Visualizing the state vector of a qubit

Another visual representation of a qubit and its states is the Bloch sphere, named after Felix Bloch. The Bloch sphere is a three-dimensional ordinary sphere that's generally used as a geometrical representation of the qubit. By this, we mean the sphere can represent the qubit states as a point anywhere on the surface of the Bloch sphere.

Conventionally, the north pole of the Bloch sphere represents the state, while the south pole represents the state. Any point on the surface of the Bloch sphere can represent the linear combination of states as a unit vector from the center (origin), as we described previously, to the surface of the Bloch sphere.

Since we have the quantum mechanical constraint that the total probability of the vector must equal to 1, we get the following formula:

The vector can then only rotate along the Bloch sphere by using the following representation:

Here...