Established physics · Quantum Mechanics 101 · Lesson 3 of 15

Interference and physical path records

For coherent alternatives ending in the same outcome, add amplitudes first and take the squared magnitude second.

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Picture it

Send one photon at a time through a balanced two-route interferometer. Mirrors bring the route alternatives together at a final beam splitter, with detectors A and B beyond it. A phase adjustment in one route changes how often each detector clicks. Each run still gives one event. The apparatus lets you change the distribution by changing the relationship between alternatives.

Why amplitudes must be added

Ordinary positive probabilities cannot cancel. Complex amplitudes can. When indistinguishable alternatives lead to the same event, their contributions combine before the probability is calculated. At a dark output the combined amplitude can vanish even though each alternative contributes a nonzero amplitude to that output. This is the physical job of phase, not an extra bookkeeping decoration.

A local cancellation is not lost probability

If one output becomes dark, probability is redistributed across the complete set of outputs. In the ideal lossless two-output apparatus, A and B always add to 100%. Real apparatus can have additional loss channels. An analysis of a real experiment must account for those outcomes too; it cannot assume that missing detections are automatically interference cancellation.

Which-route information changes the joint state

Let another system acquire a distinguishable record of the route. The photon alternatives now correlate with different record states. If those records are fully distinguishable and ignored, interference between the route alternatives disappears in the photon’s output statistics. Nobody must inspect the record. The important change is a physical correlation, not human attention.

Interferometers and double slits share a rule

A double-slit screen has many possible detection regions, with geometry and diffraction shaping the distribution. Our two-output interferometer is simpler: it concentrates the comparison into two output probabilities. Both use coherent amplitude addition. The drawing below is a map of apparatus alternatives; it is not a claim that a particular classical trajectory has been measured along each drawn line.

The order of the calculation
  1. Identify coherent alternatives
  2. Add their amplitudes
  3. Square the combined magnitude

A distinguishable record changes whether the alternatives contribute coherently to the reduced output statistics.

Try the example on paper

Phase 0°: A = 100%, B = 0%. Phase 90°: both 50%. Phase 180°: A = 0%, B = 100%. With a fully distinguishable ignored path record, both outputs are 50% at every phase.

Make it concrete

At a chosen outcome, contributions +0.3 and −0.3 cancel to zero amplitude. Their separate squared magnitudes are 0.09 and 0.09, but adding those probabilities would give the wrong coherent result. For a continuous screen these are density contributions in consistent units. The full distribution, integrated over all outputs, remains normalized.

The precise version

For contributions a₁ and a₂, P = |a₁ + a₂|² = |a₁|² + |a₂|² + 2 Re(a₁* a₂). In the balanced, lossless interferometer, P(A) = (1 + cos φ)/2 and P(B) = (1 − cos φ)/2. A fully distinguishable path record removes the cross term in this example, yielding 1/2 at each output.

Watch the trap

Do not explain a dark output by saying that two negative probabilities cancel. Probabilities stay nonnegative. It is the amplitudes that can have opposite phases.

Keep this

Coherence determines whether alternatives interfere. Physical route records can remove that interference even when unread.

Words for this lesson: interference · amplitude · phase · coherence · measurement · decoherence

Check your understanding

Reason through each question before opening its answer and explanation.

1. What must be combined before calculating a probability for indistinguishable routes?

  1. The number of observers
  2. The amplitudes
  3. The already-squared probabilities
Show answer and explanation

The amplitudes

Amplitudes carry phase. Adding them before squaring preserves interference.

2. Two contributions to one outcome are +0.3 and −0.3. What is their combined probability density?

  1. 0.18
  2. 0
  3. 0.6
Show answer and explanation

0

The amplitudes cancel: |0.3 − 0.3|² = 0.

3. Which change can remove interference even if nobody reads a detector?

  1. Closing the laboratory door without changing any relevant interaction
  2. Giving the photon a different label
  3. A physical interaction records which route
Show answer and explanation

A physical interaction records which route

A distinguishable path record changes the physical correlations; consciousness is unnecessary.

4. In the ideal apparatus, phase 90° gives which outputs?

  1. A = 0%, B = 0%
  2. A = 100%, B = 100%
  3. A = 50%, B = 50%
Show answer and explanation

A = 50%, B = 50%

Cosine of 90° is zero, so both normalized output probabilities are one half.

5. If destructive interference makes A dark, what happens to the total probability?

  1. It becomes negative
  2. It disappears from the theory
  3. It is redistributed among the other complete outcomes
Show answer and explanation

It is redistributed among the other complete outcomes

Normalization applies to the full outcome set. Local cancellation does not erase total probability.

6. What does this two-output demonstration calculate?

  1. Probabilities of an ideal balanced interferometer
  2. Measured trajectories through both routes
  3. The complete diffraction envelope of a real double-slit screen
Show answer and explanation

Probabilities of an ideal balanced interferometer

It illustrates the common amplitude rule using two output channels, with its apparatus assumptions stated explicitly.

Further reading