## Wavefunctions of the crystal - Bloch Theorem

The Hamiltonian of a semiconductor crystal has the translation symmetry

with being a reciprocal lattice vector.

The Bloch theorem states that the wavefunctions have two “good” quantum numbers, the band index n and a reciprocal vector such that the wavefunctions of the crystal may be written as:

where exhibits the periodicity of the crystal.

It may also be written as:

**Proof of Bloch’s theorem**

Translation operator

If **R** belongs to the Bravais lattice, the operator will commute with the Hamiltionian

and therefore

These two operators form a common set of commuting observable, we can therefore write the wavefunctions using eigenfunctions of both:

Consid´erons maintenant les propri´et´es de c(R)

et donc

To simplify somewhat the notation, a_{i} with i = 1, 2, 3 the basis vectors of the primitive Bravais lattice, and b_{i} the basis vector of the reciprocal lattice. We can always define (because c(a_{i}) is always normalized to unity:

If now **R** is a translation of the Bravais lattice, it can be written as

then,

Using the orthogonality relations between the direct and reciprocal basis vectors, written as

Considering the product between the vector **R** of the real space and the vector k of the

reciprocal space, written in the basis of the b_{i}:

and then

It was in fact the result we had obtained for c(**R**), that was written as:

setting naturally x_{i} = k_{i}. The wavefunctions may well be written as:

that is one of the equivalent formulations of the Bloch theorem.

## Squirrel cage motors

Modern electrical motors are available in many different forms, such as single phase motors, three-phase motors, brake motors, synchronous motors, asynchronous motors, special customised motors, two speed motors, three speed motors, and so on, all with their own performance and characteristics.

## AC Motor control - Frequency converter

A complete installation with a frequency converter controlled motor consists of a series of different components which should all be selected carefully for a given application.The components in an installation are selected according to the actual application, starting with selecting the right pump for the application.

## Faraday’s Law of Electromagnetic Induction

We have seen previously that when a DC current pass through a long straight conductor a magnetising force, H and a static magnetic field, B is developed around the wire. If the wire is then wound into a coil, the magnetic field is greatly intensified producing a static magnetic field around itself forming the shape of a bar magnet giving a distinct North and South pole.

## Electric motors make things go

Electric motors make things go. Better motors mean better go power. What constitutes “better” and what to do about it, is the subject of a great deal of investigation. Better motors generally mean 2 things, higher efficiency and greater torque produced for a given amount of input power.

## Direction of rotation and markings of terminals Three-phase motors

The windings are connected in star(Y)-connection or in delta(Δ)-connection according to IEC 60034-8. This is done by wiring the terminal board as shown in the wiring diagram to the right. The marking of the terminal board is also defined in IEC 60034-8.

Read more: Direction of rotation and markings of terminals Three-phase motors

## Resistance

An electron traveling through the wires and loads of the external circuit encounters resistance. Resistance is the hindrance to the flow of charge. For an electron, the journey from terminal to terminal is not a direct route. Rather, it is a zigzag path that results from countless collisions with fixed atoms within the conducting material.