Re: hole spin

Hi, Judy,


> It does, thanks, including the pictures. But there's a certain lack
of
> consistency between this and the other reply. Does that mean when I'm
> considering commutation relations I treat a hole as if it were an
> electron with different momentum and spin?


Igor's picture is the usual introductory one in all treatments of the
solid state. You have weakly-interacting electrons in states of fairly
well-defined energy. The states form bands, and some energies do not
correspond to states at all (the band gap). There is an important value
of the energy, the Fermi energy E_F. States below the Fermi energy are
essentially completely occupied, while states above the Fermi energy
are essentially empty. This picture can be understood in terms of the
Schr=F6dinger one-electron equation. The wavefunction psi is defined at
each point in space and time.

However I think you want to go to a deeper level. Am I correct? The
commutation relations you are talking about in your question,
presumably of creation and annihilation operators, do not correspond to
anything in Igor's picture.

Now, even though Igor's one-electron picture is fine for many purposes,
electrons *do* interact with each other via the electromagnetic field.
In Schr=F6dinger wave mechanics, to handle a system of N electrons you
have to deal with functions of 3N variables, antisymmetrized to give
Slater determinants, etc. etc. But N is a very large number, possibly
1e23. What to do?

Recasting the problem in second quantization form doesn't initially
help you, because it's just another way of saying the same thing as the
Schr=F6dinger wave equation. The action of the creation and annihilation
operators on the N-electron Slater determinant can be written down
explicitly. Second quantization notation has the formal advantage that
the resulting equations of motion are the same for all N. However if
you want to do calculations with it, you have to "unpack" it back to
Schr=F6dinger form and decide what value of N you are interested in. If
it's 1e23 you hit a brick wall.

For many years I was confused by the concept of creation operators,
because the word seems to suggest that each c+ is a sort of magic wand
which - ping! - creates another electron. By implication, the electron
it creates must in some sense be an eigenstate, and a lot of treatments
go to a great deal of effort to ensure this. However of course a
creation operator is a mathematical object, not an agent. It might help
your calculations a bit if you choose the orbital "created" by c+ to be
an eigenstate, but, since electrons do interact with one another, this
can only be the case for a specific value of the state vector it
operates on. In general, if you operate on the state vector first with
some other creation operator, then your carefully crafted creation
operator no longer gives you an eigenstate. So why bother? You can in
fact create electrons at a point {wavefunction delta(x,y,z)}, or a
plane-wave {wavefunction exp[2*pi*i*(kx+ly+mz+ft)]}, or with any
mathematical function that takes your fancy. In general, the state
created from an initial eigenstate by this operator will be a
superposition of many eigenstates.

In Igor's picture the functions you choose are the usual Bloch waves.
In metals and the usual high-dielectric-constant semiconductors,
electrons do not interact strongly, so each creation operator operating
on an eigenstate produces something that is almost an eigenstate.
However the discrepancy doesn't matter at all and is fully taken
account of in the math.

Half my difficulties with QM have stemmed from its nonstandard and
confusing uses of language. Basically, creation and annihilation
operators give you a language for talking about systems with a large
number of electrons which is less clumsy than the original Schr=F6dinger
language.

Now we come to the clever bit. We start from Igor's picture and
consider a system at absolute zero temperature, so that all states
below E_F are full and all states above E_F are empty. Formally,
creation operators and annihilation operators for fermions are
symmetrical. We swap over creation and annihilation operators for all
states below E_F.

In the original Schr=F6dinger picture you build up any particular state
of the system by starting from scratch - the vacuum - and then
operating on it in sequence by 1e23 creation operators. In the new
picture, we start from the system which already has 1e23 electrons in
it, and then operate on it with a much smaller number of the new
'creation' operators. By poetic licence, the state you start with can
be called 'the vacuum' of the new representation, even though it is
absolutely crammed full of matter.

In principle, if you wanted to, you could describe each state in the
new representation by the Slater determinant of the wavefunctions of
each of the creation operators. This would be a little bit artificial,
and in a sense it would just a label for the state, but nevertheless
mathematically homeomorphic to the horrendously complicated original.
In fact, what you end up dealing with mainly are Green's functions, or
propagators, which are built up by combinations of functions symbolized
by Feynman diagrams.



> (so that hole and electron
> anticommute, but could both be in the same energy level, assuming the
> situation made that possible?) I think so, if only by the formal fact
> that the hole operator is the electron annihilator, but it's not
quite
> clear.


No, because the choice of whether to take the original creation
operator, or the original annihilation operator, for the creation
operator in the new representation is determined by the energy of the
Bloch wave in relation to the Fermi level. There is no way that you can
have your cake and eat it too.

I'm not sure if this is the answer you wanted. You may have been
happier with Igor's picture, and you may have chosen the words you did
just because they are buzzwords. Either way, I will await your response
with interest.

Cheers,

Zigoteau.

.

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