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.