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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### 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
- The Harmonic Oscillator III
- 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

- 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

### Physical Chemistry Online Course

### Transcription: The Harmonic Oscillator III

*Hello, welcome back to www.educator.com 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 polynomials*0293

*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 www.educator.com.*1587

*We will see you next time, bye.*1589

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.