Hello, everyone. Welcome back.
I'm Charles Clark. And in this lecture, we're gong to look
at quantum theory old and new. we'll conclude our discussion of the Bohr
model, show what it predicted. give a demonstration of its mnemonic
value for when you have to infer things about atomic spectra or quantum mechanics
from the imperfect knowledge. And then deal with actual solution of the
Schrodinger equation for atomic molecular systems and look at some of the
approximation methods, and the analytic results that are needed to give a modern
account of atomic structure. So, let's proceed.
Now as before, we have some original scientific literature available with a
commentary that you might look at the one thing that will be quite helpful I think
is, of course, original paper here. then another topic that we'll treat is
the application of the Bohr theory, early important application of the Bohr theory
to the discovery of deuterium. There's actually two things two items
there. The second one is written for popular
audience. And so it's either more accessible of the
two. The first one contains some of the
original literatures original papers from the physical review and then a memoir
written by one of the co-discoverers. So, do have a look at that if you're
interested. So, returning to where we left off at the
end of the previous lecture we had solved by rather elementary techniques for the
really complete description of the classical mechanics of an electron and a
nucleus interacting through the Coulomb interaction.
And then we found these constant motion angle atom which is a characteristic of
any central, any force that depends any potential, any potential interaction that
depends only upon the distance between the particles.
And then this an additional Runge-Lenz vector.
And with that Runge-Lenz vector, we're able to write a very simple expression
for the, the orbit of the of the, the orbit executed by the interparticle
separations. So, it's an, an orbit of the distance
between the electron and the nucleus. And these are for, for energies less than
0, which means the the, the system never escapes to infinity.
the the orbit is a, is a generalized ellipse, including the special case of a
circle, which occurs when A is equal to 0.
then for A greater than 0 the the orbit clearly escapes to infinity.
and that's that can be seen well that, that, that can be shown by look, using
this identify and, and replacing replacing a denominator there.
Right, but there are no restrictions whatever imposed by classical mechanics
on the possible values of the energy. It can be any number, any, any real
number of, you know, a large, a negative number very large magnitude, which
corresponds to the electron orbiting very closely to the nucleus.
Or a very large positive number, in which case, the electron, the nucleus separate
to infinite seperationals or they go to very large separation of the large times.
It was about a hundred years ago to the day at the time I'm recording this that
Neils Bohr conceived of a solution to this problem on application of the
quantization idea to the motion of a particle in orbit about you know, in
particular, the motion of the electron and hydrogen atom.
So in, in this case he states it in terms of the motion of the electron.
And it really has to do with the, the motion of the center of mass.
But since the nucleus of the the hydrogen atom is nearly 2,000 times as massive as
the electron it, it makes sense to just think of , of it of be, as being
infinitely massive. And all the kinetic energy and angular
momentum is associated with the motion of an electron.
And so, Bohr proposed that the electrons in the hydrogen atoms or the the, the,
the stationary state of a hydrogen atom that were inferred from atomic spectra,
corresponded to circular orbits, Runge-Lenz vector equals zero.
And that in, in such an orbit, the angular momentum, L, would be equal to an
integer times the reduced Planck's constant.
And he, he expresses that condition here. You can read it on page 15 of his paper.
Now as we'll see, this had an enormous effect.
it was accepted very quickly. it led to Bohr's getting the Nobel Prize
in Physics in 1922 explicitly for his understanding the structure of atoms and
their radiative properties. And I have to say before going much
further, that Bohr's idea is not, no longer accepted now.
It's not consistent with the Schrodinger equation as we know it.
But it, it contains the germ of a, of an important idea, which is the the use of
the Planck constant which, was originally devised to explain the thermodynamics of
the electromagnetic field. The, this constant which had the units of
angular momentum, should actually be associated with quantization of the
motion of of elementary systems. And that idea still is correct.
So this, this relationship appropriately understood, is still valid.
And furthermore, the, the Bohr model in its for, own form gave an incredibly
precise explanation of many different facts.
Now, the value of the Bohr model, at least to me, is that it's easy to
understand and to reproduce. All you have to remember is the orbit's
are circular and they're quantized in units of Planck's constant.
And then, this, this gives an amazing number of useful results, which actually
scale in the same way as the, as the results of modern quantum theory.
So, for example here's, here's the here's a restatement.
We have circular orbits with quantized angular momentum, that means that A is
equal to 0. So instantly, from this equation here
that relates the, the magnitudes of the Runge-Lenz vector, the angular momentum
and the energy. we can see that E is negative and it just
by sub, substitution here by substituting, L is nh bar, or L squared
is nh bar squared. we get, we get this simple form which in
modern terms it's convenient to write in, in this way.
So Z is the, the charge, Alpha is the so-called fine structure constant given
by E squared over h bar Z. And this is, this just, and the mnemonic
for this is E is equal to mc squared, this is the Einsteinian expression for
rest mass energy of a particle reduced mass mu.
And, in fact, this very type of the way that I've written this here, is something
that falls out of a Dirac equation from a hydrogen atom, which we may discuss very
briefly later. Okay, now just to get your own sense of
familiarity with this, this equation, I'm going to pose a brief question to you.
Another use of the Bohr model is to help in clarifying the definition of some of
the fundamental constants. so, in particular, what, we, we replaced
the, the, you know, arbitrary reduced mass by the mass of the electron.
So, in other words, we're, we're now using the Bohr model to describe a system
where the, the nucleus has infinite mass, and only the electron is moving.
So, with that choice the energy of the nth Bohr orbit, which is given by this
expression we've seen previously takes the form of minus 1 over n squared R
infinity hc. Where R infinity is the Rydberg constant,
which is about 10 million inverse meters. This is one of the most precisely
measured of the fundamental constants and subject to very active interest today.
the quantity R infinity h bar c has the dimensions of energy and so the this says
E, En is minus 13.606 approximately, eV divided by n squared.
So again, with the with the choices Z equal 1 and mu equal the mass of the
electron, we have an expression for the radius f the nth Bohr orbit.
So, this is something that works for you generally.
The radius goes as the square of the principal quantum number.
This is a relationship that enables you to make pretty good estimates of the
actual size of excited states of atoms. And the constant of proportionality is
the Bohr radius a0 which is 53 picometers approximately.
this is the or in older terminology, we would say is about half an Angstrom.
and it's a good measure of the general size of atoms in their ground states.