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.