=\frac

\frac

+V(\hat x) </math>

where,

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

R represents the regular spacing between atoms.

What is the definition of an "inverse lattice"?

An inverse lattice is also known as a 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.

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

\le k \le \frac

</math>

This range is called the first Brillouin zone (BZ). It is the Wigner-Seitz 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.

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

The form of the energy eigenfunction is:

<math>u_

=e^

f</math>

where f 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.

What is the Hamiltonian for a periodic system?

<math> \hat H (\hat x, \hat p)=i\hbar\frac

+V(\hat x) = \frac

=\frac {-\hbar^2}

+V(\hat x) </math>

where,

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

R represents the regular spacing between atoms.

If we plug in our eigenfunction <math>u_

=e^

f</math> into the 1-D Hamiltonian:

<math>[\frac {-\hbar^2}

\frac

f=Ee^

</math> for f --> <math>f_