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M2L4c.txt
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M2L4c.txt
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#
# File: content-mit-8-421-2x-subtitles/M2L4c.txt
#
# Captions for 8.421x module
#
# This file has 155 caption lines.
#
# Do not add or delete any lines.
#
#----------------------------------------
So let's talk about the hydrogen atom.
The energy levels of the hydrogen atom
are described by the Rydberg formula.
This actually follows already from the simple Bohr model.
But of course also from the Schrodinger equation.
And it says that the energy levels, let me write it
in the following way.
It depends on the electron mass, the electron charge, h bar
square.
It has a reduced mass correction.
And then n is the principal quantum number.
It scales as 1 over n square.
So this here is the reduced mass factor.
This here is called the Rydberg constant r.
Sometimes with the index infinity
because it is the Rydberg constant which
describes the spectrum of a hydrogen atom
where the nucleus has infinite mass.
If you include the reduced mass correction
for the mass of the proton, then this factor
which determines the spectrum of hydrogen,
is called the Rydberg constant with an index h for hydrogen.
You find the electronic eigenfunctions
as the solution of Schrodinger's equation.
And the eigenfunctions have a simple angular part which
are the spherical harmonics.
We're not talking about that.
But there is a radial part, radial-way function.
If you solve it, if you find those wave functions,
there are a number of noteworthy results.
One is, in short form, the spectrum is the Rydberg
constant divided by n square.
I want to talk to you about your intuition
for the size of the hydrogen atom
or for the size of hydrogen-like atoms.
So what I want to discuss is several important aspects
about the radius, or the expectation value,
of the position of the electron.
And it's important to distinguish
between the expectation value for the radius
and the inverse radius.
The expectation value for the radius
is a little bit more complicated-- 1/2 1 minus l
times l plus 1 over n square.
Whereas the result for the inverse radius is very simple.
What I've introduced here is the natural length scale
for the hydrogen atom which is the Bohr radius.
And just to be general, mu is the reduced mass.
So it's close to the electron mass.
The one thing I want to discuss with you and you should--
we will need later on for the discussion of quantum
defects for field ionization and other
processes is we have to know what the size of the wave
function is.
And so usually if you a wave your hands,
you would see the expectation value of 1
over r is 1 over the expectation value of r.
But there are now some important differences.
I first want to sort of ask you why
is the expectation value of 1 over r, why does it have
this very, very simple form?
[Student]: The virial theorem.
The virial theorem.
Yes.
We know that there is a very simple form for the energy
eigenvalues.
It's 1 over n square.
Well energy is columb energy e squared over r.
So if the only energy of the hydrogen
atom that columb energy, it's very clear
that 1 over r, which is proportional to the columb
energy, has to have the same simple form as the energy
eigenvalue.
Well, there is a second contribution
to the energy in addition to columb energy,
this is kinetic energy.
But due to the virial theorem, the kinetic energy
is actually proportional to the columb energy.
And therefore, the total energy is proportional to 1 over r,
and therefore 1 over r has to scale exactly as the energy,
since the energy, until we introduce fine structure is
independent of l, only depends on the principle
quantum number n.
We find there's only an n-dependence. But if you
would ask what is the expectation value
for the radius, you find an l-dependence
because you're talking about a very different quantity.
Let me just summarize what we just discussed.
We have the virial theorem, which in general
is of the following form.
If you have a potential energy which
is proportional to radius to the n,
then the expectation value for the kinetic energy
is n over 2 times the expectation
value for the potential energy.
The most famous example is n equals
2, the harmonic oscillator.
You have an equal contribution to potential energy
of the spring and kinetic energy.
Well here for the columb problem,
we discussed n equals minus 1.
And therefore, the kinetic energy
is minus 1/2 times the potential energy.
So, this factor of 2 appears now in a number of relations.
And it's as follows.
If you take the Rydberg constant,
the Rydberg constant in cgs units,
is well that's the columb energy at the Bohr radius.
But the Rydberg constant is 1/2 of it.
So the Rydberg constant is 1/2 of another quantity
which is called 1 Hartree.
We'll talk on Monday about atomic units,
about a fundamental system of units.
And the fundamental way-- the fundamental energy
of the hydrogen atom-- the fundamental unit of energy
is whatever you can-- whatever energy you
can construct using the electron mass, the electron charge and
hbar.
And what you get is 1 hartree.
If you ever wondered why the Rydberg is 1/2 hartree, what
happens is in the ground state of hydrogen,
you have 1 hartree [? verse ?] of columb energy.
But then because of the virial theorem, you have minus 1/2
of it is kinetic energy.
And therefore the binding energy in the n
equals 1 ground state, which is 1 Rydberg,
is 1/2 of the Hartree.
So this factor of 1/2 of the virial theorem
is responsible for this factor of 2-- for those 2 energies.
I usually prefer SI units for all calculations.
But there are certain relations where
we should use CGS units.
Just as a side remark, if you want to go to SI units,
you simply replace the electron charge e square by e square
divided by 4 pi epsilon_0.
So I've discussed the hydrogen atom.
It's also insightful and you should actually
remember that or be able to rederive it for yourself,
how do things depend on the nuclear charge z?
Well if you have a nuclear charge z,
the columb energy goes up by, well,
if you have a stronger attraction,
if you would go to helium nucleus or even a more highly
charged nucleus and put 1 electron in it.
Because of the stronger columb attraction, all
the length scales are divided by z.
So everything is small enough by a factor of z.
So what does that now imply for the energy?
Well, you have a columb field which is z times stronger.
But you probe it now,
at a z times smaller radius, so therefore the energies scale
with z square.