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Suppose a particle's time evolution in a 2D $k$-space of first Brillouin zone is as shown in the figure. How can we interpret the motion of the particle in $x$-space?

Any hint for interpretation is useful for me.

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Consider that using a k quantum number means that there is no definite position of your particle (it is a consequence of Heisenberg uncertainty principle).

However, electrons in solids are described as wave packets narrowly centered around k in momentum distribution.

Said so, in Aschroft, N. W., and N. D. Mermin. "Solid State Physics (Brooks-Cole, Belmont, MA, 1976)." Appendix E you find how to relate the crystal momentum (k vector) to the actual velocity of your electron.

In general, to relate velocity and real space motion you need to integrate $$ \vec x(t) = \int_{\rm t_0}^{\rm t} \, dt' \, \vec v(t) \equiv \vec x(t) = \int_{\rm t_0}^{\rm t} \, dt' \, \vec v((k(t)). $$

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