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♪ [slow jazz music] ♪

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[Prof. Lamb] Well, we have Brian
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 
the Bohr model of the atom,

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we mentioned that there's 
a more sophisticated model

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that better describes the way 
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 
quantum mechanical model of atomic structure.

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

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

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

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

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[Brian] Wasn't it why electrons could 
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
are what we call "quantized."

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Now, the solution to this mystery
began to reveal itself

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

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It was then reasoned that if light can 
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 
that led to an understanding

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of how electron energy levels
in atoms are quantized.

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Would you like to see how?

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

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

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We're not going to do this in 
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
visualize things in three dimensions.

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

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[engine revving]

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

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

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

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

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

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Second is the fact that we can create
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
moving horizontally, don't we?

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However, if the wave reflects back on itself,

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under certain circumstances,
the reflected waves and the incoming waves

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

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This is called a standing wave.

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You've actually seen 
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
wavelengths can produce standing waves.

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These are wavelengths that fit neatly
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 
to the length of the string would do the job.

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

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When we pluck the string,
there are many wavelengths produced,

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but the wavelengths that fit along 
the string in whole numbers

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reflect back and forth off 
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
each other as they bounce back and forth.

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

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and that's the note and 
its overtones that we hear.

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So the string produces one sweet single tone.
[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
the guitar string around on itself like this.

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[no audio]

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Now notice: The only way
a standing wave will form

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

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

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

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

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Now, if we expand the size 
of the circle just enough

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so that one full wavelength is 
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 
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,
not as a particle, not as a tiny dot,

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

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then the electron would 
destroy itself in all orbits except

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

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In other words, the electrons that exist in these
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
mechanical model of the atom.

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

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

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that is, points exactly between the crests
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 
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 
simple idea into three dimensions.

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[to Brian] Now, Brian, let's replace
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 
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 
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 
between yellow and red.

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

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You might think of the yellow
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
because everything stays in a line

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

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

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

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If the length of the string--
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
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
in the crests and troughs

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

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

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then Brian, what about the areas 
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
intensity of the redness or yellowness

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doesn't instantly drop to
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 
on the string (the crests and troughs)

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

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

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

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

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with greatest intensity at the peaks and 
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 
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
and the lowest point.

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

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Pure yellow, pure red.

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Half-way in-between, 
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 
standing wave would look like.

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

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

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

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This would correspond, not to a wave in a line
(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 
and troughs using two colors.

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In this two-dimensional system,

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we see the waves as rings
of crests and troughs.

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

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As you said, they're at the point where
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 
see pure yellow and pure red.

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[Brian] Okay.

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[Prof. Lamb] Now we're ready 
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
tend to look more like shells in a sphere.

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You might thing of the waves 
like the layers in an onion

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or the dolls inside dolls of 
a Russian matryoshka doll.

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From the outside, this system would 
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
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
and there are only certain radii

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at which the electron wave doesn't 
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 
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 
of pure yellow and pure red.

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And because of destructive interference,

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the electron intensity drops
between these shells until

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

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

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Well, you now should have a 
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,
it is a much simpler picture than the real thing:

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

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We call his model the Schrödinger model. 
(Surprise!)

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And it's based on the idea that the 
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 
of onions or matryoshka dolls

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but in terms of the mathematical equation
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 
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 
then three dimensions; but in fact,

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

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Instead, it uses waves based on equations
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 
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, 
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 
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,
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
us a one-dimensional standing wave).

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Now, these variables are called 
the quantum numbers,

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and they govern the shape and 
the size of the standing wave.

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It turns out that one of Schrödinger's standing 
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
we call the s-type standing wave form.

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Oh, and while we're at it, I guess we'd better
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
besides the spherical s-type.

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

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

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

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

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and decided to call these--
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
imagine a standing wave of this type forming,

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

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

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

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[Brian] Okay.

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

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You can see it from different angles.

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Alternatively, you might imagine 
that this standing wave

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

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

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

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[Brian] Oh, yeah.

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[video is paused during emergency 
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
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 
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, 
and one along the z-axis.

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[no audio]

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

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They are the next type,

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

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

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

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

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

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The most important quantum number
is called "n," and it's easy to envision

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what that quantum number 
stands for or corresponds to.

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It's called the principal quantum number
and it corresponds to Bohr's orbit numbers:

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those numbers that we're already
familiar with from our energy well.

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It turns out that not all wave forms or
orbitals are allowed in all Bohr orbits.

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Or, in other words, only certain orbitals
are allowed for a given value of n.

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Fortunately, the orbitals that ARE allowed

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fall into a neat little pattern
hat's easy to remember.

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Let's take a look here.

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For n=1, you'll see that only the simplest 
orbital is allowed. That's the s-type orbital.

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[Brian] Oh, okay.

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

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And since the p-type come in sets of three
we show it that way here on the diagram.

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Now, for n=3, we add the d-type,
so we can have 3 s, 3 p, and 3 d.

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Brian, I bet you can't guess
what types are allowed for n=4.

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What do you think?

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[Brian] I don't know. I'd have to say 
s, p, d, and-- Isn't there one other one?

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[Prof. Lamb] Oh, yeah. Okay.

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There's got to be another one,

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and it turns out that it's 
called the f-type orbitals.

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[Brian] Oh, okay.

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[Prof. Lamb] They come in sets of seven.

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Okay, now we've seen what the orbitals look like

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and how many are allowed in each 
principal quantum number or Bohr orbit.

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One last thought before we go:

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For a 1-electron atom 
(hydrogen, for example),

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the energies of all the orbitals
with a given n value

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(that is, with the same n value) are the same.

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They have the same energy.

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And you see, that's how we've 
drawn them here on the diagram,

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but here's the important point.

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It turns out that they're NOT the 
same energy in multi-electron atoms,

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and that's very important
to the chemistry of those atoms.

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But that's an important topic for another day.

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[Brian] Great, I'm excited. [chuckles]
[Prof. Lamb] Good.

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♪ [synthesizer jazz music] ♪

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END