I'm working with the following Hamiltonian

$\hat{H}=\int\mathrm{d}\mathbf{x}\sum_{\sigma\in\left\lbrace\uparrow,\downarrow\right\rbrace}\hat{\psi}_\sigma^\dagger(\mathbf{x})\left[-\frac{\hbar^2\nabla^2}{2m}+V(\mathbf{x})\right]\hat{\psi}_\sigma(\mathbf{x})$

(plus a term concerning an interaction potential, which I can do myself when I understand this part.)

The $\hat\psi^\dagger(\mathbf{x})$ and $\hat\psi(\mathbf{x})$ are the fermionic field creation and annihilation operators. $V(\mathbf{x})=V(\mathbf{x}+a\mathbf{\hat{e}}_i)$ is a periodic potential with cubic symmetry, where $a$ is the lattice constant. The minima of the potential are denoted by $\mathbf{x}_\mathbf{j}$, with $\mathbf{j}=j_x \mathbf{\hat{e}}_x+j_y \mathbf{\hat{e}}_y+j_z \mathbf{\hat{e}}_z$ the vector that labels the minima. The coordinates are chosen in such a way that $\mathbf{x}=0$ coincides with a minimum.

The single particle Schrödinger eqn. is solved according to Bloch's theorem by eigenfunctions $\phi_{\mathbf{k},m}(\mathbf{x})$ with energies $\epsilon_{\mathbf{k},m}$. The Wannier functions $W_m(\mathbf{x}-\mathbf{x_j})$ that describe a localized particle at site $\mathbf{x_j}$ are determined by

$\phi_{\mathbf{k},m}(\mathbf{x})=\frac{1}{\sqrt{N}}\sum_\mathbf{j}\mathrm{e}^{i\mathbf{k}\cdot\mathbf{x_j}}W_m(\mathbf{x}-\mathbf{x_j})$

The aim is to write the Hamiltonian in terms of Wannier functions and the creation/annihilation operators, $\hat{c}_{\mathbf{j},m,\sigma}^\dagger$ and $\hat{c}_{\mathbf{j},m,\sigma}$, that create/annihilate an electron with spin state $\left|{\sigma}\right\rangle$ at lattice site $\mathbf{x_j}$ in the band with index $m$.

To do so, I started by expanding the field operators in terms of the Bloch functions and the operators $\hat{c}_{\mathbf{j},m,\sigma}^\dagger$ and $\hat{c}_{\mathbf{j},m,\sigma}$, i.e. I wrote

$\hat{\psi}(\mathbf{x})=\sum_{\mathbf{j},\mathbf{k},m}\hat{c}_{\mathbf{j},m,\sigma}\phi_{\mathbf{k},m}(\mathbf{x})$

Plugging this into the Hamiltonian gives me

$\hat{H}=\sum_{\mathbf{j},\mathbf{j'}}\sum_{\mathbf{k},\mathbf{k'}}\sum_{m,m'}\sum_\sigma\hat{c}_{\mathbf{j},m,\sigma}^\dagger\hat{c}_{\mathbf{j'},m',\sigma}\int\mathrm{d}\mathbf{x}\;\phi_{\mathbf{k},m}^*(\mathbf{x})\left[-\frac{\hbar^2\nabla^2}{2m}+V(\mathbf{x})\right]\phi_{\mathbf{k'},m'}(\mathbf{x})$

Next, I write this in terms of the Wannier functions:

$\hat{H}=\sum_{\mathbf{j},\mathbf{j'}}\sum_{\mathbf{k},\mathbf{k'}}\sum_{m,m'}\sum_\sigma\hat{c}_{\mathbf{j},m,\sigma}^\dagger\hat{c}_{\mathbf{j'},m',\sigma}\int\mathrm{d}\mathbf{x}\;W_{m}^*(\mathbf{x}-\mathbf{x_j})\mathrm{e}^{-i\mathbf{k}\cdot\mathbf{x_j}}\left[-\frac{\hbar^2\nabla^2}{2m}+V(\mathbf{x})\right]W_{m}(\mathbf{x}-\mathbf{x_{j'}})\mathrm{e}^{-i\mathbf{k'}\cdot\mathbf{x_{j'}}}$

However, this is the point where I'm stuck. The final result should be something like

$\hat{H}=\sum_{\mathbf{j},\mathbf{j'}}\sum_{m,m'}\sum_\sigma t_{\mathbf{j},\mathbf{j'};m,m'}\hat{c}_{\mathbf{j},m,\sigma}^\dagger\hat{c}_{\mathbf{j'},m',\sigma}$

where

$t_{\mathbf{j},\mathbf{j'};m,m'}=\int\mathrm{d}\mathbf{x}\;W_{m}^*(\mathbf{x}-\mathbf{x_j})\left[-\frac{\hbar^2\nabla^2}{2m}+V(\mathbf{x})\right]W_{m}(\mathbf{x}-\mathbf{x_{j'}})$

I need to get rid of the two exponentials somehow, but I can't get it. I think I might have done something wrong in the expansion of $\hat{\psi}_\sigma$, but I don't know what.

Thanks in advance.

I'm working with the following Hamiltonian

$$\hat{H}=\int\mathrm{d}\mathbf{x}\sum_{\sigma\in\left\lbrace\uparrow,\downarrow\right\rbrace}\hat{\psi}_\sigma^\dagger(\mathbf{x})\left[-\frac{\hbar^2\nabla^2}{2m}+V(\mathbf{x})\right]\hat{\psi}_\sigma(\mathbf{x})$$

(plus a term concerning an interaction potential, which I can do myself when I understand this part.)

The $\hat\psi^\dagger(\mathbf{x})$ and $\hat\psi(\mathbf{x})$ are the fermionic field creation and annihilation operators. $$V(\mathbf{x})=V(\mathbf{x}+a\mathbf{\hat{e}}_i)$$ is a periodic potential with cubic symmetry, where $a$ is the lattice constant. The minima of the potential are denoted by $\mathbf{x}_\mathbf{j}$, with $$\mathbf{j}=j_x \mathbf{\hat{e}}_x+j_y \mathbf{\hat{e}}_y+j_z \mathbf{\hat{e}}_z$$ the vector that labels the minima. The coordinates are chosen in such a way that $\mathbf{x}=0$ coincides with a minimum.

The single particle Schrödinger eqn. is solved according to Bloch's theorem by eigenfunctions $\phi_{\mathbf{k},m}(\mathbf{x})$ with energies $\epsilon_{\mathbf{k},m}$. The Wannier functions $W_m(\mathbf{x}-\mathbf{x_j})$ that describe a localized particle at site $\mathbf{x_j}$ are determined by

$$\phi_{\mathbf{k},m}(\mathbf{x})=\frac{1}{\sqrt{N}}\sum_\mathbf{j}\mathrm{e}^{i\mathbf{k}\cdot\mathbf{x_j}}W_m(\mathbf{x}-\mathbf{x_j})$$

The aim is to write the Hamiltonian in terms of Wannier functions and the creation/annihilation operators, $\hat{c}_{\mathbf{j},m,\sigma}^\dagger$ and $\hat{c}_{\mathbf{j},m,\sigma}$, that create/annihilate an electron with spin state $\left|{\sigma}\right\rangle$ at lattice site $\mathbf{x_j}$ in the band with index $m$.

To do so, I started by expanding the field operators in terms of the Bloch functions and the operators $\hat{c}_{\mathbf{j},m,\sigma}^\dagger$ and $\hat{c}_{\mathbf{j},m,\sigma}$, i.e. I wrote

$$\hat{\psi}(\mathbf{x})=\sum_{\mathbf{j},\mathbf{k},m}\hat{c}_{\mathbf{j},m,\sigma}\phi_{\mathbf{k},m}(\mathbf{x})$$

Plugging this into the Hamiltonian gives me

$$\hat{H}=\sum_{\mathbf{j},\mathbf{j'}}\sum_{\mathbf{k},\mathbf{k'}}\sum_{m,m'}\sum_\sigma\hat{c}_{\mathbf{j},m,\sigma}^\dagger\hat{c}_{\mathbf{j'},m',\sigma}\int\mathrm{d}\mathbf{x}\;\phi_{\mathbf{k},m}^*(\mathbf{x})\left[-\frac{\hbar^2\nabla^2}{2m}+V(\mathbf{x})\right]\phi_{\mathbf{k'},m'}(\mathbf{x})$$

Next, I write this in terms of the Wannier functions:

$$\hat{H}=\sum_{\mathbf{j},\mathbf{j'}}\sum_{\mathbf{k},\mathbf{k'}}\sum_{m,m'}\sum_\sigma\hat{c}_{\mathbf{j},m,\sigma}^\dagger\hat{c}_{\mathbf{j'},m',\sigma}\int\mathrm{d}\mathbf{x}\;W_{m}^*(\mathbf{x}-\mathbf{x_j})\mathrm{e}^{-i\mathbf{k}\cdot\mathbf{x_j}}\left[-\frac{\hbar^2\nabla^2}{2m}+V(\mathbf{x})\right]W_{m}(\mathbf{x}-\mathbf{x_{j'}})\mathrm{e}^{-i\mathbf{k'}\cdot\mathbf{x_{j'}}}$$

However, this is the point where I'm stuck. The final result should be something like

$$\hat{H}=\sum_{\mathbf{j},\mathbf{j'}}\sum_{m,m'}\sum_\sigma t_{\mathbf{j},\mathbf{j'};m,m'}\hat{c}_{\mathbf{j},m,\sigma}^\dagger\hat{c}_{\mathbf{j'},m',\sigma}$$

where

$$t_{\mathbf{j},\mathbf{j'};m,m'}=\int\mathrm{d}\mathbf{x}\;W_{m}^*(\mathbf{x}-\mathbf{x_j})\left[-\frac{\hbar^2\nabla^2}{2m}+V(\mathbf{x})\right]W_{m}(\mathbf{x}-\mathbf{x_{j'}})$$

I need to get rid of the two exponentials somehow, but I can't get it. I think I might have done something wrong in the expansion of $\hat{\psi}_\sigma$, but I don't know what.

Thanks in advance.