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♪ [slow jazz music] ♪

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[Prof. Lamb] Well, we have Brian[br]here with us today. Welcome, Brian.

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[Brian] Thank you.

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[Prof. Lamb] Brian, do you remember

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that when we talked about [br]the Bohr model of the atom,

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we mentioned that there's [br]a more sophisticated model

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that better describes the way [br]electrons behave in atoms?

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[Brian] I think so.

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[Prof. Lamb] Well, our objective here today

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is to begin learning about this modern [br]quantum mechanical model of atomic structure.

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First, let's recall that Niels Bohr...[br][Animated Bohr] Hi.

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[Prof. Lamb] ...was able to explain the line [br]specter of light emitted and absorbed by an atom

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with his relatively simple planetary [br]model of electrons circling the nucleus.

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But, do you remember? What was it that [br]Bohr WASN'T able to explain with his model?

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[Brian] Wasn't it why electrons could [br]adopt certain orbits but not others?

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[Prof. Lamb] Yeah, that's exactly right.

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Put in other words, he wasn't able to explain

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why the energy levels of the electron[br]are what we call "quantized."

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Now, the solution to this mystery[br]began to reveal itself

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when scientists realized that light can be[br]considered both a particle and a wave.

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It was then reasoned that if light can [br]exhibit both particle and wave behavior,

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maybe other things can too.

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Things like electrons, for example.

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And it was this last logical leap [br]that led to an understanding

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of how electron energy levels[br]in atoms are quantized.

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Would you like to see how?

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[Brian] Let's do it.[br][Prof. Lamb] Okay.

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[Prof. Lamb] To get started, let's see if [br]we can visualize the electron as a wave.

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We're not going to do this in [br]one single step, but gradually.

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Indeed, what you're about to hear

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is going to tax your ability to think and[br]visualize things in three dimensions.

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So I'd suggest you get your [br]cerebral cortex into high gear.

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[engine revving]

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We'll start out simple and proceed [br]step-by-step to the full theory.

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Let's start with a familiar kind of wave.[br]Here it is: our old friend, the sine wave.

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There are two things about this [br]wave we need to focus in on.

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First is the fact that it can be represented[br]by a mathematical function, and here it is.

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I'm hoping it looks familiar from [br]one of your basic math classes.

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Second is the fact that we can create[br]a standing wave from this sine wave.

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Now, what's a standing wave? Let's see.

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Normally, we think of a wave like this[br]moving horizontally, don't we?

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However, if the wave reflects back on itself,

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under certain circumstances,[br]the reflected waves and the incoming waves

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may line up to produce a [br]phenomenon that looks like this.

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This is called a standing wave.

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You've actually seen [br]standing waves like this before.

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Consider a guitar string.

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For a given guitar string length, only certain[br]wavelengths can produce standing waves.

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These are wavelengths that fit neatly[br]on the string with no string left over.

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For example,

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a wave whose wavelength is exactly equal [br]to the length of the string would do the job.

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That's the kind of standing wave[br]we're showing on the string right now.

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When we pluck the string,[br]there are many wavelengths produced,

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but the wavelengths that fit along [br]the string in whole numbers

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reflect back and forth off [br]the two ends of the string

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and are reinforced by each other that way.

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This produces the standing wave.

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Other wavelengths destructively interfere with[br]each other as they bounce back and forth.

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So only the wavelengths which fit in [br]whole numbers along this string survive,

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and that's the note and [br]its overtones that we hear.

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So the string produces one sweet single tone.[br][sound of a plucked guitar string]

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The same standing wave phenomenon

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could be generated if we could somehow bend[br]the guitar string around on itself like this.

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[no audio]

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Now notice: The only way[br]a standing wave will form

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is if the path of the wave is the correct length[br]to accommodate a whole number of wavelengths.

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If the circular path is too long or too short,[br]the wave interferes and cancels itself like this.

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When the path isn't just the right length,[br]the crests and troughs of the wave don't line up.

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and we say the wave is destructively[br]interfering with itself, see?

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Now, if we expand the size [br]of the circle just enough

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so that one full wavelength is [br]added to its circumference,

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then once again, a standing wave is formed.

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Now, can you see how this approach might help [br]explain the quantum nature of the Bohr orbits?

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I know it's hard to do,

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but if we were to think of the electron,[br]not as a particle, not as a tiny dot,

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but as some sort of wave moving [br]around the circular orbit,

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then the electron would [br]destroy itself in all orbits except

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ones where the circumference of the orbit [br]is a whole number of wavelengths, right?

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In other words, the electrons that exist in these[br]particular orbits but not in orbits in-between.

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Isn't that neat? And what you've just heard

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is the basic idea of the quantum[br]mechanical model of the atom.

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It doesn't seem to be really[br]all that hard, does it?

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Notice that in our electron standing waves,[br]there would be nodes;

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that is, points exactly between the crests[br]and troughs where the wave has no amplitude.

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Interesting, wouldn't you say?

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So what was all the fuss about [br]getting our brains in high gear?

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Well, we're not done yet.

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We've got to, now, project this relatively [br]simple idea into three dimensions.

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[to Brian] Now, Brian, let's replace[br]the familiar form of sine wave

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with another that's less familiar.

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For those of you who know the terms,

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we're going to replace the transverse wave [br]you just saw with a compression wave.

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Here it is.

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Now, at first glance, maybe this doesn't [br]look like a wave to you, but it really is.

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The wave isn't in an up-and-down motion

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but in the change of color [br]between yellow and red.

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Think of it as a yellow and red wave[br]moving along a string.

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You might think of the yellow[br]as the wave crests, perhaps,

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and the red as the troughs.

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This is a truly one-dimensional wave[br]because everything stays in a line

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and the line doesn't bend into a [br]second dimension to form the wave.

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Now, of course, we can bend this string around[br]on itself, just like the one you saw earlier.

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Again, think of the electron [br]now as being this wave.

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If the length of the string--[br]that is, the circumference of the orbit--

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is a whole number of wavelengths,

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then the electron will form a standing[br]wave around the circumference

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and won't destroy itself.

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Now, just for our purposes here today,

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let's think of the electron intensity

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being the strongest in this string[br]in the crests and troughs

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(that is, where the color [br]is pure red or pure yellow).

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If that's where the electron's [br]presence or intensity is strongest,

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then Brian, what about the areas [br]in-between pure yellow and red?

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What would those represent, do you suppose?

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[Brian] Clearly, the magnitude or[br]intensity of the redness or yellowness

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doesn't instantly drop to[br]zero outside these points.

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It more fades from one color to the other.

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[Prof Lamb] That's exactly right.

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And so, we don't think of the electron

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as having intensity ONLY at those points [br]on the string (the crests and troughs)

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but that the electron intensity gradually [br]drops to zero at the node and then rises again.

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Now remember, we're not thinking of [br]the electron here as a particle right now.

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We're thinking of it as [br]a wave (a standing wave).

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So the electron's standing wave [br]is smeared out along this string

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with greatest intensity at the peaks and [br]troughs and zero intensity at the nodes.

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Now, it's a little hard to see the nodes here.

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Brian, where do you suppose the nodes [br]are in this yellow and red representation?

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[Brian] Well probably half-way between

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the highest point on the line[br]and the lowest point.

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[Prof. Lamb] Yeah, the points where[br]the color is the most pure.

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Pure yellow, pure red.

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Half-way in-between, [br]there would be a node.

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Now let's go to the tricky part, shall we?

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We don't live in a one-dimensional world,

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so we need to consider

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what a three-dimensional [br]standing wave would look like.

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That would represent an electron [br]in the real world of the atom.

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But it's hard to do; it's hard to imagine,[br]so we need to proceed in steps.

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Let's first see if we can imagine a [br]two-dimensional standing wave, shall we?

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This would correspond, not to a wave in a line[br](like we saw last) but to a wave in a plane.

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Again, we're gonna use a compression wave,

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and we can see the crests [br]and troughs using two colors.

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In this two-dimensional system,

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we see the waves as rings[br]of crests and troughs.

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The nodes, of course, will also be in the [br]shape of rings, but they're hard to see.

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As you said, they're at the point where[br]the yellow and red are of equal intensity,

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about half-way (in fact, exactly half-way)

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between the points where we [br]see pure yellow and pure red.

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[Brian] Okay.

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[Prof. Lamb] Now we're ready [br]to take the last step, this time,

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to a three-dimensional standing wave.

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Instead of rings in a plane, this system would[br]tend to look more like shells in a sphere.

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You might thing of the waves [br]like the layers in an onion

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or the dolls inside dolls of [br]a Russian matryoshka doll.

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From the outside, this system would [br]just look like a solid sphere,

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but if we cut a cross-section of the sphere,

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we would see this beautiful[br]standing wave-like pattern.

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We can then imagine the electron in this form.

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The nucleus sits at the center of the sphere[br]and there are only certain radii

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at which the electron wave doesn't [br]partially or completely destroy itself.

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These are the distances form the center

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where we see pure yellow [br]or pure red in our depiction.

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You might say that the electron intensity

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is highest at the distances [br]of pure yellow and pure red.

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And because of destructive interference,

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the electron intensity drops[br]between these shells until

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exactly half-way between pure yellow [br]and pure red, it drops down to zero.

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This latter distance represents a node[br]where the electron has no intensity at all.

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Well, you now should have a [br]fairly good conceptual picture

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of how the modern wave model of the atom works.

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As difficult as this is to grasp, however,[br]it is a much simpler picture than the real thing:

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the wave model of the electrons in atoms[br]developed by a scientist named Schrödinger.

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We call his model the Schrödinger model. [br](Surprise!)

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And it's based on the idea that the [br]electron can be thought of as a

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three-dimensional standing wave.

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These waves aren't described in terms [br]of onions or matryoshka dolls

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but in terms of the mathematical equation[br]that describes the standing wave.

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Now, one caveat: Up to this point,

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we've just been describing the general [br]principle of treating an electron as a wave.

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We started with a one-dimensional sine wave

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then moved it into two dimensions [br]then three dimensions; but in fact,

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the Schrödinger model doesn't actually[br]use a simple sine wave like this at all.

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Instead, it uses waves based on equations[br]that are much more sophisticated than y=sin x.

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Yet despite that complexity, just keep in mind

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that these so-called wave equations [br]or wave functions of Schrödinger

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are just really variations on this kind of equation.

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Here's the simplest Schrödinger wave function.

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As is common in equations, [br]these wave equations contain many variables

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and it turns out that some of these variables

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adjust the size and position [br]of the wave in certain ways.

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And here's the key:

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For the wave to be a standing wave,[br]these variables can only have certain values

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(just as the length of the circular string

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could only have certain values to give[br]us a one-dimensional standing wave).

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Now, these variables are called [br]the quantum numbers,

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and they govern the shape and [br]the size of the standing wave.

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It turns out that one of Schrödinger's standing [br]waves looks a lot like our simple 3D sine wave.

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with a spherical shape.

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Such standing waves belong to a class[br]we call the s-type standing wave form.

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Oh, and while we're at it, I guess we'd better[br]give these kinds of standing wave forms a name.

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We'll call them orbitals.

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So the s-type orbitals are spherical in shape.

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Well, it turns out that there are other shapes

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for orbitals or standing wave forms[br]besides the spherical s-type.

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Here's another orbital shape here.[br]I wonder what we ought to call it.

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Brian, what does it look like to you?[br][Brian] It kind of looks like a dumbbell to me.

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[Prof. Lamb, chuckling] Yes, it does.[br][Animated man groans]

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[Prof. Lamb] Unfortunately, [br]someone got there before us though

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and decided to call these--[br]not "d-type" for dumbbell

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but "p-type" orbitals.

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Now, if you're wondering how you might[br]imagine a standing wave of this type forming,

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think of this orbital like a sphere,[br]say a balloon. I have a balloon here.

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If we were to twist this balloon in the [br]middle (form a node in the middle),

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we'd end up with two lobes [br]that look like this, wouldn't we?

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[Brian] Okay.

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[Prof. Lamb] That's what a[br]p-type orbital looks like.

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You can see it from different angles.

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Alternatively, you might imagine [br]that this standing wave

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is formed by spinning our friend, [br]the sine wave, around its axis.

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If you had a sine wave like this,[br]if you spun it around its axis,

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it would generate in space[br]these two lobes, wouldn't it?

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[Brian] Oh, yeah.

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[video is paused during emergency [br]alert on the bottom of the screen]

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[Prof. Lamb] Now, about p-orbitals,

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it's interesting that they always[br]come in sets of three, like blind mice.

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[Brian chuckles]

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[Prof. Lamb] The three look the same

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except that they differ in [br]their orientation in space.

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One is lined up along the x-axis,

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one along the y-axis, [br]and one along the z-axis.

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[no audio]

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Okay, it turns out that there[br]are also d-type orbitals.

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They are the next type,

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and they come, not in sets[br]of three but in sets of five.

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Here they are. Kinda cute, huh?[br][Brian chuckles] Sure.

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[no audio]

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[Prof. Lamb] Okay, now let's take a minute

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to talk about these quantum numbers[br]that show up in the wave functions

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(that is, these equations that define[br]the shape of the standing wave).

0:17:19.803,0:17:25.237
The most important quantum number[br]is called "n," and it's easy to envision

0:17:25.237,0:17:29.352
what that quantum number [br]stands for or corresponds to.

0:17:29.352,0:17:34.919
It's called the principal quantum number[br]and it corresponds to Bohr's orbit numbers:

0:17:34.919,0:17:40.554
those numbers that we're already[br]familiar with from our energy well.

0:17:40.554,0:17:47.751
It turns out that not all wave forms or[br]orbitals are allowed in all Bohr orbits.

0:17:47.751,0:17:53.635
Or, in other words, only certain orbitals[br]are allowed for a given value of n.

0:17:53.635,0:17:57.087
Fortunately, the orbitals that ARE allowed

0:17:57.087,0:18:00.319
fall into a neat little pattern[br]hat's easy to remember.

0:18:00.319,0:18:02.756
Let's take a look here.

0:18:02.756,0:18:08.371
For n=1, you'll see that only the simplest [br]orbital is allowed. That's the s-type orbital.

0:18:08.371,0:18:10.159
[Brian] Oh, okay.

0:18:10.159,0:18:15.991
[Prof. Lamb] For n=2, two types [br]are allowed: the s and the p type.

0:18:15.991,0:18:20.097
And since the p-type come in sets of three[br]we show it that way here on the diagram.

0:18:20.097,0:18:30.325
Now, for n=3, we add the d-type,[br]so we can have 3 s, 3 p, and 3 d.

0:18:30.325,0:18:33.774
Brian, I bet you can't guess[br]what types are allowed for n=4.

0:18:33.774,0:18:34.675
What do you think?

0:18:34.675,0:18:42.575
[Brian] I don't know. I'd have to say [br]s, p, d, and-- Isn't there one other one?

0:18:42.575,0:18:43.926
[Prof. Lamb] Oh, yeah. Okay.

0:18:43.926,0:18:46.101
There's got to be another one,

0:18:46.101,0:18:49.428
and it turns out that it's [br]called the f-type orbitals.

0:18:49.428,0:18:50.286
[Brian] Oh, okay.

0:18:50.286,0:18:52.563
[Prof. Lamb] They come in sets of seven.

0:18:52.563,0:18:55.412
Okay, now we've seen what the orbitals look like

0:18:55.412,0:18:59.980
and how many are allowed in each [br]principal quantum number or Bohr orbit.

0:18:59.980,0:19:03.162
One last thought before we go:

0:19:03.162,0:19:06.964
For a 1-electron atom [br](hydrogen, for example),

0:19:07.214,0:19:11.418
the energies of all the orbitals[br]with a given n value

0:19:11.418,0:19:14.317
(that is, with the same n value) are the same.

0:19:14.317,0:19:16.247
They have the same energy.

0:19:16.247,0:19:19.556
And you see, that's how we've [br]drawn them here on the diagram,

0:19:19.556,0:19:22.864
but here's the important point.

0:19:22.864,0:19:27.163
It turns out that they're NOT the [br]same energy in multi-electron atoms,

0:19:27.163,0:19:30.882
and that's very important[br]to the chemistry of those atoms.

0:19:31.237,0:19:33.452
But that's an important topic for another day.

0:19:33.452,0:19:37.153
[Brian] Great, I'm excited. [chuckles][br][Prof. Lamb] Good.

0:19:37.153,0:19:51.572
♪ [synthesizer jazz music] ♪

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END