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Quantum Mechanics: All the Equations in One Place

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
  • Quantum Mechanics Equations 0:37
    • De Broglie Relation
    • Statistical Relations
    • The Schrӧdinger Equation
    • The Particle in a 1-Dimensional Box of Length a
    • The Particle in a 2-Dimensional Box of Area a x b
    • The Particle in a 3-Dimensional Box of Area a x b x c
    • The Schrӧdinger Equation Postulates
    • The Normalization Condition
    • The Probability Density
    • Linear
    • Hermitian
    • Eigenvalues & Eigenfunctions
    • The Average Value
    • Eigenfunctions of Quantum Mechanics Operators are Orthogonal
    • Commutator of Two Operators
    • The Uncertainty Principle
    • The Harmonic Oscillator
    • The Rigid Rotator
    • Energy of the Hydrogen Atom
    • Wavefunctions, Radial Component, and Associated Laguerre Polynomial
    • Angular Component or Spherical Harmonic
    • Associated Legendre Function
    • Principal Quantum Number
    • Angular Momentum Quantum Number
    • Magnetic Quantum Number
    • z-component of the Angular Momentum of the Electron
    • Atomic Spectroscopy: Term Symbols
    • Atomic Spectroscopy: Selection Rules

Transcription: Quantum Mechanics: All the Equations in One Place

Hello, welcome back to educator.com and welcome back to Physical Chemistry.0000

We finished our discussion of Quantum Mechanics and when we get to thermodynamics at the end of that portion,0006

I had a lesson where all the equations will put in one place.0012

I decided to the same thing for quantum mechanics.0016

That is what this lesson is.0019

It is going to be a quick review of quantum mechanics.0020

Just talking about what equations we actually dealt with and what are the important things, we saw a lot of mathematics.0023

This lesson distills all of that into what is essential.0030

We had Le Broglie relation, basically it is the wave length of a particular particle of mass M travelling in a certain velocity V.0040

If you take planks constant and divide by the MV, you are going to get the actual wavelength of that particle, creating the particle as a wave.0050

We had the statistical relations as we would talk about in the beginning.0061

If you have a bunch of values, the average value of X is going to be the integral,0065

this –infinity to infinity is just over the particular space that you happen to be dealing with, of X × P of X DX.0072

This average value of X² = this and this is variance0081

which is this sigma² is the one thing - the other thing².0087

That is always going to be greater than or equal to 0.0092

P of X is the probability density.0095

P of X DX is the probability.0098

We will see what probability density and probability are for the wave function in just a moment.0102

These are just reminders of what it is that we did.0108

We have the Schrödinger equation, this is the equation that we set up0112

for any particular quantum mechanical system that we are interested in.0114

Depending on what this potential energy is, we set the equation, we solve the equation, and we get a wave function ψ.0118

From that wave function, we apply different operators to that wave function to extract information.0126

That is all we are really doing in quantum mechanics.0132

We find, we set up the Schrödinger equation, we solve the Schrödinger equation and we get the actual ψ wave functions.0135

We get the energies for each one of those wave functions and we do things0142

with that wave function depending on what we want to extract that information.0146

The solution ψ of X describes a particle of mass M moving in a potential field V of X.0151

The Schrödinger equation expressed as an Eigen value problem.0159

The Hamiltonian operator applied to the wave function ψ is going to equal some scalar E × ψ, where H is the Hamiltonian operator.0162

This is the Hamiltonian operator in one dimension.0176

This is the Hamiltonian operator in two dimensions.0179

This is the Hamiltonian operator in three dimensions.0182

These are partial derivatives.0187

The particle in a one dimensional box, this was the first quantum mechanical system that we discussed.0191

The ψ, the wave equations for this ended up being so a particle in the one dimensional box of length A,0198

particle moving back and forth, where is it? How fast it is going?0205

This describes it, the energy is this.0208

The wave function is this and is the quantum number takes on values 1, 2, 3, 4, 5, 6, 7.0212

It could be in the ψ 1 state, ψ 2 state, ψ 3 state, ψ 4 state.0219

It could be in a bunch of quantum states, that is the whole idea.0224

The particle in a two dimensional box of the area A × B, A by B.0230

2 dimensional box, 2 dimensions.0234

This is the equation for the wave function.0238

X goes from 0 to A, Y goes from 0 to B.0243

You have two quantum numbers M sub X and M sub Y.0246

The energy is equal to this.0249

The N may vary independently, you do not have to have 1-1, 2-2, 3-3.0252

It can be 5, 14, 3, 8, they vary independently, the quantum numbers.0257

The particle in a 3 dimensional box, we are just multiplying the individual equations for a 1 dimensional box.0263

It is this whole thing, the function of 3 variables.0271

X goes from 0 to A, Y goes from 0 to B, Z goes from 0 to C.0276

I have 3 quantum numbers, they vary independently.0280

This is the expression for the energy of a particle in a 3 dimensional box.0283

The postulates of the quantum mechanics.0292

The Schrödinger equation of the first postulate.0294

The Schroeder equation is not necessarily something that we derive.0297

We talked about one of the early lessons.0301

I think the 2nd or 1st one, we talk about the possibility of where it might come from but it is a postulate of quantum mechanics.0304

It is just is, this is one of our postulates.0311

That is where our starting points, it is not something we proved or derived.0313

We just accepted it as fact and we move from there.0316

For a given quantum mechanical system, the Schrödinger equation is solved to yield a function ψ,0320

which depends on the coordinates of the system.0325

Either one dimensional X, 2 dimensional Y, 3 dimensional XYZ, and so on.0327

The state of the quantum mechanical system is completely determined by this wave function.0331

All information about the system can be derived from this wave function.0336

The normalization condition is this.0339

The integral of ψ conjugate × ψ is equal to 1.0346

Always integrate over the entire appropriate space.0350

In the case of a particle in a 1 dimensional box, you are going to integrate from 0 to A.0352

In the case of a 2 dimensional box, you are going to do a double integral 0 to A and 0 to B.0356

In the case for the particle in a 3 dimensional box, you are going to do a triple integral 0 to A, 0 to B, 0 to C.0361

In the case of the hydrogen atom, you are going to have R, θ, φ.0370

R is going to be from 0 to infinity.0374

θ is going to be from 0 to π.0377

φ is going to be from 0 to 2 π, things like that.0380

Whatever space you happen to be dealing with.0382

This is a general symbol for the integral, make sure use the appropriate single, double,0386

or triple integral for a specific problem depending on whether the system is 1 dimension, 2 dimension,0390

or 3 dimension respectively.0395

Notice, you do not see any DX or DV here.0397

This is just a very abstract notation for the integral.0401

The integral itself depends on the specific problem, single, double, or triple.0405

The probability density is this ψ star ψ conjugate × ψ, that is the probability density.0412

The probability itself that the particle would be found in the region.0419

A little differential volume region center to XYZ is ψ * ψ DX.0424

I’m actually going to change this, this should be DV.0431

DX is for just a 1 dimensional problem DV.0435

This gives us the actual probability that you will find it in that little differential volume.0440

Therefore, the normalization condition says that the total probability of finding a particle somewhere is 1.0 or 100%.0444

If you add up the probabilities, let us say we have some 2 dimensional box.0454

If I add up the probability that is going to be somewhere in the box.0458

All the probabilities add to 1.0461

I will find it somewhere in the box.0464

For every observable in classical mechanics, velocity, kinetic energy, that corresponds a linear Hermitian operator in quantum mechanics.0468

Linear means that if I have two functions, if I add them together and then operate on it,0482

it is the same as if I operate on each separately and add it.0488

It means the operation in addition, you can switch the order.0492

It also means that if I take some function, multiply by a constant, and then operate on it,0495

what I can do is operate on the function first and then multiply by constant.0500

That is what linear means.0504

This is the definition of linear.0506

It is probably one of the single most important definitions in all of science.0507

Hermitian means the integral of F*, it means if I operate on G and then multiply on left by F*,0513

I get the same answer.0524

If I operate on F, if I take the conjugate of that and then multiply by GE.0526

That is what Hermitian means.0531

In any measurement of a classical observable associated with a given quantum mechanical operator,0536

the only values observe will be the Eigen values associated with the Eigen functions satisfying this.0541

In other words, if I have an operator and if I operate on the kinetic energy operator.0547

If I'm dealing with a particular quantum mechanical system, this equation, the only values that I actually0555

find when I measure the kinetic energy of this system are going to be those values.0561

The A sub N, the Eigen values, that is always says.0565

The average value of an observable corresponding to an operator is given by this.0571

If the function is an Eigen function of the operator, this is what I will get.0580

I will see these.0583

If a particular wave function for a given system does not happen to be an Eigen function of the operator0586

that I’m interested in, I'm going to get different values but I can take an average of all those different values, that is what this says.0594

The average value is going to be integral of ψ star and F star divided by the integral of ψ * ψ.0601

If a wave function is normalized, in other words if this integral is equal to 1 then it just becomes this.0605

The 1 in the denominator goes away.0616

Very import definition.0617

The average value of an observable associated with the particular operator is equal to the integral of ψ *, the operator ψ.0620

We did a lot of these problems especially for the hydrogen atom.0631

The Eigen functions of a quantum mechanical operator are orthogonal, in other words perpendicular.0634

For the Eigen value problem, this, the integral to ψ sub N * ψ sub P for two different Eigen functions0644

A = 0 and L = 0, when N does not equal P.0653

The commutator of two operators A and B is this.0658

Apply A first then apply B.0664

Apply B first then apply A.0666

If you do that, you end up subtracting one from the other and you end up getting 0, we said that the operators commute.0668

It means it does not matter which order you operate on a function.0674

We said that the operators commute.0680

If it does not equal 0 then they do not commute, that means that going to be A first then B,0681

you get a different answer than if you are going to do B first than A.0687

They do not commute.0690

Commutation is intimately related to the uncertainty principle.0695

We have the uncertainty principle.0702

This is a general expression of the uncertainty principle.0704

The uncertainty of a measurement of A × the uncertainty of measurement of0709

B is going to be greater than or equal to ½ the absolute value.0711

The integral ψ * AB of ψ, this is just equation that you should know.0715

This is the general expression of the uncertainty principle.0723

If the operators commute then this integral is 0, then the sigma AB is just 0.0726

That means that you can determine each one of those A and B to any degree of accuracy you want0731

but if this is not 0 then you cannot determine it to any degree of accuracy you want.0743

In the case of momentum in a 1 dimensional case.0748

For a momentum and position, those two operators, the position operator and0752

the momentum operator, the linear momentum operator, they cannot commute.0758

Which means that this thing on the right is not equal to 0,0762

which means that if you are precise value for linear momentum, you are going to be very uncertain as to where your particle is.0765

If you want to know where your particle is, you are going to be very uncertain how fast the particle is going.0775

There is a balance that you have to strike that you cannot go below that number.0779

That is what this says.0784

The sigma are the uncertainties, the standard deviations for each observable.0786

The sigma definition is this.0790

The sigma is just this thing, the square root of that.0792

We discussed the harmonic oscillator.0801

The energy of the harmonic oscillator was given by this equation or ν was equal to this.0804

K is the force constant and μ is the reduced mass.0808

The equations of the wave functions for the harmonic oscillator are given by this thing.0814

α = this thing and H sub R, those are the Hermite polynomials.0818

The ψ sub X are non degenerate, there is no degeneracy for the harmonic oscillator.0827

The rigid rotator, the energies for the rigid rotators are these.0834

J takes on these values, the quantum number for the rigid rotator.0838

The degeneracy for the quantum mechanical system in state J is equal to 2J + 1.0841

That means if J was equal to 4 that means there are 9 states that have that same energy.0847

For rotation, there is degeneracy.0853

For vibration, there is no degeneracy.0856

The wave functions for the rigid rotator will be discussed under the hydrogen atoms.0858

Here I gave you the energy, I’m not going to do the wave functions yet.0862

I’m going to discuss them in a minute when we talk about the hydrogen atom, which is what we did when we did the lessons.0865

The hydrogen atom, the energy of the hydrogen atom is equal to this, where E is the proton charge 1.602 × 10⁻¹⁹ C.0872

E is not the order number E.0880

The wave function for the hydrogen atom, for an electron and a hydrogen atom is equal to this thing.0887

3 quantum numbers N, L, M, there is a radial component, there is a spherical component.0894

The radial component is a function of R.0899

The spherical component is a function of 2 variables θ and φ.0902

This is the equation for the radial component.0907

The L are the associated Laguerre polynomials.0911

The S is the angular components are also called the spherical harmonics.0919

These are the equations for them.0924

When we did lessons, we actually listed the first 5 to 10 of those.0927

The P sub L superscript absolute value of M of cos θ, those are the associated Legendre functions.0933

We have a principal quantum number N.0943

The energy of the hydrogen atom is a function of N only.0946

We have the angular momentum quantum number.0951

The angular momentum of the electron is a function of L only.0954

It is given by this, whatever value of L is, just put in here and it will give the magnitude of the angular momentum.0958

It will give you the direction, it will give the magnitude.0965

We do not know the direction, the angular momentum cone, L is what we call J when discussing rigid rotator.0968

When we talk about the rigid rotator, the J, the quantum number, that is actually L for the hydrogen atom.0975

The magnetic quantum number, you will see as M or M sub L.0983

M that depends on L, there are 2 0 + 1 values for every value of M compared to the degeneracy J.0987

2 J + 1 for the rigid rotator.0997

The Z component of the angular momentum of the electron is a function of M only.1000

The energy of the electron is given by the principal quantum number N.1005

The angular momentum of the electron is given by the angular momentum quantum number L.1009

The Z component of the angular momentum, in other words the projection of the1014

angular momentum vector on the Z axis is given by this.1017

That is controlled by M, this is of course in the absence of any magnetic field.1024

We also talked about atomic spectroscopy.1035

We talk about term symbols.1038

This is the expression for the basic term symbol for a poly electron, for a many electron atom.1041

When an atom of hydrogen is also hydrogen.1053

This term symbol tells me what state it is in.1056

L is the total orbital angular momentum quantum number.1060

S is the total spin angular momentum quantum number.1063

And J is the total angular momentum quantum number.1067

Remember, we have angular momentum that comes from the orbit of the electron itself.1070

We have an angular momentum that comes from the spin of the electron itself.1074

And the total angular momentum is just the sum of those two.1077

And a term symbol tells me what state it is in.1080

The selection rules δ L + or -1, δ S is 0, δ J is 0 + or -1.1084

The J = 0, the J = 0 transition is not allowed.1092

That brings us to the end, thank you so much for joining us here at www.educator.com1097

for this quick review of all the equations that are necessary for quantum mechanics.1100

We will see you next time, bye.1103