Relationship between crystal momentum and true momentum Most textbooks make it clearly that crystal momentum is not true momentum. However, in a lot of literature, crystal momentum is treated as true momentum. Here's two examples:


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*Rashba spin splitting. It is known that spin orbit coupling in solids can be mathematically expressed as $\nabla V\times \vec{p}\cdot \vec{v}$. If $\nabla V$ is along $\vec{z}$ direction, it can be simplified to $\vec{z}\times \vec{p}\cdot \vec{v}$. Then, in most literature, it is directly converted to $\vec{z}\times \vec{k}\cdot \vec{v}$ and $\vec{k}$ here is the crystal momentum.

*$k\cdot p$ method in heterostructures. In heterostructures, a common method to derive the Hamiltonian is to replace crystal momentum in original $k\cdot p$ Hamiltonian by $i\hbar\nabla$, which is the true momentum operator. For example, if some system is described by a $k \cdot p$ Hamiltonian $H(k_x, k_y)$. If we cut a strip along $y$ direction and break translational periodicity in $x$ direction, It is common to use $H(-i\hbar\partial_x, k_y)$ to describe this strip.
To summarize, it seems crystal momentum can sometimes be treated as true momentum. Why is that?
 A: A eigenstate of a crystal hamiltonian can be written as a Bloch function in space representation
$$
\psi(\mathbf{r}) = e^{i\mathbf{k}\mathbf{r}} u_\mathbf{k}(\mathbf{r})
$$
$u$ is periodic with respect to the unit cell. The momentum is now given by
$$
\langle \psi|\hat{\mathbf{p}} |\psi\rangle = -i\hbar \int  e^{-i\mathbf{k}\mathbf{r}}u_\mathbf{k}^*(\mathbf{r}) \nabla_r\; e^{i\mathbf{k}\mathbf{r}} u_\mathbf{k}(\mathbf{r}) \;\text{d}^3r\\
=\hbar \mathbf{k} - i\hbar \int u_\mathbf{k}^*(\mathbf{r}) \nabla_ru_\mathbf{k}(\mathbf{r}) \;\text{d}^3r
$$
If we now assume that the second term is small or vanishes due to symmetry reasons we can set $\langle\hat{p}\rangle = \hbar \mathbf{k}$ 
A: I have asked a professor about this and he gave me the answer.
After replacing $\mathbf{k}$ by $-i\nabla$ in $H(h\mathbf{k})$, we are actually getting a new Hamiltonian that acts on envelop of wave functions.
To make this answer relatively complete, I will briefly introduce the main steps focusing on only one band.
Suppose the band we are interested in has a dispersion $E(\mathbf{k})$, we can write its wave function as $\psi_{\mathbf{k}}=e^{i\mathbf{k}\cdot r}u_{\mathbf{k}}(r)$, where $u_{n\mathbf{k}}$ is periodic in space. A wave package can therefore be constructed from Bloch states as follow:
$$\psi'(r,t)=\sum_{\mathbf{k}~\text{near}~0}c(\mathbf{k},t)e^{i\mathbf{k}\cdot r}\psi_{\mathbf{0}}(r)$$
We can define $F(r,t)=\sum_{\mathbf{k}~\text{near}~0}c(\mathbf{k},t)e^{i\mathbf{k}\cdot r}$ as the envelop function of the wave packet. Since $\mathbf{k}$ is near $\mathbf{0}$, $F(r,t)$ is slow varying with respect to $r$.
It can be shown that by substituting $\mathbf{k}$ by $-i\nabla$ in $E_n(\mathbf{k})$, we can get a dynamic equation of $F(r,t)$:
$$E(-i\nabla)F(r,t)=i\hbar\frac{\partial F(r,t)}{\partial t}$$
Generally, substituting $\mathbf{k}$ with $-i\nabla$ in a $\mathbf{k}\cdot \mathbf{p}$ Hamiltonian yields an "effective" Hamiltonian that acts on envelop functions.
For more reference, here's a book: http://link.springer.com/book/10.1007%2Fb13586
