M2L7c.txt

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Let's go back to the helium atom,
and look at the special symmetry, which
comes, of course, because operators
are symmetric with regards to the two electrons.
What we have is, we have the total spin
is the sum of the spins of electron one and electron two.
All spin operators the spin operator S and its components,
are symmetric against particle exchange.

In other words, if you couple to the spin
through a magnetic field, you couple to the two electrons
symmetrically.
And therefore, you have the selection
rule that only symmetric and symmetric, and anti-symmetric
and anti symmetric states couple.
Or, to say the same thing-- to say
the same result in different words--
both the spatial and the spin symmetry, each of them
can be symmetric and anti-symmetric.
Of course, the product of them has
to be anti-symmetric because we're talking about fermions.
So both of these symmetries are good quantum numbers.

Or, more formally, all observables commute
with the particle exchange operator.
So in any arbitrary observable commute.

So the commutator of the particle exchange operator
vanishes.
OK, but now what is special to the approximation in which we
have treated helium, as long as wavefunctions
and operators separate into spin-dependent and
space-dependent parts-- of wavefunction
had a spatial part, and then we multiply it
with a spin wavefunction.
So the wavefunction factorize, separated into a spin
part and a spatial part.
Also, the operator, if you have operator for total spin,
if you have operator for dipole moment,
the operators are either addressing
the spatial degree of freedom, or the spin degree of freedom.
So as long as this is the case, we
have an even extended conservation law for both spin
exchange, for exchange of the particles
in the spatial domain, and the spin domain.
So as long as wavefunctions and operators
separate into spin and space dependent parts,
then the particle exchange operator
can be regarded as the product of an exchange operator, which
only acts on the spatial wavefunction,
and on the spin part of the wavefunction.
And both quantum numbers, for spin exchange,
symmetric, anti-symmetric, and for exchange of particles
in the spatial part of the wavefunction,
symmetric or anti symmetric, are conserved.

So often we discuss symmetries and the reason
why something is conserved, because then we can think
about, how do we violate it?
Often you can say, conservation laws
are often written down just as a way to think about it.
How can we break them?
And in this case, we can break this symmetries, which
doesn't allowed transitions between singlet and triplet,
we have to break them to get an intercombination lines.
So in order to get a singlet to triplet transition,
is only possible when we violate the assumptions
we've just discussed.

And, of course, an assumption was
that the wavefunction factors into spin
part and a spatial part, but when spin and spatial
wavefunctions are mixed, for instance by spin orbit
coupling, then the assumptions we've discussed
are no longer valid.
So what we need is, for instance a mixing of spin and spatial
wavefunctions by spin orbit coupling.

Well, as we will learn later today,
spin orbit coupling is actually a relativistic effect,
it naturally arises in the dirac equation,
and it scales with the nuclear charge as Z to the four.
So therefore in Helium it is very, very weak.
Therefore, the triplet ground state is extremely long-lived.
The lifetime in Helium is 8,000 seconds.
It's one of the most longest-lived metastable states
you can imagine in which you find in atomic physics.
Now, since spin orbit coupling becomes rapidly stronger
with nuclear charge, you would expect the metastable lifetime
to be much shorter for the analogous
state in the other rare gases.
And indeed, the other rare gases are on the order of 40 seconds.
You have other atoms where you have singlet and triplet lines.
We need two electrons, and the two spins
can from singlet or triplet.
So that naturally also happens in two atoms which
are in the second column of the periodic table,
magnesium, calcium, strontium, have intercombination lines.
And they are actually very relevant in current research
because these are candidates for atomic clocks.
In those atoms, magnesium, calcium, strontium,
physics is more complicated.
The line width is typically kilohertz.
But that's probably just what you want.
If the line width is kilohertz, the lifetime is millisecond,
you still have a matrix element that you can drive
the transition and observe it.
For an atomic clock, you don't want a transition
which is thousands of seconds, or half the age
of the universe, because it's too weak to be observed, too
weak to be driven.
So intercombination lines of kilohertz
lined with a few hundred Hertz or such,
are almost ideal for optical atomic clocks.
Any questions?