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.