Single Particle

We assume to have a Hilbert space with a complete basis of wave vectors $\vert\nu\rangle$ corresponding to wave functions $\langle {\bf r} \sigma \vert\nu\rangle$ including the spin,

$$\vert\nu\rangle \leftrightarrow \psi_{\nu}({\bf r}\sigma)=\langle {\bf r}\sigma\vert\nu\rangle.$$ (1.5)

Examples:

• harmonic oscillator, $\vert\nu\rangle=\vert n\rangle$ with $n=0,1,2,...$ and the harmonic oscillator wave functions $\psi_n({\bf r})$.
• two-level system with $\vert\nu\rangle$ and $\nu =+$ and $\nu =-$.
• hydrogen atom with $\vert\nu\rangle=\vert nlsjm\rangle$.

The last example shows that $\nu$ is a `multi-index' (index `vector').