Relationship between crystal momentum and true momentum

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 $\langle\hat{p}\rangle = \hbar \mathbf{k}$

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