Quantum amplitudes and interference

For a discrete outcome $j$, let $c_j$ be its complex amplitude in a normalized state. The Born rule assigns probability $|c_j|^2=c_j^*c_j$. The star means complex conjugation, so the answer is real and nonnegative even when the amplitude contains $i$. Amplitudes can cancel; probabilities cannot be negative. Normalization requires the probabilities of a complete set of mutually exclusive outcomes to sum to one.

For a particle on a continuous line, the wavefunction $\psi(x)$ supplies an amplitude density instead. Its squared magnitude has units of inverse length. A detector covering an interval measures an integral of that density, not its value at a mathematical point. A density can exceed one without violating probability rules: the integral over any region remains between zero and one. Always identify whether your outcome labels are discrete or continuous before interpreting a graph.

Suppose two paths reach the same detector and the experimental arrangement preserves their coherence. Add their amplitudes before taking the squared magnitude. The resulting cross term depends on relative phase. Opening a second path can therefore reduce the detection rate at a particular output, even though it provides another alternative. Probability conservation concerns all outputs together, so reduced counts in one port appear elsewhere in an ideal lossless interferometer.

This rule needs its conditions. If the paths leave reliably distinguishable records in an environment or a path detector, the reduced description of the output generally loses the interference term. Merely choosing not to read the record does not restore coherence. Conversely, not knowing a classical coin result is not automatically a coherent superposition. Preparation, coupling, and distinguishability determine which calculation is appropriate; a person's awareness is not a variable in the Born rule.

Multiplying every amplitude by the same phase $e^{i\alpha}$ leaves every Born probability unchanged. That overall phase labels the same physical pure state. Changing one component relative to another is different: a later measurement can recombine the components and reveal the change. A phase is therefore not a hidden probability. It is information about how components will combine in an appropriate experimental comparison.

Our ideal two-port example chooses output amplitudes $A_0=(1+e^{i\phi})/2$ and $A_1=(1-e^{i\phi})/2$. Beam-splitter phase conventions can change these formulas' appearance, but not their predictions when used consistently. They give $P_0=\cos^2(\phi/2)$ and $P_1=\sin^2(\phi/2)$. These are a controlled two-mode model, not a simulation of every optical component or of a relativistic particle field.