Probability Amplitude Calculator
Calculate quantum probability amplitudes, Born rule probabilities, phase angles, and normalization.
About this calculator
This calculator computes the Born-rule probability and geometric properties of a quantum probability amplitude α = Re + i·Im, treated as a single complex number rather than a full multi-term wavefunction. Probability |α|² is the sum of the squared real and imaginary parts -- the actual physically measurable probability the Born rule assigns to finding a quantum system in the state this amplitude describes -- and it is always non-negative regardless of the signs of Real Part and Imaginary Part. Magnitude |α| is simply the square root of that probability, i.e. the amplitude's length in the complex plane. Phase Angle is the argument of α -- the angle the vector (Re, Im) makes with the positive real axis -- and it plays no role in Probability or Magnitude at all: phase is invisible to a single measurement of this state, but it is exactly what produces interference when two or more amplitudes are added together before squaring, which this single-amplitude calculator does not model.
Normalized Re and Normalized Im rescale the amplitude to unit length (Re/|α|, Im/|α|) without changing its direction in the complex plane, useful when you want the direction alone rather than an arbitrary overall scale. Inner Product ⟨α|α⟩ is Probability again, restated in bra-ket notation, since taking the inner product of a state with itself always returns |α|². This calculator does not check or enforce normalization -- the Real Part and Imaginary Part you enter need not already satisfy |α|² = 1 -- nor does it model superposition of multiple basis states, so it cannot answer questions about interference or entanglement between two or more amplitudes.
Inputs
Results
Probability |α|²
1
Magnitude |α|
1
Phase Angle
53.13°
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 |
|---|---|---|---|
| 0.4 | 0.52 | 0.7211 | 33.69° |
| 0.6 | 0.72 | 0.8485 | 45° |
| 1.2 | 1.8 | 1.3416 | 63.43° |
| 2 | 4.36 | 2.0881 | 73.3° |
What each input means
- Real Part (Re)
- Real component of the amplitude
- Imaginary Part (Im)
- Imaginary component of the amplitude
How this is calculated
Formula
P = |α|² = Re² + Im²Worked example, using the default values
- Identify Input ParametersReal Part (Re) = 0.6, Imaginary Part (Im) = 0.8 = 2 input(s) provided
- Calculate Probability |α|²Probability |α|²1 = 1
- Calculate Magnitude |α|Magnitude |α|1 = 1
- Calculate Phase AnglePhase Angle53.13 = 53.13
- Calculate Normalized ReNormalized Re0.6 = 0.6
- Calculate Normalized ImNormalized Im0.8 = 0.8
Engine last updated . Checked against 4 independently-derived tests — how we verify calculators. Built by Paul Gunder, a software engineer, not a licensed financial, medical, or legal professional.
Frequently Asked Questions
What's the difference between Probability and Magnitude?
Magnitude |α| is the length of the amplitude vector -- an ordinary number that can run as high as roughly 14.1 given this calculator's ±10 input range on each component. Probability |α|² is Magnitude squared, and it's the number the Born rule uses to answer "how likely is this outcome" for a genuinely normalized single-amplitude state. Because squaring erases sign information, Probability is always non-negative even when Real Part or Imaginary Part is negative.
Does increasing Imaginary Part always increase Phase Angle?
No -- Phase Angle is the argument atan2(Im, Re), and its response to Imaginary Part depends on the sign of Real Part: with Real Part held positive, increasing Imaginary Part does increase Phase Angle, but with Real Part held negative the same increase in Imaginary Part decreases Phase Angle instead. The same conditional relationship holds in reverse for Real Part's effect on Phase Angle. There is no single "increasing this component always moves Phase Angle this way" rule across the full ±10 input range on each component -- only Probability and Magnitude, which depend on the squared values, behave the same way regardless of sign.
What do the Normalized Re and Normalized Im outputs represent?
They rescale the amplitude to unit magnitude by dividing Real Part and Imaginary Part by |α|, so Normalized Re squared plus Normalized Im squared equals 1 (aside from rounding) regardless of how large the original Real Part or Imaginary Part were. This is the same normalization step a physicist performs before treating a raw amplitude as a proper quantum state whose Probability output must sum to 1 across every possible outcome.
Does entering a large Real Part or Imaginary Part mean the state is unphysical?
Not by itself -- the calculator accepts any values from -10 to 10 on each component so you can explore the math freely, but a genuinely normalized single quantum state needs Magnitude no greater than 1 (equivalently, Probability no greater than 1). If your inputs produce a Magnitude above 1, the Normalized Re and Normalized Im outputs show the properly normalized version pointing in the same direction in the complex plane.
Can this calculator show quantum interference?
No -- interference comes from adding two or more amplitudes together before squaring the combined result, and the cross-terms that produce constructive or destructive interference only appear once multiple amplitudes are combined. This calculator evaluates a single amplitude α = Re + i·Im in isolation, so it reports that one term's own Probability and Phase Angle but cannot model how it would combine with a second amplitude.
Related Calculators
The questions that sit next to this one — chosen by subject, including calculators filed under a different category.
Quantum Measurement Calculator
Calculate quantum measurement probabilities, expectation values, variance, and wave function collapse.
Quantum PhysicsWave Function Calculator
Calculate quantum wave function properties including probability density, momentum, energy, and phase/group velocities.
Quantum PhysicsQuantum Tunneling Calculator
Calculate quantum tunneling probability, transmission coefficient, and penetration depth through a potential barrier.
More in Science & Physics.