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Lecture Comments (4)

1 answer

Last reply by: Professor Hovasapian
Thu Aug 18, 2016 6:51 PM

Post by Joseph Cress on August 18, 2016

What is the main reasoning for normalizing a function and how do we know if the function is already normalized or not.

Thank YOU

1 answer

Last reply by: Professor Hovasapian
Fri Feb 26, 2016 12:57 AM

Post by bohdan schatschneider on February 9, 2016

The math doesn't works out for your Hermitian example using the momentum operator.  The pi should be squared within the ^1/4 bracket and this becomes a pi^1/2 which then cancels out all of the other pi^1/2's.  You did it for the left and right hand sides!  I spent 30 min trying to figure out where I went wrong dang-it.  Otherwise, good example.

The Postulates & Principles of Quantum Mechanics, Part 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 Postulates & Principles of Quantum Mechanics, Part III 0:10
    • Equations: Linear & Hermitian
    • Introduction to Hermitian Property
    • Eigenfunctions are Orthogonal
    • The Sequence of Wave Functions for the Particle in a Box forms an Orthonormal Set
    • Definition of Orthogonality
    • Definition of Hermiticity
    • Hermiticity: The Left Integral
    • Hermiticity: The Right Integral
    • Hermiticity: Summary

Transcription: The Postulates & Principles of Quantum Mechanics, Part III

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

Today, we are going to continue our discussion of the Postulates and principles of Quantum Mechanics.0004

Let us jump right on in.0008

Postulate 2, we saw that before, says that the quantum mechanical operators are linear and that are hermitian.0013

Let us go ahead and recall what we mean.0022

The definition of linear says that if I operate on the sum of the wave functions,0031

it is the same as operating on them individually and then adding them afterward.0040

Then also this is the second part of ψ and ψ equals ψ × operator ψ.0048

That is what linear means.0058

You can add the wave functions, operate on it, you will get the same answer if you operate on them individually and add afterward.0058

You are switching order.0065

This says that if you take some wave function and multiply by a constant and then operate on it,0068

it is the same as operating on the function first and then multiplying by a constant.0072

This is very deep property.0076

Not all functions, not all operators actually share this property so this is very special.0078

And the definition of hermitian is this.0084

Let me just go ahead and write it out and then we will go ahead and talk about it a little bit.0096

It equals the integral of G, the operator complex conjugate, and * complex conjugate, where F and G are any two wave functions.0102

What this says that is, if I have two wave functions F and G, if I were to operate on one let us say G,0136

If I were to operate on one of them and multiply by the complex conjugate of the other and then take the integral,0147

I end up getting the same answer as if I take this one, if I take the complex conjugate of the first one, in this order,0154

And if I actually operate on it with a complex conjugate operator and then multiply by G and integrate that.0165

Do not worry about what this means in terms of the deeper mathematics and the deeper implications of the structure of what is going on0175

with these operators and with these functions, this is just the definitions.0183

In Quantum mechanics, if you do not entirely grasps what is going on, it is fine.0186

Quantum mechanics is entirely mathematical.0191

You can treat it mathematically, you could deal with it as some as a bunch of symbols that you are going to manipulate0194

and you will get the answer that you seek.0199

These idea of hermitian or hermiticity will grow as you get a little bit more experience with the functions and the operators.0201

You will get a better sense of it.0210

Now, this is just the definition.0211

We want to work with the definition.0214

Here is why this hermitian property is important.0222

An operator that satisfies this equation, the hermitian property guarantees that the Eigen values of0229

the operator in question are always real numbers.0252

Any operator that satisfies this equation, this hermitian property, it guarantees that the Eigen values0264

of that operator are going to be real numbers.0270

You remember that Eigen value equation, C = A sub N C sub N.0273

It guarantees that these are real.0283

We know that operators can be complex and wave functions can be complex.0285

But this hermitian property guarantees that the Eigen values are real.0289

Why do the Eigen values have to be real?0293

Let us write that question.0297

Why do the Eigen values have to be real?0298

Why do we care that the Eigen values be real?0300

Let us write EV.0307

Because from other postulates, the previous postulates,0313

we know that when we make measurements what we observe are the Eigen values of the operator in question.0327

For example, if we are measuring the kinetic energy of a particular particle, we measure the kinetic energy.0362

The value that we measure is actually going to be one of the Eigen values of the kinetic energy operator,0372

operating on the particular wave function for that particle, for that quantum mechanical system.0376

We know this already.0380

What we observe are the Eigen values.0381

Measurements that we observe are the Eigen values of the operator.0384

If we are observing some kinetic energy 1.9 × 10⁻¹⁸ J, that has to be a real number.0396

What we observe, what we measure in the real world, they have to be real numbers.0407

They cannot be complex numbers.0412

They cannot be imaginary numbers, they have to be real.0412

This is why the Eigen values have to be real because if we are observing them, they are real.0415

If we are observing them, they must be real because we can see them and we can actually measure them.0423

Again, we see that both operators and wave functions can be complex.0450

The hermitian property guarantees that at least the Eigen values ore always guaranteed to be real, that is why.0456

We have seen that operators and wave functions can be complex.0468

The hermitian property guarantees that the Eigen values are always real numbers.0493

Again, we have C sub N = A sub N C sub N, these are real.0522

This can be complex, this can be complex.0530

That is going to be real because that is what we are going to observe.0533

Our hermitian operator, now that we know we are dealing with operators that are always going to be hermitian.0539

We are going to be dealing with hermitian operators so you do not have to worry about that.0547

Our hermitian operators give rise to a sequence of Eigen functions.0551

C sub 1 C sub 2 C sub 3, this equation when you solve it for C, you can get a sequence of wave functions.0567

Not just one function but many, the different states of the quantum mechanical system.0577

Each wave function represents a particular quantum state of that system.0581

They are sequence of Eigen functions.0586

These Eigen functions display another incredible property.0596

They are orthogonal.0624

In other words, they are perpendicular in some sense.0629

You are accustomed to vectors being perpendicular.0634

We think of a vector being perpendicular to another.0638

You remember from your study of vectors in physics when you take the dot product of two vector and when the dot product equals 0,0641

That means that the vectors are perpendicular.0647

Perpendicularities are a geometric notion.0651

It is something that is dependent on a picture.0653

We understand perpendicular, the deeper underlying mathematics we call it orthogonal.0656

Vectors are orthogonal when their dot product equals 0.0662

Functions are orthogonal when their integral equals 0.0666

Again, the Eigen functions display another incredible property.0672

They are orthogonal, they are perpendicular.0675

Here is the definition, if ψ sub M and ψ sub P,0678

Ψ are just integers, ψ sub 1, ψ sub 3, ψ sub 2, ψ sub 5.0685

The 2nd Eigen function, the 5th Eigen function, that is all those are.0691

If ψ sub M and ψ sub P are two Eigen functions of a hermitian operator, then the integral of ψ sub M conjugate × C sub N is equal to 0, always.0696

If it is a hermitian operator, the sequence of functions that it generates,0726

if I take any two of those functions, multiply them together, making one of the complex conjugate.0730

If I take the integral it is always going to equal 0.0736

This is a profoundly deep and important property.0738

If it is analogous, two vectors being perpendicular, that is all it is.0741

Orthogonal is the mathematical definition of perpendicular or perpendicular to geometric notion.0745

We have to see it.0750

Orthogonal was a deeper algebraic property.0751

This is just a mathematical definition.0755

You take the function, you take another function, you take the complex conjugate one of the other functions,0757

You multiply the two wave functions and integrate it over the particular region of interest and0763

you are going to get 0 always, if the operator is hermitian.0769

Any set of functions that satisfies the integral ψ sub N ψ sub B = 0 for N not equal to P is said to be orthogonal, the set itself.0777

I have a set of functions that are Eigen functions for the kinetic energy operator.0812

That set of functions is called an orthogonal set.0817

If the actual wave functions, if the ψ sub N also happen to be normalized, we will always deal with normalized functions, by the way.0825

If they are not normalized, we will normalize them first to always deal with them to make it easier.0839

If they happen to be normalized, the set is called orthonormal.0843

Just a little bit of vocabulary here.0848

The set is called orthonormal.0850

For those of you who have taken linear algebra or perhaps read a little bit about matrices,0854

determinants, and vector spaces in some of your sideline reading,0859

You realize that we are using the same terms.0863

We talk about an orthogonal set of vectors.0865

We talk about an orthonormal set of vectors, it is the same thing.0868

They are completely analogous, orthonormal.0871

The sequence of wave functions for the particle in a box is an example of an orthonormal set.0876

Recall, the ψ sub N for the particle in a box of X, we will just keep it as particle in a 1 dimensional box, is equal to 2/ A ^½ sin N π/ A × X.0911

N goes from 1234567.0929

All of these, they actually form an orthonormal set.0934

You can confirm for yourself either with a software or by hand.0937

You can confirm for yourself and we will do some when we actually start doing some of the problems, do not worry about that.0942

The integral from 0 to A of 2/ A¹/2 × the sin of N π/ A × X × 2/ A¹/2 of sin of P π/ A × X DX is going to equal 0,0955

Where N does not equal P.0987

In other words, there is going to be ψ sub 5, ψ sub 9, N is 5 and P is 9.0989

As long as your N and P are not equal, this is going to equal 0 always.0996

Now the definition of orthogonality straightforward.1004

The definition of orthogonality is straightforward.1010

There is nothing strange about it mathematically.1021

It is very straightforward.1025

It just says take the integral of ψ sub N * ψ sub P, and it is always going to equal 0.1026

Take a function and your sequence, take an Eigen function, take another Eigen function, take the complex conjugate of it.1033

If it is real, it is going to be the same.1041

Multiply them together and integrate, you are going to get 0.1043

Very straightforward.1045

The definition of hermitian or hermiticity is a little more complicated.1047

This is the one we want to do an example of.1053

We want to see how it is we are going to work with this integral.1058

The definition of hermiticity or hermitianess, whatever you like, there is no propriety when it comes to verbal descriptions of these things.1061

Do not get caught up in the vocabulary of the word, wording of it.1075

The definition of hermiticity is more complicated.1078

It says the integral of F* A ̂ G is equal to the integral of GA ̂* F*.1086

Let us pick an operator, let us pick a couple of wave functions, and let us see how to work with this particular definition.1106

Let us confirm that the left side is actually equal to the right side.1112

Just playing with some functions because we are working symbolically and very abstractly.1115

Let us bring it down to earth a little bit well.1124

As much as possible we are given the nature of the mathematics.1127

For our operator, I should go to blue, I’m not exactly sure.1130

That is fine, as change of pace let us go to blue.1136

For our operator, let us go ahead and pick the linear momentum operator which is equal to - I H ̅ DDX.1140

Again, I’m just going to go ahead and continuously use the partial differential notation instead of the DDX notation.1154

When we are in 1 dimension, it is just the regular derivative but the partial derivatives that is the actual definition.1161

I’m just going to use that and I hope that you will forgive me if I use that notation, it is a little lazy to see but it does the same thing.1168

For our function F, let us go ahead and pick this particular function 0 of X = Α/ π¹/4 E ⁻Α X²/ 2.1177

For G, I'm going to pick another sequence, this is a specific type of wave function.1199

I will you what in just a minute.1204

Let us go, 4 Α/ π¹/4 X × E ⁻Α X²/ 2.1208

These two wave functions, they just happen to be ψ 0 and ψ 1 are the first two in the sequence of wave functions.1235

You will see these functions again in a little bit, do not worry about it.1261

These two are the first 2 in the sequence of wave functions for the quantum mechanical harmonic oscillator.1264

Remember, the Hooke’s law you have mass attached to a spring, back and forth, that is the harmonic oscillator.1278

These are the wave functions of the quantum mechanical harmonic oscillator.1288

I will write this down.1294

Α in this equation is a parameter that is related to the identity of the species that make up1295

the particular quantum mechanical harmonic oscillator.1315

For all practical purposes, we do not lose any generality if we just go ahead and take Α = 1.1319

We take Α= 1 without losing generality because it is just a constant,1328

Not doing anything to it will not affect anything so we can take it equal to 1.1338

Ψ sub 0 of X is going to equal 1/ π¹/4 E ^- X²/ 2.1344

Our ψ sub 1 of X is going to equal 4/ π¹/4 XE ^- X²/ 2.1360

X is going to be greater than or equal to –infinity and less than or equal to +infinity.1375

Let us go ahead and deal with the left integral first.1385

Again, this is going to get a little mathematically heavy but there is going to be nothing here1388

that you should not be able to actually follow because it is just algebra.1392

The left integral, we just want to go through this to get some practice.1399

The left integral is going to be, we said that it is the integral of F * Α ̂ G was going to be,1403

F* is going to be ψ sub 0 *, A was our linear momentum operator, G was our ψ sub 1.1417

This is the integral that we are going to solve first.1430

The first thing we are going to do is we are going to base on this, the integrand,1434

we are going to operate on ψ sub 1 with a linear momentum operator.1438

We are going to deal with this part first.1444

Ψ sub X hat of ψ sub 1 = - I H ̅ DDX of 4/ π¹/4 × X E ⁻X²/ 2.1448

That is going to equal - I H ̅ 4/ π.1470

When I take the derivative of this, this is going to be a derivative of the product.1488

This × the derivative of that + that × the derivative of this.1492

I will just going to go ahead and skip just one line here.1496

It is going to end up being E ⁺X²/ 2 - X² E ⁻X²/ 2.1510

We need to multiply what we just got with ψ sub*.1526

Ψ sub 0*, this is a real function, it is not a complex function.1531

The complex conjugate is the function itself.1537

Ψ sub 0 which is 1/ π¹/4 E ⁻X²/ 2.1541

The integrand ψ sub 0*, this ψ sub 1 when I take this, multiply it by that, I end up with.1553

That is fine, I will go ahead and write it out.1572

It is going to be 1/ π¹/4 E ^- X²/ 2 × - I H ̅ 4/ π¹/4 × E ^- X²/ 2 - X² E ^- X²/ 2.1574

Again, all we are doing here is just a bunch of algebra.1600

The left integral is going to be - I H ̅, when I put everything together 4/ π²¹/4 × the integral from –1607

infinity to + infinity of E ^- X² - X² E ^- X² DX.1638

When I multiply all this out, when I multiply this constant by this constant, I multiplied this function ×1650

this function of this function, distributing this is straight multiplication algebra,1657

I end up with this integral.1661

What I end up with is the following.1663

I end up with - I H ̅ × 4/ π¹/4 × π¹/2 – π ^½/ 2.1665

The integral itself, how I went from the integral to this thing right here, this is the answer to the integral on the previous page.1685

Just go ahead and let your math software do that.1695

If you do not what to use your math software, go ahead and just look in the table of integrals.1698

That table of integrals will show you that that particular integral ends up being this right here.1703

Do not worry about the actual integration part, that is not quite as important as the actual what is happening.1709

When we solve this, we end up with - I H ̅¹/2/ 2.1714

That was the left integral.1725

Let us go ahead and do the right integral.1728

We are hoping that the right integral is going to end up equally also this.1734

The right integral, because again what we are doing is we are just working with1740

the definition of the hermitian property just to get a sense of how to work with it when we are given a function,1742

when we are given an operator.1748

So the right integral is going to be G A ̂* F*.1750

That is going to be the integral of ψ sub 1 × the linear momentum operator in the X direction conjugate × ψ sub 0 conjugate.1760

The conjugate of the linear momentum operator, the linear momentum operator was –I H ̅ DDX so the complex conjugate of that is positive I H ̅ DDX.1773

Notice the minus is gone because it is complex conjugate.1797

The complex conjugate of - I is + I.1808

The ψ sub 0 * is the same as the ψ sub 0 because it is a real function.1816

P ̂* sub X of ψ sub 0 * is going to equal + I H ̅ DDX of 1/ π¹/4 E ^- X²/ 2.1826

When we actually work out this derivative, we end up with – I.1855

Let me work this one out.1875

Equals +I H ̅ × 1/ π¹/4, the derivative with respect to X of that is going to be E ^- X²/ 2 × -X.1878

This negative now comes out here so we end up getting our negative I H ̅ back, we end up with the 1/ π¹/4 and we get XE ^- X²/ 2.1906

We need to multiply the ψ sub 1 × what it is that we just got ψ sub 0*.1925

That is going to equal 4/ π¹/4 X E ^- X²/ 2 × what we just got which is - I H ̅ × 1/ π¹/4 X E ⁻X²/ 2.1935

This is going to equal - I H ̅ 4/ π²¹/4 X² E ^- X².1960

The right integral is - I H ̅ × 4/ π¹/4 × the integral for - infinity to infinity of X E ^- X² DX.1974

Just look this up in the table or have your mathematical software do it and it ends up equaling - I H ̅ 4/ π¹/4.2000

And this particular integral, N sub equaling π ^½ / 2.2011

When I multiply this out, I get - I H ̅¹/2/ 2, which ends up being exactly the same thing that I got for the left integral.2018

This whole exercise was just practice using that property, that hermitian property.2035

Based on the definition of hermitian, this operator which is – I H DDX is a hermitian operator.2047

Again, we will only be working with linear hermitian operators.2061

Once again, we will close it out by saying this hermiticity, this property of hermiticity guarantees that for P ̂ of ψ sub N = A sub N ψ sub N,2071

guarantees that the particular A sub N are real numbers.2098

Are members of the set of real numbers.2106

And it also guarantees that the integral of ψ sub N* ψ sub P is equal to 0 when N is not equal to P.2110

Hermitian and orthogonal.2125

Thank you so much for joining us here at

We will see you next time, bye.2131