Sunday, 31 May 2009

Antisymmetry

Antisymmetry, second quantization and things related to it are something I kind of stayed away so far. And now that I have finally taken a look at it, I notice how nice the math behind it actually is. And the important things that many things work in general and do not require orbitals.

We consider functions of the following form where A is some set and N a natural number[1]:

$\Psi:A^N\rightarrow\mathbb{C}$

Let V be the vector space of all such functions. And let's look at operators [2] acting on V. First we could consider the Transposition operator

$\hat{T}_{ij}:V\rightarrow V$
$\Psi(r_1,\ldots,r_i,\ldots,r_j,\ldots,r_N)\mapsto\Psi(r_1,\ldots,r_j,\ldots,r_i,\ldots,r_N)$
$1\leq i< j\leq N$

This is a well defined mapping and the application is straight forward, e.g.

$\hat{T}_{12}(e^{x_1-x_2}+x_1)=e^{x_2-x_1}+x_2$

And if you think about it some more you notice that it is linear.

And if I am correct it is both Hermitian and orthogonal with the usual skalar product. This would mean that all the eigenvalues are either 1 or -1. Either way, we are interested in the eigenspaces corresponding to the eigenvalue -1, i.e. functions that are antisymmetric with respect to the transposition. The intersection of all these eigenspaces VA is the set of functions that are antisymmetric with respect to all the transpositions. It is a vector space since it is formed as a intersection of vector spaces.

$V_A=\bigcap_{1\leq i < j \leq N}V_{-1}(\hat T_{ij})=\{\Psi \in V:\hat T_{ij}(\Psi)=-\Psi, 1\leq i < j \leq N\}$

All fermion wave functions that comply with the Pauli principle have to be taken out of this space.

Next it helps to be a little bit more general and to introduce the permuation operator (related to a permuation σ out of the symmetric group SN) as a generalization of the transposition operator

$\hat{\sigma}:V\rightarrow V$
$\Psi(r_1,\ldots,r_N)\mapsto\Psi(r_{\sigma(1)},\ldots,r_{\sigma(N)})$
$\sigma\in S_N$

This operator is also linear and I think also Hermitian and orthogonal.

Then you could write the antisymmetric space also like this

$V_A=\{\Psi \in V:\hat \sigma(\Psi)=sgn(\sigma)\Psi, \sigma \in S_N\}$

since every transposition is a permutation with negative sign and every permutation can be formed out of transpositions and the sign corresponds to how many you take.

Well let's define one more operator, the antisymmetrizer

$\mathcal{A}:V\rightarrow V$
$\Psi\mapsto\frac{1}{n!}\sum_{\sigma\in S_N}sgn(\sigma)\hat\sigma(\Psi)$

You can show that this operator is a projection operator. This is equivalent to the condition that applying it twice is the same as applying it once, since the vector is already projected after the first time. The proof is also straight forward. Applying it twice leads to a double sum

$\mathcal{A}\circ\mathcal{A}=\frac{1}{(n!)^2}\sum_{\sigma,\pi\in S_N}sgn(\sigma\pi)\widehat{\sigma\pi}=$

But this just means that you sum n! times over all permutations, and it turns out

$=\frac{1}{(n!)^2}n!\sum_{\tau\in S_N}sgn(\tau)\widehat{\tau}=\mathcal{A}$

And next you could prove that the space it projects into, is just the VA we had before. You have to show that any projected function fullfils the condition for VA above and that every function which fullfills the condition remains unchanged which is both quite straight forward.

Maybe I should also add why these results are nice. I did have the expression for the Slater determinant before. But by considering that this is nothing but a (normalized) projection of an orbital product into the antisymmetric space it is easier to deal with it. It is no longer some weird expression but nothing but a linear operator. And things like adding Slater determinants are much clearer when you consider this. And that is why it is cool.

And actually as a next step you can consider commutation. These operators commute with the Hamiltonian because they are nothing but relabelling of equivalently treated electrons. Only for that reason you know that eingenfunctions of the Hamiltonian which are also antisymmetric even exist. You restrict the Hamiltonian eigenfunctions to the ones with a -1 eigenvalue of the antisymmetrizer. In the next step you may also do some restrictions according to spatial symmetry - the interacting spaces will transform like irreducible representations of the symmetry group. And finally spin. In the spin case the interacting spaces transform like representations of the unitary group (and I am kind of trying to understand why that is).

[1] Physically spoken N is the number of particles and A is the set of possible coordinates of the particle.

[2] As I probably said before: An operator is a function that makes a function out of a function. Or if you don't like the word "function", it is a mapping between two vector spaces.