X × ψ sub R = the integral from –infinity to infinity of X ψ sub R².1367

That is odd because F of X = X is odd.1397

Add an odd function × an even function, you have a integral of odd function.1405

This integral is equal to 0.1409

The average value of X =0.1417

If I take a 1,000, a 1,000,000 measurements, on average I’m going to find that the particle spends time in the middle,1421

simply because it is spending an equal amount of time on this side and an equal amount on this side.1432

It averages out to 0.1437

The average momentum is going to be the integral –infinity to infinity.1440

Remember, ψ sub R complex × the momentum operator which is - I H ̅.1448

The average value of any particular thing that we are trying to measure 1464

is going to equal the wave function × the operator of the wave function, that is the definition.1468

- infinity to infinity ψ sub R complex - I H ̅ DDX of ψ sub R.1479

If ψ sub R is even, it implies that the derivative is odd.1495

And if ψ sub R is odd, that implies that ψ sub prime R is even.1504

In either case, ψ sub R × the momentum operator acting on ψ sub R is odd.1514

If this is even, this part is odd.1533

If this is odd, this part is even.1536

In either case, I have an odd × an even function which is an odd function.1539

In either case, the integrand is going to be odd.1543

Therefore, the average momentum is also 0.1545

All that means is that it is spending as much time going this way, as it is going this way.1552

It is oscillating back and fourth.1556

That is another preference for going this way or that way.1558

On average, I might find it going this way or this way.1560

At any given moment, I might find it going this way or this way.1566

On average, it is going in both directions or going 0.1568

The harmonic mechanical oscillator, the average value of X is 0 and the average value of P is 0.1576

Thank you so much for joining us here at

We will see you next time, bye.1589