## Tuesday, 1 January 2008

### Spin

It took me three books (Atkins, Kutzelnigg, Levine) to finally understand spin. Not its physics (I don't like physics) but what it does in quantum mechanics.

The problem is that you only have spin if you include relativity into QM (and no-one wants to do that [1]). Without relativity, spin has to be introduced as an axiom.

I think a simple way to understand it, is the following:

A one particle wave function θ(x,y,z) without spin depends on the three spatial coordinates. It's a function
$\theta:\mathbb R ^3 \rightarrow \mathbb C$

The spin-assumption is that the function depends on a fourth coordinate, called ms, that can take on the values 1/2 and -1/2 (or "up" and "down"). In other words Ψ(x,y,z,ms) is a function

$\Psi:{\mathbb R} ^3 \times \{-\frac{1}{2}, \frac{1}{2}\} \rightarrow \mathbb C$

Integration over the whole configurational space is now understood as three integrations and one summation:

$\int~\Psi(x,y,z,m_s) ~d\tau = \sum_{m_s=-\frac{1}{2}}^{\frac{1}{2} }\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\Psi(x,y,z,m_s)~dxdydz$

Of course you can go on with this and define an N particle wavefunction:

$\Psi:({\mathbb R} ^3 \times \{-\frac{1}{2}, \frac{1}{2}\})^N \rightarrow \mathbb C$

where

$\Psi(x_1, y_1, z_1, m_{s_1}, x_2 ..., m_{s_N})^*\Psi(x_1, y_1, z_1, m_{s_1}, x_2, ..., m_{s_N})$

is the (differential) probability of meeting particle 1 at (x1, y1, z1) with spin ms1 while particle 2 is at (x2, y2, z2) with spin ms2, and so on.

What does a wave function with spin look like?

Since the typical approximation for the Hamiltonian does not include spin, we can assume the wave function to be separable:

$\Psi(x,y,z,m_s)=\theta(x,y,z)\sigma(m_s)$

where θ is the spatial function from above, and σ is a function whose domain consists of only two elements.

$\sigma: \{-\frac{1}{2}, \frac{1}{2}\} \rightarrow \mathbb C$

Apparently if we have two linear independent functions of that kind, we can make any other such function as a linear combination of those two. We choose two such functions called α and β with the additional requirement that they form an orthonormal basis (of the spin-function-space):
$\sum_{m_s=-\frac{1}{2}}^{\frac{1}{2}}\alpha(m_s)^*\alpha(m_s)=1,~\sum_{m_s=-\frac{1}{2}}^{\frac{1}{2}}\alpha(m_s)^*\beta(m_s)=0
\sum_{m_s=-\frac{1}{2}}^{\frac{1}{2}}\beta(m_s)^*\alpha(m_s)=0,~\sum_{m_s=-\frac{1}{2}}^{\frac{1}{2}}\beta(m_s)^*\beta(m_s)=1$

These conditions can be satisfied by the following (the 4 values have to form a unitary matrix):

$\alpha(\frac{1}{2})=1,\alpha(-\frac{1}{2})=0
\beta(\frac{1}{2})=0,\beta(-\frac{1}{2})=1$

Now θ(x,y,z)α(ms) is called a spin orbital with α-spin. It vanishes unless the spin is 1/2 (if α(ms) is chosen the way shown). With the α(ms) and β(ms) chosen orthonormally you can easily evaluate integrals involving spin orbitals (easy as far as the spin-part is concerned). But I don't feel like showing that today.

[1] Except of course for Dirac.

Shawn Wilkinson said...

Heh, I remember it took me awhile to understand the mathematics of spin as well. But I put fault on my p.chem professor for charging through the mathematics to get to the spectroscopy section.

Good intro to the topic, though.

CY said...

Find it funny that you dislike physics but love the maths. I'm a physics lover but find maths quite disdainful. Quite the opposite eh? ;)

Keep on writing!

Felix said...

thanks, nice to hear that you liked it, and that i did not write it just for myself

cy: in chemistry i like the "softness", intuition. in maths i like the abstract logic. physics does not have enough of either :)

Shawn Wilkinson said...

blasphemy, felix! pure blasphemy!

although at the moment, I don't feel like defending what encompasses probably nearly 50% of my academic career and research interests. lulz!