.

a) Use the spin-up/spin-down notation and write down the four spin wave functions (triplets and singlet) for two spin $1/2$ particles. Assign total spin $S$ and total spin projection $M$ to each of them.

b) Use the triplets and singlet from a) and briefly explain the meaning of `entanglement'.

c) Consider the following four spin-orbitals

$$\psi_1({\bf r}) \vert\uparrow\rangle,\quad \psi_1({\bf r}) \vert\... ...i_2({\bf r}) \vert\uparrow\rangle,\quad \psi_2({\bf r}) \vert\downarrow\rangle.$$ (3.2)

(i) For a system of two electrons, explicitly write down the four basis states, i.e. the four Slater determinants (each is an antisymmetric linear combination of products of these spin-orbitals).

(ii) Explicitly write down the single-triplet basis as linear combinations of the basis states from (i).