For a one-qubit system, a general pure state $∣ψ⟩$ can be written as a linear combination of $∣0⟩$ and $∣1⟩$:

where $0≤ϕ<2π$ and $0≤θ<π$. When interpreted as the polar and azimuthal angles in spherical polar coordinates, the angles $ϕ$ and $θ$ uniquely identify a point on a unit sphere called the Bloch sphere. With the one-to-one correspondence between the points on the Bloch sphere and the pure state space of a one-qubit system, the Bloch sphere can be used to visualize any pure state of a one-qubit system.

Bloch Sphere provides an interactive way to visualize the pure one-qubit states on the Bloch sphere by manipulating the $ϕ$ and $θ$ angles as well as pseudo-rotating the sphere by clicking/touching and dragging.

See also

On the nature of the wave function

The wave function is an abstract mathematical concept and cannot be "measured" directly. So what is it then?

Dirac notation for quantum states

How to read the bra–ket notation?

The Hadamard gate

The definition of the Hadamard gate and some of its properties.