Quantum Physics Foundations: Probability Amplitudes and Interference of Alternatives

1) Probability vs. Probability Amplitude (Plain Language)

In everyday reasoning, you assign a probability to an outcome: a number between 0 and 1 that tells you how often you expect it to happen in repeated trials. If there are multiple ways to get the same outcome and those ways are independent “alternatives,” you often add probabilities.

Quantum predictions use a different intermediate quantity: a probability amplitude. You can think of an amplitude as a “pre-probability” number that can be positive, negative, or more generally have a direction-like aspect (phase). The key rule is:

• Add amplitudes for indistinguishable alternatives.
• Convert to probability only at the end: probability = (magnitude of total amplitude)².

So the workflow is:

1. List the alternatives that could lead to the same outcome.
2. Decide whether those alternatives are indistinguishable (no information exists that could tell them apart, even in principle within the setup).
3. If indistinguishable: add their amplitudes to get a total amplitude.
4. Square the magnitude of the total amplitude to get the probability.

In symbols (kept minimal):

Indistinguishable alternatives A and B leading to outcome O:  a_total(O) = a_A(O) + a_B(O)  Probability:  P(O) = |a_total(O)|^2

Contrast with the “ordinary probability addition” rule (used when alternatives are distinguishable):

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Distinguishable alternatives A and B leading to outcome O:  P(O) = P_A(O) + P_B(O)

The entire phenomenon of quantum interference is the gap between these two rules.

2) Phase Intuition: Why Amplitudes Can Cancel

To understand interference without heavy math, treat an amplitude like an arrow (a “phasor”) that has:

• Length: how strong that alternative contributes.
• Direction: its phase, which controls whether it reinforces or cancels another contribution.

When you add amplitudes, you add these arrows head-to-tail. The resulting arrow is the total amplitude. The probability is proportional to the square of the total arrow’s length.

Phasor-style pictures (text diagrams)

Constructive interference (same direction):

a1:  ---->      a2:  ---->      sum:  -------->  (bigger length)  => higher probability

Destructive interference (opposite directions):

a1:  ---->      a2:  <----      sum:  (almost zero)           => low probability

Partial interference (angled):

a1:  ---->      a2:    />       sum:  (somewhere between)     => intermediate probability

Phase is what rotates the arrow. If two alternatives arrive “in step” at a detector position, their arrows align and the probability increases. If they arrive “out of step,” arrows oppose and the probability decreases.

A minimal numeric example (no complex numbers required)

Use a simplified model where amplitudes are just real numbers that can be positive or negative (this captures the idea of phase as “same” vs “opposite”):

• If alternative 1 gives amplitude +1 and alternative 2 gives amplitude +1, total amplitude is +2, probability is (2)^2 = 4 (large).
• If alternative 1 gives amplitude +1 and alternative 2 gives amplitude -1, total amplitude is 0, probability is 0^2 = 0 (cancellation).

In real quantum systems, phase is continuous (not just “+” or “−”), so you can smoothly move between reinforcement and cancellation.

3) Two-Path Rule: Why Blocking or Marking a Path Changes the Distribution

Consider a setup with two possible paths from a source to a detection screen (call them path 1 and path 2). We focus on a particular point x on the screen.

Step-by-step prediction when paths are indistinguishable

1. Assign an amplitude from each path to point x: a1(x) and a2(x).
2. Add them: a_total(x) = a1(x) + a2(x).
3. Convert to probability: P(x) = |a_total(x)|^2.

Because a1(x) and a2(x) can have different phases depending on x, the sum can be large at some positions (bright fringes) and small at others (dark fringes). The distribution across the screen is shaped by interference.

What changes when you block a path?

If you block path 2, then a2(x)=0 for all x. The rule becomes:

a_total(x) = a1(x)  =>  P(x) = |a1(x)|^2

Interference disappears because there is nothing to add. Practically, the pattern becomes the “single-path” distribution (often broader and without alternating bright/dark structure).

What changes when you “mark” a path (which-path information)?

Marking means the setup leaves some record correlated with the path (for example, a detector click, a polarization tag, a timing tag, or any physical trace in the environment). The crucial point is not whether a human reads the record, but whether the alternatives become distinguishable in principle within the physical setup.

When paths are distinguishable, you do not add amplitudes across those alternatives for the same final outcome on the screen. Instead, you add probabilities:

P(x) = P1(x) + P2(x) = |a1(x)|^2 + |a2(x)|^2

Notice what is missing: the cross-term that creates interference. If you expand the indistinguishable case, you get:

|a1 + a2|^2 = |a1|^2 + |a2|^2 + (interference term)

Marking removes the interference term because the alternatives are no longer combined as amplitudes for the same indistinguishable event.

A concrete compare-by-reasoning example

Suppose at a particular screen point x0 the two path contributions are equal in strength but opposite in phase (perfect cancellation when unmarked):

• Unmarked: a1(x0)=+1, a2(x0)=-1 ⇒ a_total=0 ⇒ P=0 (a dark fringe).
• Marked (distinguishable): P=|+1|^2 + |-1|^2 = 1 + 1 = 2 (not dark anymore).

This is why “which-path” marking can fill in dark fringes: it changes the rule from amplitude addition to probability addition.

4) Normalization: “Total Probability Equals 1”

Any prediction must respect that the particle is detected somewhere (within the set of outcomes you are considering). That requirement is normalization:

• For discrete outcomes (e.g., detectors A, B, C): P(A)+P(B)+P(C)=1.
• For a continuous screen position x: the probabilities across the screen add up to 1 when integrated over all positions you count as “detection.”

Normalization is not an extra physical effect; it is a consistency condition that constrains allowable amplitudes.

Practical normalization in a two-detector toy model

Imagine two detectors, Left (L) and Right (R). You compute total amplitudes:

a(L) = a1(L) + a2(L)   and   a(R) = a1(R) + a2(R)

Then probabilities:

P(L) = |a(L)|^2   and   P(R) = |a(R)|^2

Normalization demands:

P(L) + P(R) = 1

If your raw amplitude assignments produce P(L)+P(R) ≠ 1, that means your amplitudes are not scaled consistently (or you omitted outcomes). In practice, you can enforce normalization by rescaling amplitudes by a common factor so that the final probabilities sum to 1.

Example: rescaling to satisfy total probability 1

Suppose your reasoning gives:

• a(L)=2 ⇒ P(L)=4
• a(R)=1 ⇒ P(R)=1

Total is 5, not 1. A simple fix is to scale both amplitudes by 1/√5:

• a'(L)=2/√5 ⇒ P'(L)=4/5
• a'(R)=1/√5 ⇒ P'(R)=1/5

Now P'(L)+P'(R)=1. This illustrates how normalization constrains predictions: you can’t arbitrarily increase all amplitudes without changing the total probability budget.

5) Mini-Labs: Compute-by-Reasoning Exercises (Qualitative Fringe Visibility)

These exercises are designed so you can answer by applying the rule “add amplitudes if indistinguishable; add probabilities if distinguishable,” plus phase intuition. No detailed calculations are required.

Mini-Lab A: Blocking one path

Setup: Two-path interference pattern is visible on a screen.

Change: Block path 2 completely.

Predict:

• Fringes: disappear (no alternating bright/dark from two-path interference).
• Overall spread: becomes the single-path distribution from path 1.
• Brightness: total detection rate may drop (fewer particles reach the screen), but the key qualitative change is loss of fringe structure.

Mini-Lab B: Perfect which-path marking

Setup: Two paths open; initially unmarked, fringes present.

Change: Add a perfect marker that makes path 1 and path 2 distinguishable (even if you do not read it).

Predict:

• Fringe visibility: goes to zero (pattern becomes the sum of two single-path intensity profiles).
• Dark fringes: fill in (positions that were near-zero due to cancellation become nonzero).

Mini-Lab C: Partial marking (imperfect which-path information)

Setup: Two paths open; you add a weak marker that sometimes reveals the path, but not always (or leaves only a faint trace).

Reasoning guide: The more distinguishable the alternatives become, the less you are allowed to combine them as a single indistinguishable amplitude contribution.

Predict:

• Fringe visibility: decreases but does not necessarily vanish.
• Bright fringes: become less bright; dark fringes: become less dark.
• In the limit of zero marking: full visibility returns. In the limit of perfect marking: visibility goes to zero.

Mini-Lab D: Phase shift in one path

Setup: Two paths open and unmarked; fringes present.

Change: Insert a device in path 2 that changes its phase (think “rotates the phasor arrow” for that path) without revealing which path was taken.

Predict:

• Fringe visibility: stays high (paths remain indistinguishable).
• Fringe positions: shift across the screen (places that were constructive can become destructive and vice versa).

How to reason: You are still adding amplitudes, but the relative phase between a1(x) and a2(x) changes, so the locations of alignment/cancellation move.

Mini-Lab E: Unequal path strengths

Setup: Two paths open and unmarked.

Change: Attenuate path 2 so its amplitude is smaller in magnitude than path 1 (but still indistinguishable).

Predict:

• Fringe visibility: decreases (cancellation is less complete when one arrow is shorter).
• Dark fringes: no longer reach zero (because a small arrow cannot fully cancel a much larger one).
• Average brightness: shifts toward what path 1 alone would produce, but with residual modulation from interference.

Mini-Lab F: “Mark then erase” (conceptual)

Setup: You mark the paths (destroying interference), but later you apply a process that removes the distinguishing record so that, for a selected subset of events, the alternatives become indistinguishable again.

Predict:

• Without erasing: no fringes (probabilities add).
• With erasing (for the selected subset): fringes can reappear in that subset (amplitudes add again), even though the full unfiltered dataset may still show no fringes.

Reasoning: The rule depends on whether the alternatives are distinguishable in the relevant description of the outcome. If you condition on information that makes the alternatives indistinguishable again, you restore amplitude addition for that conditioned set.

Quick self-check table: which rule applies?