Bloch state

A Bloch state is an energy eigenstate of a lattice-periodic Hamiltonian that is simultaneously an eigenstate of every lattice translation. It can be written as ${\psi }_k(x)=e^{ik\cdot x}{u}_k(x)$ with lattice-periodic $u_k$.

Crystal momentum is the translation quantum number $\hslash k$ of a Bloch state. Since $k$ and $k+G$ give the same translation character whenever $G$ belongs to the reciprocal lattice, crystal momentum is defined modulo reciprocal-lattice vectors.

A charge $-e$ starting at a band minimum when the field $E_0(\sin \omega t,\cos \omega t)$ is applied acquires the displayed crystal momentum. Its orbit is a shifted circle. For masses $m_x,m_y$, the real-space coordinates are proportional to $(\sin \omega t-\omega t)/m_x$ and $(\cos \omega t-1)/m_y$: an elliptic oscillation plus a uniform drift. Omitting the integration constant changes the physical initial condition and reduces the maximum light-axis kinetic energy by a factor of four.

A lattice-translation-invariant interaction conserves total crystal momentum modulo a reciprocal lattice vector $G$. On a periodic chain of $M$ sites with unit spacing, the discrete Fourier transform supplies ${\sum }_{j=1}^M{e}^{i(k_1+k_2-k_3-k_4)j}=M{\delta }_{k_1+k_2,k_3+k_4(\mathrm{mod}2\pi )}$. This Fourier orthogonality proves the selection rule directly; it permits transfer by a reciprocal lattice vector rather than requiring equality of unrestricted momenta.