For more information, please see full course syllabus of Physical Chemistry

For more information, please see full course syllabus of Physical Chemistry

### 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
- Quantum Mechanics Equations
- 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

- 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

### Physical Chemistry Online Course

### 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 variance *0081

*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 up *0112

*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 things *0142

*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 actually *0555

*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 operator *0586

*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 functions *0644

*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 of *0709

*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 want *0731

*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 and *0752

*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 the *1014

*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.com*1097

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

*We will see you next time, bye.*1103

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