Welcome back everyone. 
I'm Charles Clark. 
We're now going to start to look at the 
spectrum of the hydrogen atom in detail. 
We'll review the historical background. 
we'll see what classical mechanics 
yields. 
And we'll go into the old quantum theory, 
always trying to understand the world and 
take steps to reproduce actual 
experimental data. 
Now just to remind you, our focus is on 
the spectrum of hydrogen. 
Here's the emission spectrum. 
and you can see I'm sorry, [SOUND] you 
can see here this red line at 656 
nanometers, which I'll remind you, is the 
same as this line seen in the Fraunhofer 
spectrum. 
this is one of the first things to be 
explained by Bohr's theory. 
so we're going to look at the things 
that, that govern the, the spectrum of 
hydrogen in this region and we'll, we'll 
show how they pertain to others. 
Again, just keep in mind, we're we're 
staying within the visible spectrum here. 
Here are our, our three reference lasers. 
So, I'd like to, you know, to just keep 
those the, the wavelengths and 
frequencies in mind. 
Now, just a reminder of why this system 
was very hard to understand by classical 
mechanics. 
So, it seems that atoms have these 
invariable properties that are 
demonstrated by the spectral lines. 
They don't change at all. 
one sees the same state of an atom on the 
surface of the, in the, in the atmosphere 
of the sun as one does for something in 
gas in a, in a terrestrial laboratory. 
And it's hard to understand how that can 
be if you think of things from standpoint 
of simple Newtonian, Newtonian mechanics. 
So, in classical mechanics you can assign 
any initial value of position velocity. 
Let's say, one particle class mechanics. 
A particle can have any position or 
velocity at an initial time. 
And so the interaction with its 
environment can also change the position 
and velocity in some random way. 
It's very difficult to see how you could 
have permanent structures. 
There are a lot of very interesting ideas 
that were put forward in those days. 
Here's, here's one that appeals to me the 
idea that atoms could be vortex rings in 
some kind of pervasive fluid. 
This was worked on quite a bit by Lord 
Kelvin William Thompson. 
And the, the rationale for that was that 
it was if Helmholtz had found that there 
was a conservation law in effect for 
vortices, and that you could get actually 
fairly complex structures that would last 
for long periods of time. 
So maybe an atom was some sort of vortex 
structure in a fluid. 
In fact the, the appearance of vortex 
rings in atomic gases has returned to be 
of interest in recent years due to the 
the development of the Bose-Einstein 
condensations which which are subject of 
a later talk. 
This is a theoretical paper from last 
year that shows the the generation of 
vortex knots in in a Bose-Einstein 
condensate and shows how they, they 
preserve their form to some extent as 
time evolves. 
We won't discuss that in detail now, but 
it turns out that this is not, this is 
not the solution to the problem of atomic 
structure for quite a fundamental reason. 
So, the problem with continuum models of 
the structure of atoms, the structure of 
the, the, the spectra that we're seeing. 
Is that the the electron and the protons 
were discovered and they seem to behave 
very much like particles that behave, 
behave as classical mechanics suggests. 
They're subject to electrical magnetic 
forces. 
Now, a good understanding of the size of 
atom was obtained in the 19th century. 
And it's pretty much consistent with what 
we know now. 
This was, this was developed on the 
understanding of the kinetic theory of 
gases. 
Even experiments like a lack of spreading 
of, of a drop of oil on a big pond allows 
you to estimate the upper bound or atomic 
sizes. 
And the experiment report, analysis of an 
experiment reported in 1911 by Rutherford 
was quite shocking at the time. 
Rutherford showed that almost all of the 
mass in an, in an atom was concentrated 
in a very small part of its volume, 
almost, almost in a point compared to the 
size of the atom itself. 
And the, the only possible conclusion was 
that atoms are mostly empty space with 
electrons orbiting the nucleus under the 
influence of long range Coulomb 
interactions. 
So indeed, that's not such a strange 
system compared to others we now of. 
For example, if you think the solar 
system as just containing the sun and the 
earth, the sun contains most of the mass 
and the orbit the earth is revolving 
around the sun in a regular way in a 
regular period. 
And and, and most, most of the space in 
between them is empty. 
However, it's rather difficult from the 
standpoint of classical mechanics to 
reconcile that, that idea with the atoms 
that are sort of thought to be like 
little, little balls that are, you put 
together in solids and close packing 
format, or something like that. 
So we're going to start now to see, 
since, since it appears to be the case 
that the, the Coulomb interaction between 
particles is the dominating feature of 
atomic structure. 
We're going to, we're going to see what 
classical mechanics actually says about 
that in detail because lessons learned 
there are applicable to the quantum 
system. 
So in particular, we're going to look at 
hydrogen-like systems, by which I mean to 
say, a system of one electron and one 
nucleus. 
Now, the nucleus could be hydrogen helium 
plus. 
So, hydrogen has a has a nuclear charge 
of one unit of electronic charge. 
Helium, the neutral helium atom has two 
units and nor, normally has two electrons 
about it. 
So, this would be the spectrum of the 
structure of helium plus, the first ion 
of helium which is something that is 
present in the sun, very important object 
in, in astrophysics. 
And in fact there's been quite a number 
of studies of these one electron ions, 
all the way up to Uranium 91 plus. 
that sounds like a lot. 
But that, that, in fact, has been 
observed experimentally in at Livermore 
Laboratory. 
Another system that has been studied 
extensively in recent years that's, 
that's of this type is Positronium. 
That's the bound state of the positron, 
the antielectron, and the ordinary 
electron. 
that's a group at the University of 
California at Riverside as one of the 
world leaders in that area. 
And even more remarkable to me the system 
called antiprotonic helium. 
This is a system where you have a, an 
antiproton, which is is like the proton, 
but with a negative charge bound to the 
nucleus of the helium atom. 
This has been studied in some detail at 
the CERN Laboratory near Geneva, 
Switzerland. 
Now, just to remind you about the, I was 
talking about all those elements. 
Here's a periodic table. 
You can get a free high resolution copy 
of this at this address. 
So, here's hydrogen in the upper corner. 
Here would be helium with, with one 
electron, lithium two plus, going all the 
way down, where is it, to uranium. 
So many, many, I don't, I can't name them 
all. 
But many of the the atoms that you see 
here have been stripped down to have all 
their electrons stripped off but one. 
the reason for doing that for the case of 
uranium, for example, is to understand 
the effects of relativity in atomic 
structure, relativity in quantum 
electrodynamics. 
Okay, so let's see what classical 
mechanics tells us about these 
hydrogen-like systems. 
So, we have, a single nucleus with a 
position denoted by r sub n the mass, 
capital M, a charge of plus Ze. 
Z is the atomic number. 
So, it would be one for hydrogen two for 
helium, three for lithium and so on. 
And then e is what we call the specific 
charge. 
That's the the absolute value of the 
charge of the electron. 
Then there's the electron with a position 
vector, described by position vector r 
sub e little mass, little m, and a 
negative charge of minus e. 
We're going to keep, we're going to keep 
the masses in play represented explicitly 
in this treatment because there's not 
such a big simplification from changing 
it to something else. 
And one of the exercises that we want to 
do is to go through the process that led 
to the discovery of deuterium. 
It's a very important event both from the 
standpoint of scientific development of 
nuclear physics and also the development 
of nuclear energy and nuclear weapons. 
And it was a discovery that was made by 
looking at the Balmer series of hydrogen. 
Okay, now here are the Newton's equation. 
It's Ma, mass times the second 
derivative, the acceleration of the 
nuclear coordinate. 
And then, these are the, this is 
Coulomb's law. 
So, there's two class of the mechanical 
class of, there is a equation of motion 
for the nuclear and coordinate which is 
just inversely propose, it's, the force 
on it is in the direction of the electron 
nuclear distance. 
And it's inversely proportional to the 
square of that distance. 
You see, there is a factor of r from the 
vector and a factor of of r cubed below. 
So, that's 1 over r squared Force Law. 
And then, according to Newton's what is 
it, third law, there's an equal but 
opposite force on the electron. 
So, you have the same thing there. 
Now, I, the I'm using this, this 
coordinate r here which is the separation 
between the nucleus and the electron 
coordinates because that is in fact the 
relevant length for, that describes the 
electrical Force Law. 
What you can do now is you can get, you 
divide this, you divide, okay, take this 
equation and divide it by 1 over M, 
multiply it by 1 over M, you multiply 
this equation by 1 over little, 1 over 
little m. 
And you subtract them and so you get an 
equation of motion for the the electron 
nucleus separation alone. 
So, you see on this side, you have the 
second derivative of that is just a 
function of itself. 
So, we've now managed to get an equation 
that can be solved directly for the the 
separation between the two particles. 
So, from, from this equation, we go 
directly here. 
You see, we're just taking the mass, this 
is an, in, this, this term here has, is 
the dimension. 
This term here has the dimensions of 1 
over mass. 
So, we just invert it to get what's 
called the reduced mass. 
And now, we have a simple Newtonian 
equation for the motion of the of the 
relative distance between the, the 
electron and the nucleus. 
Now, we define the momentum, p, 
associated with this coordinate. 
It's just the mass times the velocity. 
Usual, the usual definition. 
And this then allows us to rewrite the 
equations of motion as a, a pair of 
couple first order equations in time. 
So, here, here's, here's that equation we 
just made up. 
That's that r dot is 1 over mu times p. 
And then p dot is just taken from, from 
this equation. 
because it, P dot is our double dot and 
so it's just minus Ze squared r over r 
cubed. 
Now, why do we reduce this to first-order 
equations, you might ask. 
This is actually something that's come 
up, in the discussion forums. 
And so, I want to make a statement here 
about why it's essential to have 
equations of motion that are first-order 
in time. 
So, when we say, as physicists, that we 
understand a system, that we can 
determine it's future from what we know 
at the present. 
That means, if we know the state of a 
system arbitrary system of s at some 
given time t, where we want a framework 
that gives us the ability to calculate or 
predict what the state would be like at a 
short time later, t plus delta t. 
So, that seems to me to be a very basic 
idea of a dynamical theory. 
Now well, if you have a, you have, you 
have a system at state t its state at a 
sub, a slightly later time, delta t, 
delta t being very small, is going to be 
approximately given by the original state 
here plus the time interval times the 
derivative of the system, the time 
derivative of the system. 
Now, this is a loose way of speaking. 
Let's, let's show explicitly how that's 
represented in quantum mechanics. 
And this is what you saw in Victor's 
lectures. 
Schrodinger's equation must be 
first-order in time. 
Why? 
Because quantum mechanics tells us that 
all the information about the system that 
we can have is containance wavefunction. 
So, in other, the, the wave function 
fully describes a system in a given time. 
Therefore, if we want to find the, the 
wavefunction at a slightly greater time, 
we have to be, you know, we have to be 
able to take its, its time derivative. 
We have to have some operator that 
advances it in time and experiment tells 
us that this is the operator. 
It's one minus i times delta t times the 
Hamiltonian operator divided by h bar. 
So, if you, if we apply this, and delta t 
is, is, is sufficiently small, then we'll 
get the state would, to apply that to 
science, state of time t will get it to 
get it at a, at a time t plus delta t. 
In fact, this is an approximation of e to 
minus i delta t H over h bar to the 
first-order. 
So that, this would be the, this would be 
the full propagator. 
But that's, that's the lowest order 
approximation. 
Now, to make sure that you grasp that 
idea here's a question for you to answer. 
Now, I hope some of you are able to 
answer the previous question correctly. 
And this is a, is a subject that came up 
in the forum on several occasions. 
And it is a very good question. 
Why isn't Schrodinger's equation 
second-order in time, like the other wave 
equations? 
So, the traditional wave equations that 
you, you're taught in school for a 
vibrating string or for the 
electromagnetic field, if you, if you 
take college level physics, are generally 
packed for your convenience as 
second-order equations, but began life as 
first-order equations. 
So, you know, the the old gag Abbot asked 
Costello, where does milk come from? 
Costello says, oh, it comes from a 
bottle. 
Which Abbott says, oh, you're silly, the 
bottle comes from a shop. 
When you consider, consider first 
principles, milk actually comes from a 
shop. 
So the real underlying equa, equations 
that we deal with. 
Sometimes it's most convenient to think 
of them in terms of a second-order 
equations, there, they can, they can be 
decoupled and solved more conveniently in 
second order form, but they're always 
derived from first-order equations. 
And that is the, that is the form that 
we're going to use enough to treat the 
hydrogen atom in classical mechanics now.