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Energy and wave function of a free electron
For a free particle the potential is $$V(\mathbf{r}) = 0$$. The Schrödinger equation for such a particle, like the free electron, is
 * $$-\frac{\hbar^2}{2m}\nabla^2 \Psi(\mathbf{r},t) = i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},t) $$

The wave function $$\Psi(\mathbf{r},t)$$ can be split into a solution of a time dependent and a solution of a time independent equation. The solution of the the time dependent equation is
 * $$\Psi(\mathbf{r},t) = \psi(\mathbf{r}) e^{-i \omega t} $$

with energy
 * $$E = \hbar \omega $$

The solution of the time independent equation is
 * $$\psi_{\mathbf{k}}(\mathbf{r}) = \frac{1}{\sqrt{\Omega_r}} e^{i\mathbf{k}\cdot\mathbf{r}}$$

with a wave vector $$\mathbf{k}$$. $$\Omega_r$$ is the volume of space occupied by the electron. The electron has a kinetic energy
 * $$E = -\frac{\hbar^2 k^2}{2m} $$

The plane wave solution of this Schrödinger equation is
 * $$\Psi(\mathbf{r},t) = \frac{1}{\sqrt{\Omega_r}} e^{i\mathbf{k}\cdot\mathbf{r} - i \omega t} $$

For solid state and condensed matter physicists the time independent solution $$\psi_{\mathbf{k}}(\mathbf{r})$$ is of major interest. It is the basis of electronic band structure models that are widely used in solid-state physics for model Hamiltonians like the nearly free electron model and the Tight binding model. The eigenfunctions of these Hamiltonians are Bloch waves and Wannier functions. Bloch waves and Wannier functions are modified plane waves.

Empty Lattice Approximation
The periodic potential of the crystal lattice in a free electron band structure model is not more precisely defined than "periodic". Implicitly it is assumed that the potential is weak, otherwise the electron wouldn't be free, but it is just strong enough to scatter the electrons. How strong must a potential be to be able to scatter an electron? The answer is that it depends on the topology of the system how large topologically defined parameters, like scattering cross sections in three dimensions], depend on the magnitude of the potential and the size of the potential well. One thing is clear for the 1, 2 and 3-dimensional spaces that are known by man: potential wells do always scatter waves no matter how small their potentials are or what their sign is and how limited their sizes are. For a particle in a one-dimensional lattice, like the Kronig-Penney model, it is easy to substitute the values for the potential and the size of the potential well. If the values of the potential and size of the potential wells are reduced to infinitesimal values the band structure of the Empty Lattice Approximation is obtained.
 * Periodic potential

In the NFE model the Fourier transform, $$U_{\mathbf{G}}$$, of the lattice potential, $$V(\mathbf{r})$$, in the Hamiltonian, can be reduced to an infinitesimal value. The the values of the off-diagonal elements $$U_{\mathbf{G}}$$ in the Hamiltonian almost go to zero and the magnitude of the band gap $$2|U_{\mathbf{G}}|$$ collapses. The division of k-space in Brillouin zones still remains however. The dispersion relation is
 * Nearly free electron model
 * $$E = \sum_{\mathbf{G}}\frac{\hbar^2 (\mathbf{k} + \mathbf{G})^2}{2m}$$

"Free electrons" that move through the lattice of a solid with wave vectors $$\mathbf{k}$$ far outside the first Brillouin zone are still reflected back into the first Brillouin zone. See the external links section for sites with examples and figures.
 * Second, third and higher Brillouin zones

The Schrödinger equation
For a free particle the potential is $$V(\mathbf{r}) = 0$$, so the Schrödinger equation for the free electron is
 * $$-\frac{\hbar^2}{2m}\nabla^2 \Psi(\mathbf{r},t) = i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},t) $$

This is a type of wave equation that has numerous kinds of solutions. One way of solving the equation is splitting it in a time-dependent oscillator equation and a space-dependent wave equation like
 * $$i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},t) = E \Psi(\mathbf{r},t)$$

and
 * $$-\frac{\hbar^2}{2m}\nabla^2 \Psi(\mathbf{r},t) = E \Psi(\mathbf{r},t) $$

and substituting a product of solutions like
 * $$\Psi(\mathbf{r},t) = \psi(\mathbf{r}) e^{\alpha t} $$

The Schrödinger equation can be split in a time dependent part and a time independent part.

Solution of the time dependent equation
The time dependent part of the Schrödinger equation is, unlike the Klein-Gordon equation, a first order oscillator equation
 * $$i\hbar\frac{\partial}{\partial t} e^{\alpha t} = E e^{\alpha t}$$.

The complex (imaginary) exponent is proportional to the energy
 * $$\alpha = -\frac{i E}{\hbar} $$

The imaginary exponent can be transformed to an angular frequency
 * $$E = i \hbar \alpha = \hbar \omega $$

The wave function now has a stationary and an oscillating part
 * $$\Psi(\mathbf{r},t) = \psi(\mathbf{r}) e^{-i \omega t} $$

The stationary part is of major importance to the physical properties of the electronic structure of matter.

Solution of the time independent equation
The wave function of free electrons is in general described as the solution of the time independent Schrödinger equation for free electrons
 * $$-\frac{\hbar^2}{2m}\nabla^2 \psi(\mathbf{r}) = E \psi(\mathbf{r}) $$

The Laplace operator in Cartesian coordinates is
 * $$\nabla^2 = \frac{\partial^2}{{\partial x}^2} + \frac{\partial^2}{{\partial y}^2} + \frac{\partial^2}{{\partial z}^2}$$

The wave function can be factorized for the three Cartesian directions
 * $$\psi(\mathbf{r}) = \phi_x(x)\phi_y(y)\phi_z(z)$$

Now the time independent Schrödinger equation can be split in three independent aprts for the three different Cartesian directions
 * $$-\frac{\hbar^2}{2m} \frac{\partial^2}{{\partial x}^2} \phi_x(x) = E_x \phi_x(x) $$

As a solution an exponential function is substituted in the time independent Schrödinger equation
 * $$\phi_x(x) = N_x e^{\kappa x}$$

The solution of
 * $$ \frac{\partial^2}{{\partial x}^2} \phi_x(x) = \kappa^2 N_x e^{\kappa x} = -\frac{2m}{\hbar^2} E_x N_x e^{\kappa x} $$

gives the exponent
 * $$ \kappa = i k_x = i \sqrt{\frac{2m E_x}{\hbar^2}} $$

which yields the wave equation
 * $$\psi(\mathbf{r}) = N_x N_y N_z e^{i(k_x x + k_y y + k_z z)}$$

and the energy
 * $$ E = \frac{\hbar^2}{2m}(k_x^2 + k_y^2 + k_z^2) $$

With the normalization
 * $$ \int_{\Omega_r} \psi_{\mathbf{k}}^*(\mathbf{r})\psi_{\mathbf{k}}(\mathbf{r}) d\mathbf{r} = 1 $$

and the wave vector length
 * $$ k = \sqrt{k_x^2 + k_y^2 + k_z^2} $$

we arrive at the plane wave solution with a wave function
 * $$\psi_{\mathbf{k}}(\mathbf{r}) = \frac{1}{\sqrt{\Omega_r}} e^{i\mathbf{k}\cdot\mathbf{r}}$$

for free electrons with a wave vector $$\mathbf{k}$$ and a kinetic energy
 * $$E = \frac{\hbar^2 k^2}{2m} $$

in which $$\Omega_r$$ is the volume of space occupied by the electron.

The plane wave solution
The product of the time independent stationary wave solution and time dependent oscillator solution
 * $$\Psi(\mathbf{r},t) = \psi(\mathbf{r}) e^{-i \omega t} $$

give the traveling plane wave solution
 * $$\Psi(\mathbf{r},t) = \frac{1}{\sqrt{\Omega_r}} e^{i\mathbf{k}\cdot\mathbf{r} - i \omega t} $$

which is the solution for the free electron wave function.