Two Electrons

Electrons have spin $\frac{1}{2}$ and we now have to work out how the electron spin enters into the Slater determinants. The single particle wave functions for particle $1$ are products of orbital wave functions and spin wave functions,

$$\psi(\xi_1)= \psi({\bf r}_1) \vert\sigma_1\rangle_{(1)}.$$ (2.2)

For spin- $1/2$, the spin label $\sigma_1$ can take the two values $\sigma_1=\pm 1/2$ which by convention are denoted as $\uparrow$ and $\downarrow$. The two spinors have the following representation in the two-dimensional complex Hilbert space (spin-space),

$$\vert\uparrow\rangle_{(1)} =\left( \begin{array}{c} 1 0 \end... ...narrow\rangle_{(1)}=\left( \begin{array}{c} 0 1 \end{array} \right)_{(1)}.$$ (2.3)

Here, the index $_{(1)}$ means that this spin referes to particle $(1)$.

We now consider the four possibilities for the spin projections $\sigma_1$ and $\sigma_2$ and the corresponding four sets of basis wave functions,

$$\frac{1}{\sqrt{2}}\left[ \psi_{\nu_1}( {\bf r}_1)\psi_{\nu_2}( {\... ...\bf r}_2)\psi_{\nu_2}( {\bf r}_1) \vert\uparrow \uparrow \rangle_{(12)} \right]$$

$$\frac{1}{\sqrt{2}}\left[ \psi_{\nu_1}( {\bf r}_1)\psi_{\nu_2}( {\... ...f r}_2)\psi_{\nu_2}( {\bf r}_1) \vert\downarrow \uparrow \rangle_{(12)} \right]$$

$$\frac{1}{\sqrt{2}}\left[ \psi_{\nu_1}( {\bf r}_1)\psi_{\nu_2}( {\... ...f r}_2)\psi_{\nu_2}( {\bf r}_1) \vert\uparrow \downarrow \rangle_{(12)} \right]$$

$$\frac{1}{\sqrt{2}}\left[ \psi_{\nu_1}( {\bf r}_1)\psi_{\nu_2}( {\... ...}_2)\psi_{\nu_2}( {\bf r}_1) \vert\downarrow \downarrow \rangle_{(12)} \right].$$ (2.4)

Here,

$$\vert\uparrow \downarrow \rangle_{(12)} \equiv \vert\uparrow \rangle_{(1)}\otimes\vert\downarrow \rangle_{(2)}$$ (2.5)

is a product spinor, i.e. a spin wave function with particle (1) with spin up and particle (2) with spin down, and corresp[ondingly for the other product spinor.

We can now re-write the basis states Eq. (III.2.4) by forming linear combinations of the `mixed' spinors (exercise: check these !),

$$\psi_S(\xi_1,\xi_2) = \psi^{\rm sym}_{\nu_1,\nu_2}({\bf r}_1,{\bf r}_2) \vert S\rangle$$ (2.6)

$$\psi_{T_{-1}}(\xi_1,\xi_2) = \psi^{\rm asym}_{\nu_1,\nu_2}({\bf r}_1,{\bf r}_2) \vert T_{-1}\rangle$$ (2.7)

$$\psi_{T_{0}}(\xi_1,\xi_2) = \psi^{\rm asym}_{\nu_1,\nu_2}({\bf r}_1,{\bf r}_2) \vert T_{0}\rangle$$ (2.8)

$$\psi_{T_{+1}}(\xi_1,\xi_2) = \psi^{\rm asym}_{\nu_1,\nu_2}({\bf r}_1,{\bf r}_2) \vert T_{+1}\rangle.$$ (2.9)

Here, the symmetric and antisymmetric orbital wave functions are defined as

$$\psi^{\rm sym}_{\nu_1,\nu_2}({\bf r}_1,{\bf r}_2) = \frac{1}{\sqrt{2}}\left[ \psi_{\nu_1}( {\bf r}_1)\psi_{\nu_2}( {\bf r}_2) + \psi_{\nu_1}( {\bf r}_2)\psi_{\nu_2}( {\bf r}_1) \right]$$ (2.10)

$$\psi^{\rm asym}_{\nu_1,\nu_2}({\bf r}_1,{\bf r}_2) = \frac{1}{\sqrt{2}}\left[ \psi_{\nu_1}( {\bf r}_1)\psi_{\nu_2}( {\bf r}_2) - \psi_{\nu_1}( {\bf r}_2)\psi_{\nu_2}( {\bf r}_1) \right].$$ (2.11)

Furthermore, the spin wave functions are defined as

$$\begin{array}{ccc} \vert S\rangle =& \frac{1}{\sqrt{2}}\left[... ...rrow \uparrow \rangle &\mbox{\rm Triplet State}\\ \end{array}.$$ (2.12)