Probability Amplitude Calculator

Calculate quantum probability amplitudes, Born rule probabilities, phase angles, and normalization.

Real Part (Re)

Imaginary Part (Im)

Tab to move between fields

Probability |α|²

Magnitude |α|

Phase Angle

53.13°

Normalized Re0.6

Normalized Im0.8

Inner Product ⟨α|α⟩1

How to Use This Calculator

Enter the Real Part (Re) and Imaginary Part (Im) of the quantum probability amplitude α.
The Probability |α|² is the Born rule result — the actual measurable probability of finding the system in this state.
Magnitude |α| must be ≤ 1 for a normalized amplitude; if it exceeds 1, renormalize the state vector first.
Phase Angle (°) represents the relative phase, which matters for quantum interference but not for individual measurements.
Use Normalized Re and Im outputs when you need the unit-normalized components for superposition calculations.

How the result changes with Imaginary Part (Im)

Imaginary Part (Im)	Probability \|α\|²	Magnitude \|α\|	Phase Angle
-8	64.36	8.0225	-85.71°
-3	9.36	3.0594	-78.69°
3	9.36	3.0594	78.69°
8	64.36	8.0225	85.71°

P = |α|² = Re² + Im²

Identify Input Parameters
Real Part (Re) = 0.6, Imaginary Part (Im) = 0.8 = 2 input(s) provided
Calculate Probability |α|²
Probability |α|²
1 = 1
Calculate Magnitude |α|
Magnitude |α|
1 = 1
Calculate Phase Angle
Phase Angle
53.13 = 53.13
Calculate Normalized Re
Normalized Re
0.6 = 0.6
Calculate Normalized Im
Normalized Im
0.8 = 0.8

Engine last updated August 5, 2026. Checked against 4 independently-derived tests — how we verify calculators.

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