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 Hydrogen Atom VII

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 Hydrogen Atom VII 0:12
- p Orbitals
- Not Spherically Symmetric
- Recall That the Spherical Harmonics are Eigenfunctions of the Hamiltonian Operator
- Any Linear Combination of These Orbitals Also Has The Same Energy
- Functions of Real Variables
- Solving for Px
- Real Spherical Harmonics
- Number of Nodes

### Physical Chemistry Online Course

### Transcription: The Hydrogen Atom VII

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

*Today, we are going to close out our discussion of the hydrogen atom.*0004

*It is going to be the 7th lesson of the hydrogen atom discussion.*0008

*Let us jump right on in.*0011

*In the last lesson, we look primarily at the S orbital, the 1S, 2S, and the S in general.*0015

*I will stick with black today.*0022

*In the last lesson, we looked primarily at the S orbitals.*0026

*The 1S, 2S, and the S orbitals, in general.*0044

*We discovered that they are spherically symmetric.*0058

*I will just write down here spherically symmetric.*0063

*What it means is that, it depends only on R, the radius of the electron from the center of the nucleus.*0069

*It does not depend on θ or φ.*0079

*Let us take at look at the P orbitals.*0091

*Now, we look at the P orbitals.*0093

*For the P orbitals, the L value is equal to 1.*0104

*The L is equal to 0.*0107

*We have a relationship between N and L and M.*0111

*We have the N = 2, we have L = 1, we have N = 0.*0118

*We have L = 1, M = + 1, and we have the L = 1, M = -1.*0126

*The 2P orbital has 1, 2, 3 sub orbitals.*0139

*You know them as PX PY and PZ.*0144

*The 2P orbital has three sub orbitals.*0154

*That is all these orbitals are, they are wave functions.*0164

*We call them orbitals.*0171

*We have something like this.*0173

*We have ψ 2, 1, 0 = to R 2,1 S 1,0.*0174

*We have ψ 2, 1, 1 that is equal to R2, 1 S1, 1.*0183

*We have ψ 2, 1 -1 that is R 2, 1 and S 1 -1.*0190

*Let us look at the S sub L superscript M.*0200

*Let us look at the S1, 0, S1, 1, and S 1 -1.*0211

*Our S1,0, which is going to be a function of θ and φ, remember these are the spherical harmonics.*0215

*These are the wave functions for the rigid rotator and they consist of a function of 2 variables, the θ and the φ.*0223

*It is going to equal 3/4 π ^½ × cos of θ and S 1, 1 of θ and φ is equal to 3/8 π ^½.*0231

*This is going to be sin θ E ⁺I φ.*0253

*Of course, we have the S1 -1, this is the function of θ and φ.*0258

*And this is equal 3/8 π ^½ and this time it is a sin θ but it is E ⁻I φ.*0264

*These are our three spherical harmonics for the N =2, L = 1.*0277

*We are concerning ourselves with the P orbitals, L = 1.*0287

*Notice, these are not spherically symmetric.*0293

*In other words, they depend on θ and they depend on φ.*0296

*This one just depends on θ but these depend on θ and φ.*0299

*Let us go over here.*0304

*I will go ahead and go to the next page.*0310

*These are not spherically symmetric because they depend on θ and φ.*0312

*We just look at the spherical harmonics but the whole wave function ψ,*0343

*because they depend on all three variables R, θ, and φ.*0346

*Notice that the S 1, 1 and the S 1-1 are complex.*0355

*In other words, they have that E ⁺I φ and E ⁻I φ term.*0372

*Let us rewrite them again, it is not a problem.*0378

*S 1, 1, I’m going to leave off the θ in the φ.*0379

*3/ 8 π ^½ sin θ E ⁺I φ and the S 1 -1 is 3/ 8 π ^½ sin of θ E ⁻I φ.*0384

*These are complex.*0404

*Recall that the S sub L superscript M, the spherical harmonics,*0407

*they are Eigen functions of the Hamiltonian operator, the energy operator.*0420

*When we say Hamiltonian, we just mean the energy operator, the Hamiltonian operator.*0436

*We will also be calling it just the plain, old, energy operator.*0440

*The total energy, we will speak of the kinetic energy operator, the potential energy operator.*0450

*But when we say the energy operator, we are talking about them combine the Hamiltonian.*0454

*Recall that these are actually Eigen functions.*0460

*What that means is this.*0463

*You remember what Eigen function means?*0465

*SLM = H ̅² × L × L + 1/ 2 I SLM.*0468

*This right here, when we take the function, one of the spherical harmonics,*0483

*when we operate on it with the Hamiltonian operator, we end up been getting the function back*0488

*but this time multiply by some constant.*0492

*This constant is the energy Eigen value.*0494

*It gives us the different values of the energy that we get when we actually measure the energy.*0498

*This right here, this is the energy Eigen value.*0504

*It depends only on L, the second quantum number.*0517

*Notice it depends only on L, because it depends only on L, L is the same for this and this.*0535

*L is equal to 1, so the energies of these two are the same.*0541

*The energy of the S 1, 1 state = the energy of the S1 -1 state.*0548

*They are degenerate.*0557

*Since these 2 orbitals are degenerate, remember what degenerate means, they have the same energy.*0561

*Degenerate have the same energy, let us right that in here.*0575

*Since they have the same energy, we know from previous lessons that any linear combination of these orbitals also has the same energy.*0583

*Because the Hamiltonian operator is linear operator, we are only dealing with linear operators in quantum mechanics.*0611

*In any linear combination of these orbitals also has the same energy.*0627

*We will give the formal version.*0648

*More formally, we have S 1, 1 and S 1 -1 are degenerate Eigen functions of the Hamiltonian operator.*0657

*Therefore, some constant × S 1, 1 + some constant × S 1 -1 of any linear combination,*0683

*any constant that I can put in for C1 and C2 is also and Eigen function of the Hamiltonian operator.*0696

*It is also an Eigen function of the Hamiltonian operator with the same Eigen value, the same energy,*0705

*which is H ̅² L × L + 1/ twice the rotational inertia.*0726

*Let us go ahead and continue on.*0740

*We need to choose a linear combination, that is going to be most convenient for us.*0743

*We can choose because we said any linear combination will do.*0752

*We just need to combine in such a way that we get something that works for us.*0759

*Makes our lives easier, makes mathematics easier, or does something else for us that we have done.*0762

*We can choose any linear combination we wish.*0767

*Normally, the following linear combinations are the ones that we use.*0782

*For the P sub X orbital, we have 3P orbitals.*0805

*Now, we are going to actually designate them PX PY PZ.*0812

*The PX, we take the following linear combination ½ ^½ × S 1, 1 + S 1 -1.*0816

*As far as the constants are concerned, we choose 1/2¹/2, 1/√2.*0831

*When we add the 2, let us say the S 1 -1 multiply by this, we end up with the following.*0839

*We end up with 3/ 4 π ^½ sin θ cos φ.*0848

*Notice the what we have done here in taking the linear combination, is we have turned a complex function,*0862

*the 2 complex functions S 1, 1 and S 1 -1, remember they were complex.*0868

*They had that E and the + or –I φ term, that is complex.*0873

*We are doing this linear combination, we turn them into real function.*0882

*We got rid of the complex part.*0886

*It is a function of θ and φ still, but it is a real function of θ and φ.*0888

*That is why we took the linear combination.*0892

*That is the only reason to take the linear combination is because in this particular case, we wanted to deal with a real function.*0896

*You will see in the minute you want to deal with a real function.*0902

*It gives us a little bit geometrical information, as far as space is concerned.*0904

*That the only reason to do so.*0909

*It is not as though we have actually, it is not like it makes it any better than the complex functions.*0911

*The PY, we take the following linear combination.*0919

*We take - I × 1/2¹/2 and this time we take S 1, 1 – S 1 -1.*0923

*That is your linear combination.*0934

*We end up with 3/ 4 π ^½ × sin θ.*0935

*This time it is a sin φ.*0944

*What we have done here, let us move on to next one.*0948

*These linear combinations of the S 11 S 1 -1 linear combinations, they change.*0953

*S 1 1 and S 1 -1 into functions of real variables, instead of functions of real and complex variables.*0969

*Let us work out the P sub X in detail so you actually see where this is coming from.*1010

*We do not just drop this linear combination on you and say this is the wave function, deal with it.*1014

*Let us actually work on this out.*1023

*Let us workout P sub X and you could do P sub Y if you want.*1026

*We know exactly why it took the form that it took.*1034

*We know exactly where it came from.*1044

*We said that P sub X is equal to ½ ^½ S sub 1, 1 + S sub 1 -1.*1054

*That is going to be equal to 1/2 ^½, S sub 1, 1 is 3/ 8 π ^½ × the sin of θ E ⁺I φ + S sub 1 -1*1068

*which is 3/ 8 π ^½ × the sin of θ E ^- I φ.*1090

*I'm going to go ahead and factor out the 3/ 8 π ^½.*1103

*I’m going to go ahead and factor out the sin θ, so I get the following.*1108

*I get ½ ^½, I get 3/8 π ^½, I get the sin θ and I'm left with E ⁺I φ + E ^- I φ.*1112

*I will go ahead and expand this out using Euler’s formula.*1134

*Let us go ahead and combine these.*1144

*I’m going to write this as 3/16 π¹/2 × the sin of θ.*1152

*This is going to be cos φ + I × the sin of φ, this is that.*1166

*That is going to be + cos of φ is going to be cos of φ – the sin – I × the sin of φ.*1174

*Remember, this is cos -φ + I × the sin – φ.*1196

*Take the minus sign with the angle φ.*1204

*Cos of - φ is cos φ, sin of - φ is - sin of φ.*1206

*That is where this part comes from.*1214

*We have I sin φ -I sin φ, that goes away.*1216

*We are left with 3/ 16 π ^½ × sin of θ and this is × 2 × the cos of φ.*1223

*When we combine the two with this, we end up with our final 3/ 4 π ^½ × sin of θ × the cos of φ.*1241

*There we go, we took our orbital S 1, 1 and S 1 -1.*1257

*We want to do something about the complex variable.*1263

*Because of the nature of the Eigen function, because they are both Eigen functions,*1266

*any linear combination of them is also an Eigen functions.*1272

*We have not lost anything if we just combine them.*1275

*If you combine them this way, we turn a complex functions into real functions.*1277

*That is the only reason to take the linear combination.*1283

*It has to confuse why did we do this?*1286

*This is the only reason to help us, we do not need to deal with the complex variable.*1288

*We want to deal with the real variables.*1292

*In this case, that is the only reason we take the linear combination.*1294

*The linear combination you choose is absolutely your choice.*1297

*If for some odd reason the particular situation that you are dealing with,*1302

*you need the function to look a certain way to help you do something, you can choose any linear combination you want.*1305

*That is the whole idea, any constant will do.*1311

*We have S 1, 0 is equal to 3/ 4 π ^½ and this is the cos of θ, that is the PZ.*1318

*S 1 1 is equal to 3/4 π ^½ sin θ cos φ, that is the PX.*1341

*S 1 -1 = 3/4 π ^½ sin θ sin φ, that is PY.*1355

*We have actually specified which one belongs to which.*1369

*L is equal to 1, for M sub L = 0, that is the PZ orbital.*1372

*For M sub L =1, that is the PX orbital.*1382

*For M sub L = -1, that is the P sub Y orbital.*1384

*Is this real spherical harmonics?*1393

*When we plot these, θ and φ, when we plot them in a 3 dimensional space,*1409

*that one plotted give us the familiar dumbbell shaped orbitals that we remember from general chemistry.*1416

*You remember the shape of the P orbital.*1447

*One along the X axis, one along the Y axis, one a long the Z axis.*1457

*I’m not going to draw it.*1463

*That is the reason why I'm not drawing it.*1464

*The drawing is in your book, if you want to see them.*1466

*You will see in a second why is that I'm actually not drawing them.*1469

*Mathematically, nothing is gained by using these real functions, instead of the complex versions the E ⁺I φ, the E ⁻I φ.*1473

*Mathematically, nothing is gained.*1488

*As far as I'm concerned, nothing is gained by using these real functions for the PY and the P sub X,*1493

*instead of the complex functions, which are the original derived functions.*1521

*The only reason they are used is because they lend the directional character or quality to the orbitals in space.*1535

*We want to be able to visualize an orbital.*1572

*By using these linear combinations in turning the spherical harmonics, at least the S 1, 1 S 1 -1*1581

*which is complex into real functions, when we plot these with θ and φ,*1588

*a 3 dimensional space we actually get dumbbell shaped orbitals.*1594

*The fact that is the only reason that this is done because it lends a directional character.*1600

*It allows us to wrap our mind around it, gives us a little bit of a picture of what the orbital looks like*1605

*or where we might find the electron.*1610

*That is the only reason that they actually done this.*1613

*Mathematically, nothing is gained by it.*1616

*Because we live in space, because the real world is a 3 dimensional space, it allows us to think geometrically about these things.*1618

*That is the only reason that this is the case.*1631

*As far as chemistry is concerned, because this is a physical chemistry course, it also helps with the geometry of the molecules.*1633

*You remember the geometry with these real functions,*1643

*the arrangement of the orbital in space correlates with a lot of what we know in terms of geometry.*1648

*This is always something that helps geometrical intuition.*1653

*There is nothing that was actually mathematically gain by doing this.*1657

*In fact, it is been my experience that when one deals with a complex functions as opposed*1660

*to the real functions, mathematically things tend to be easier.*1663

*But that is just my experience.*1667

*My own personal commentary on this.*1672

*I strongly deemphasize any pictorial representation of orbitals at this level and beyond.*1674

*Clearly, we want to tie this into what is that you have learned in general chemistry*1710

*and the shapes of the orbitals and this statistical distribution of the electron.*1714

*The electron cloud, things like that.*1718

*These pictures are still in your physical chemistry books.*1721

*Which is fine, it is not a problem but I strongly deemphasize that you actually think about them this way.*1723

*These orbitals are mathematical functions.*1729

*That is how you want think about it.*1731

*Beyond the orbitals, 1S, 2S, 2PX, 2PY, 3P, 3D, the orbitals are mathematical objects and should be considered as such.*1735

*I’m going to say it, I’m not going to write this out.*1773

*In general, I’m a big fan of using geometry in order to help your mathematical intuition.*1782

*I’m a huge fan of that.*1787

*I think you should always use geometrical arguments, geometrical notions,*1788

*to help you understand what it is that is going on.*1793

*This is one of the few cases where I think that nothing is certainly lost by doing that but you have to be a little bit careful.*1795

*This is one of few exceptions where geometrical thinking, geometrical notions,*1802

*I think actually get in the way of you really experiencing the power of what it is that is going on here*1808

*and what it is that it is actually been done here.*1818

*We have been able to reduce the representation of electrons to a mathematical function.*1820

*When we say what is an electron, the electron is this.*1824

*This is the electron, it is represented by mathematical function.*1827

*You do not want think of it as a physical object.*1831

*You want to what I think it as this mathematical thing because this mathematics that*1832

*you are actually going to manipulate, it is going to give you the answers that you seek.*1836

*For the sake of chemistry, for the sake of geometry, for the sake of the molecular orbitals*1841

*that we are going to be doing later on, it is nice to have that.*1846

*But I would not lean on it heavily, which is why I have deemphasize pictures.*1848

*These are mathematical objects.*1852

*One final comment here, with the power of mathematics it is not constrained by physical reality.*1856

*It is not constrained and limited by physical reality.*1876

*We have to place constraints and a certain physical constraints on the things that we do, in order to do derive these,*1898

*these wave functions for the different quantum mechanical systems that we have been dealing with.*1904

*Of course, this quantum behavior as we saw, arises precisely because these particles*1908

*are not free to just be wherever, move wherever.*1914

*We have to constrain these particles and the wave functions have to adhere to those constraints,*1917

*which is why quantum behavior emerges.*1923

*Quantum behavior exists only because of these physical constraints.*1925

*If the mathematics, we end up with complex functions, complex and real functions,*1929

*the power of mathematics is not constrained by reality.*1934

*Even though we have had to use reality to derive things mathematically.*1936

*The moral is do not lean too heavily on pictorial representations.*1942

*Let the math do what the math does, and trust the math more than trust your intuition.*1946

*This is a direction that you want to start leaning in as you move into higher classes, move on to graduate school.*1951

*You want to put your thinking and become more and more abstract.*1958

*You want to release of the constraints on it so you can actually think in a broader sense.*1961

*You want to be able to think of the big picture and this is the beginning of that.*1966

*One final thing, I will go ahead and close off these lessons.*1971

*We have ψ 2, 1, 0 = R 2,1 S 1, 0.*1980

*We have ψ 3, 1, 0 = R 3, 1 S 1, 0.*1987

*We have ψ 4, 1, 0, I’m just listing some of the ones where I’m not changing the N, the primary quantum number 234, 410, is 4, 1 and is S 1, 0.*1996

*Notice the S 1, 0, S 1, 0, S 1, 0.*2012

*Recall from previous lesson, the number of nodes places where there is 0 probability*2016

*for finding the electron, the number of nodes is equal to N - L - 1.*2026

*The 2P, it has 0 nodes.*2036

*2P, N is 2 L is 1, 22 -1 -1 is 0.*2044

*The 3P has 1 node, the 4P has 2 nodes, and so on.*2051

*Thank you so much for joining us here at www.educator.com.*2066

*We will see you next time, bye.*2068

1 answer

Last reply by: Professor Hovasapian

Wed Apr 20, 2016 1:22 AM

Post by Tammy T on April 18, 2016

Dear Prof. Hovasapian,

I see no information about conical, planar and angular node?

Would you please explain about these nodes? For example, what are they and what to know about them? What orbitals will have these nodes?

Thank you!