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h1.  How do we mathematically represent a crystal? 

We use the Hamiltonian for periodic potentials:

<math> \hat H (\hat x, \hat p)=i\hbar\frac{\partial }{ \partial t} =\frac{\hat P ^2}{2m}+V(\hat x) =  \frac{\hat P ^2}{2m}=\frac {-\hbar^2}{2m} \frac{ \partial^2 }{ \partial x^2} +V(\hat x) </math>

where,

<math> V(\vec r + \vec R)=V(\vec r) \rightarrow V(x+na)=V(x) </math>


R represents the regular spacing between atoms.

h1.  What is the definition of an "inverse lattice"? 

An inverse lattice is also known as a [reciprocal lattice|http://en.wikipedia.org/wiki/Reciprocal_lattice].

Specifically, for our class, reciprocal vectors form a lattice with reciprocal length dimensions. I.e., large separations between lattice points in a direct lattice lead to small separations in the reciprocal lattice.

h2. Brillouin Zone 

Any point k in the reciprocal space (not necessarily a lattice point) can be expressed in 1-D as:

<math> k'=k+m\frac{2\Pi}{a}=k+G </math>

where,

<math>-\frac{\Pi}{a} \le k \le \frac{\Pi}{a}  </math>

This range is called the first [Brillouin zone|http://en.wikipedia.org/wiki/Brillouin_Zone] (BZ). It is the [Wigner-Seitz cell|http://en.wikipedia.org/wiki/Wigner-Seitz_unit_cell] (primitive unit cell comprising a point and its nearest neighbors) of the reciprocal lattice.

This concept also extends to 2-D and 3-D cases.

h2. Properties of k

k is a number (or vector) that has units of inverse length and is related to the eigenvalues of the discrete translational operator:

<math>\hat T_a\Psi(x)=\Psi(x+a)</math>


The form of the energy eigenfunction is:

<math>u_{k+G}(x)=e^{ikx}f(x)</math>

where f(x) is a function that is periodic in the in the lattice period a.


When labeling states, it is enough to consider only the k's in the first BZ, since only those k's correspond to the distinct eigenvalues of the discrete translational operator.

h1.  What is the Hamiltonian for a periodic system? 

<math> \hat H (\hat x, \hat p)=i\hbar\frac{\partial }{ \partial t} =\frac{\hat P ^2}{2m}+V(\hat x) =  \frac{\hat P ^2}{2m}=\frac {-\hbar^2}{2m} \frac{ \partial^2 }{ \partial x^2} +V(\hat x) </math>

where,

<math> V(\vec r + \vec R)=V(\vec r) \rightarrow V(x+na)=V(x) </math>


R represents the regular spacing between atoms.


If we plug in our eigenfunction <math>u_{k+G}(x)=e^{ikx}f(x)</math> into the 1-D Hamiltonian:

<math>[\frac {-\hbar^2}{2m} \frac{ \partial^2 }{ \partial x^2} +V(\hat x)]e^{ikx}f(x)=Ee^{ikx}f(x)</math>

we get a family of energy solutions n for each k. I.e., E --><math> E_{n,k}</math> for f --> <math>f_{n,k}(x)</math>.


And since f(x) is periodic in a (our lattice period), we need only solve for the first period subject to the periodic boundary conditions.

h1.  What is the definition and eigenvalues/eigenfunctions of a system with a periodic potential? 

h2. Bloch wave

A [Bloch wave|http://en.wikipedia.org/wiki/Bloch_wave] is...

h1.  What is the conserved number associated with the discrete translational symmetry in a periodic system? 
h1.  How do we label the states in periodic systems? 
h1.  What is the form of a Fourier series in terms of the inverse lattice vector G? 
h1.  How is a periodic function represented graphically on the k axis? 
h1.  How is an arbitrary function expressed as a linear combination of plane waves and how is this description different from a Fourier series? 
h1.  How do we expand a complex function in a complex Fourier series? 
h1.  How do we generate the series of equations that link the coefficients separated by inverse lattice vectors? 
h1.  What is the resulting eigenvalues problem and how are its solution related to the eigenfunction of the periodic Hamiltonian?