FORMULON.
Quantum Laboratory · 01

Quantum State
Explorer

Move through the mathematics of a single qubit. Change its amplitudes, normalize the state, and watch its probability and Bloch-sphere representation update in real time.

“The laws of quantum mechanics themselves cannot be formulated ... without resort to the mathematical language.”

State controls

|ψ⟩ = α|0⟩ + β|1⟩
0.866
0.500

Quick states

Normalization
|α|² + |β|² = 1.000

State used by the explorer
|ψ⟩ = 0.866|0⟩ + 0.500|1⟩
P(0)75.0%
P(1)25.0%
Bloch X0.866
Bloch Z0.500
Polar angle θ60.0°
Azimuth φ0.0°

What are you seeing?

For this real-amplitude explorer, the normalized state can be written as |ψ⟩ = cos(θ/2)|0⟩ + sin(θ/2)|1⟩. The Bloch vector is (sin θ, 0, cos θ). The two measurement probabilities are P(0)=|α|² and P(1)=|β|². Try |+⟩ and |−⟩: both have equal measurement probabilities, but they point in opposite directions on the Bloch sphere.