Hello everyone, welcome back to Atomic Structure and Spectra.
I'm Charles Clark and I'll be concluding with part 5 of this lecture.
And then in the subsequent lecture we'll turn to the old quantum theory.
And the practical calculations using the wave functions to address atomic systems.
So let's get started. Before proceeding, let me make a
procedural announcement concerning the homework for this week.
It's not mathematically demanding, in most cases but it is conceptually demand,
demanding. And one part of it requires you to read a
section of Albert Einstein's 1905 paper on the photoelectric effect.
That is, can be found here in this document which is the additional
materials linked from the course home page.
Now we have in there some commentary by Cornelius Cilanchos, it's quite
informative. The original German paper of Einstein,
and then a link to the English translation that's on Wikimedia.
I should mention that in preparing this material, I found that there was a very
considerable error in the existing English translation on Wikimedia.
And I posted that information to the German student study group.
And I'm really grateful to Stephan Leger for stepping in, correcting that error,
and I think a few others as well. So, this is really, for me, a positive
outcome of the course that it's, it's giving rise to an improved version of, of
this wonderful paper. That is potentially the most the widely
accessed source of access to it given the fact that its in the public domain on
Wikimedia. Okay, so now lets proceed and I will
remind you were we left off. Again, we're on the trail of the spectrum
of the hydrogen atom. and the real focus we're going to start
up in earnest in the next, in the next lecture is on this famous Bulma Alpha
Line at 656 nanometers. again, very very close to our, our red
reference laser. this is as you may recall.
Up here, we have the Emission spectrum. And then, this is Absorption spectrum of
the sun, as seen by the 4A transform spectrometer at Kit Peak Observatory.
Now in the last lecture we began studying this electron nucleus system, the Hy,
Hydrogen-like, systems. Hydrogen, Helium plus, Lithium plus, and
some other more exotic things. And from the standpoint of, of classical
mechanics. And well, we didn't, we didn't get that
far. But there was, there's quite a number of
things associated with separating out the relative from the center of mass motion
that are generic value. And we ended up finding that the due to
the particular force law. which is the same as the, as the Coulomb
force law, which is the same as the gravitational force law.
We have a conserved quantity, the angular momentum.
So that angular momentum. Let's, let's pretend we have, actually we
do have, we have god like powers over this system.
So we can, we have the ability to prepare this system with arbitrary initial
conditions for the positions and the velocities.
So think about it that way. we're trying to understand globally
everything that can possibly happen if we fix a an initial position, an initial
velocity. So if we just set the thing up so that we
determine, we determine, we have a, we have a separation between the two
particles. We determine that at time t equals 0 and
then we also determine the relative velocities of the electron and the
nucleus. Then that, those choices determine the
value of the energy and they also determine the value of the angular
momentum. For example, if I, I, I just take the two
particles and hold them still at time t equals 0, separated by some distance, r.
then I release them, well then we have r of 0 is equal to r not, whatever value
that was. And r, sorry p of 0 vector r 0, here is
just r not P of 0 is equal to 0. So tat means in a case like that, the
energy is just now equal to minus ZE squared over r 0.
And you might, you might just think about what is the angular momentum if we have.
we have the 6 value r now we just put that in there.
Well if initially the the momentum was 0 then the angular momentum is going to be
0 and it remains that way for the entire remaining time, the system is
interacting. Now is there anything else that we can do
to get, to further, get a under command of this system?
And the answer is yes. We're heading for that answer.
And a good way to start, is for example, we found a constant of the motion.
that was made by say squaring the momentum and then another one the vector
cross of the motion with using a cross product of the position the momentum.
And let me remind you that the angular momentum defines a plane, this is in the
previous thing. defines a plane in which the, the
position and the momentum remain for all times.
So as soon as we set up our initial condition, for positional momentum, we
define an orbital plane, and the particle moves in that plane.
Now It's always a good idea to look at alternative coordinates for that plane.
So, we don't really know right now how the momentum and how the momentum and the
the position are going to evolve in time. But we do have 2 other variables that
arise quite naturally. So we're interested in vectors that are
in the same plane as the momentum and the the position.
And here are 2 choices. one is r cross l.
Now, I hope I've done this correctly, so here I'm taking L to be out of the plane
of the image. So L is pointing toward you, from your
display, and now, if we use the right hand rule to calculate r cross l from r,
my result is that this is r cross L in that direction.
And then there's another, another another vector independent of the previous one P
cross l. And as you can see I've, I have again l
out of the board. I do a cross product with P cross l.
Now that, that turns in the other direction so this is P cross l.
So these are 2, 2, 2 vectors that sort of have a parallelism with the position, the
momentum. That is their, their basically, you're
taking a, a constant vector, the angular momentum, and performing the cross
product. So in some sense they're really just, a
simple linear functions of the position momentum.
But they well let's, let's just see if they give us any additional information
on the problem. So the first thing we'll do is work out
explicitly what r cross L is. So L is r cross p, so r is r cross r
cross p, or just to simplify the just, just to put everything in terms of r in
its time derivative, it has this simple form.
Now this gives us an occasion to use a famous vector identity.
Which some of you may have encountered in in electrodynamics or other mechanics
courses. Very useful.
The back cab rule is the pneumonic given to it, among people like me.
So it's, it's actually, its relatively easier to prove.
But its a good idea just to be aware of it.
a cross a quantity b cross c is b a dot c minus c a dot b, that pronounce batcab in
American. Just for those with a mathematical bent,
this is not the same as a cross b cross c, so the cross product is not
associative. It actually you, you have to be quite
explicit in the in the, the grouping of operations in order to get the same
result. And you can, you can see this directly
from from applica, changing, changing the
order around and applying the back cap. So this is a very simple calculation when
you know how to do it. I said there's nothing very much of
intrinsic of interest in using the back cab rule.
I'm just showing the details, so you see how straight forward it is.
And so when you when you do evaluate this identity.
you get this form and then, remarkably enough, it turns out that, this r cross
L, which we just took as a basically another basis vector.
That's natural to used to describe the dynamics of the system because it lies in
the, in the same. It, it points to locations where the
particle can. Where the electron nucleus system can be
in the plane. We find that it is proportional to a, a
time derivative. just, which is the, it's the minus r
cubed times the time derivative of the unit vector of r.
So this is the unit vector of r is just the if, if the if the particle is moving
in some trajectory, then the unit vector is always the, the same length.
And it, it keeps, it keeps a track of the, the direction of the the direction
of the electron nucleus separation. But not, it's the, its independent of the
magnitude. Okay, so now let's see let's see if we
can infer anything useful. About from this addition term, addition
vector p cross l. So, what we do here first, is just to
calculate the time dirivitive of P cross l.
that's pretty easy because. p cross L dot would be p dot cross l plus
p cross l dot. But l dot vanishes, because l is constant
of motion. And so now we have we can just take the
Newton's law and plug into p dot, a substitute for p dot.
And then we get this this identity here. So, I guess you can see now that
That this, these 2 can combine to give something that's independent of time.
So you see, that p dot cross l ended up giving a something that's proportional to
the Time derivative of the, of the unit
vector. And so, here is our central result.
this is a, called the Runge-Lenz vector. It's been known under different names,
for a long period of time. There's a very informative Wikipedia
article about it. And it is,
Well, as you see, that's just as it is here.
p cross l minus ze square, e squared mu times the unit vector in r.
And it's a constant of the motion and we are going to now look at its properties
and it turns out a very important Interpretative value.
In fact, it makes it possible for us to concisely express all possible solutions
of this equation of motion for the electron nucleus system.
This is very unusual for most problems of classical mechanics.
There is no such global Solution. In this case it's what we have, what we
have is an integral system where the, the constants of the motion completely
specify the properties up to the choice of the origin of time.
So, the first thing to note, about this Runge-Lenz vector, is it enables us to
define the orbit in a remarkably simple way.
So let's say, we have the Runge-Lenz vector.
Which is a constant, sits in space. And now let's just, we'll just refer the,
electron orbit to it by the most straightforward, manner.
We know that, A is in the plane of the, of the electron
orbit. Because it was, we, we chose it, by using
from vectors that were always in that plane.
and so theres this angle phi between the position vector and the run lens vector
completely defines where the position vector is.
So, we all know how to take the scale or product r.a Is r the, this this is, this
is the vector, sorry this the vector product R.A.
R A cosign theta. And so now we just, we just work it
through. We work it through and this, this gives
us opportunity to Invoke another common vector identity a dot b cross c.
So this is something also widely used throughout physics and I suggest that you
think about what this implies for the evaluation of r dot p cross l, and
that's the subject of an in-video quiz. Well I hope that was a straightforward
exercise for some of you. And that for those who, whom it wasn't
you learned something. I mean this is for example, that identity
a.b cross c is a way of computing the volume of a tetrahedron that is defined
by the, the vectors, three vectors, a, b, and c.
Anyway, here's the result. this is just l squared and that leads to
this simple formula for the, the radius as a function of the angle phi.
So, another, another way of looking at this is that the.
Dependence of one over R. If we just invert that equation.
If we say one over r. As a function of fee.
It just, it just oscillates sinusoidally. So there's a, there's a very simple, very
simple dependence on the inverse of r so which is proportional to the, the
potential energy. So, that shows, in some sense, there's a,
there's a continual harmonic oscillating interchange of energy between the
potential and the kinetic parts, kinetic contributions of the energy.
Okay, so there's one last calculation to do.
to illustrate the essential points of this system and that is to calculate
theses, these, the, the magnitude of A is terms of of other quantities.
And so I mean, this, this just involves some, some vector very straight forward
vector. calculations who which you.
We've done, we've done those already so I'm not I just show you the result.
And so this shows us a significant thing in that.
The, the run a lens vector. Its magnitude depends both upon the
energy and the angular momentum. In a sense the run the lens factor.
It looks like we have a large number of constants in motion here.
The energy, the anglement of that vector, and the run of lens factor.
So 0[INAUDIBLE] these are 7 constants of motion that are defined on the surface
independently. Well, that's far too many.
In fact the the number of independent ones is more like 5.
That is the energy, the angular momentum, vector, and then the run of lens vector
must be perpendicular to the anglementum vectors.
So it has, it has 2 components, but in fact there's then this consistency
equation so it reduces it to 1. Okay, so now you can see some.
some possibilities here. For example, this r can go to well, it
has a possibility of going to infinity if, if A is greater than this, Ze squared
mill. Because the cosine phi can, can become
negative. So let's just let's think about what we
can, what we know about the, the, the values of these various.
these various quantities and see how that determines the form of the possible
orbits of this system. So that's a basic, test of that, of your
understanding of that, is in the next in-video quiz.
So, I hope it's clear to you that in classical mechanics, we can in fact
choose initial conditions that correspond to any possib, any energy.
Negative or positive, independent of the value the the physical parameters of the
problem the charge of the, the charge of the mass.
And that there's a very sharp dividing line.
And this choice at this, this determination of the dividing line will
remain to be important. So for energies less than 0, what do
those correspond to? Well, for example, if we start the
system, we start the system with the electron and nucleus at rest at some
fixed distance. Then we saw early in the lecture, and
it's very easy to, to see from this, that this is the only contribution to the
energy so the energy isn't, is negative there.
So, the indeed if we start the system at rest, what happens is as you'll probably
be able to figure out. It represents, it, it corresponds to
extreme of this of this denominator, and you, you get you get an elliptical
motion. well extreme case-full elliptical motion.
Now, on the other hand, if the energy is greater than 0, then the the result is
that the electron nucleus system completely dissociates.
That is the the value of the radius eventually the value of the separation
eventually gets as large as you please. So the, the, case is e less than 0,
correspond to the bound states of atoms and molecules that respond to the sharp
emission lines. And absorption lines in which we're
interested. Now for E greater than 0, we have at
least discussed some of those, those such states.
more or less in passing. So, for example the photoelectric effect,
when an atom, an electron is ejected from an atom, correspond to a state of, E, E
greater than 0. And the, is the, that's basically the
same process as the ion, ionization of gases by ultraviolet radiation.
Which is the section of Einstein's paper that's, it dealt with in the homework for
this week, and Rutherford's Scattering experiment.
So we will see in the next lecture, in the next set of lectures, how this run
the lens vector enters into quantum mechanics.
And just to be very give, give you some guidances to where we're heading.
these, these statements are modified. one significantly, the other not very
much at all. So we will see that the states of energy
less than 0 are now discrete. So quantum mechanics indicates that only
certain values of energies are possible. And these have to do with basically the
interference of the quantum waves to form.
Sort of echoing bound states. This is sort of implicit in the Feynman
path integral picture that Victor showed you.
For the continuum on the other hand, I mean there still are significant effects
of quantum mechanics, but all energies remain possible.
And we'll look at the details of that in the next lecture, so, hope to see you
return then.