What the two slit experiment tells us about decoherence

---

What the two slit experiment tells us about decoherence

1. The set-up: what we actually observe

In the double-slit experiment we send electrons (or photons, or atoms) one at a time towards a screen. Each run gives a single, localised dot on the screen. Nothing like an “interference pattern” is visible in any single event. The pattern is an ensemble fact: if we repeat the experiment many times with identically prepared electrons, the histogram of impacts converges to a stable probability distribution, and that distribution shows fringes.

This is worth stating bluntly because it stops us from talking nonsense about “seeing the wavefunction”. We never do. We infer the correct quantum description from stable statistics across many runs.

2. The state-vector description without which-path information

Let the electron’s relevant alternatives after the slits be the two emerging wavepackets, one associated with the left slit and one with the right. Call these states |L> and |R>. In the ideal symmetric case, immediately after the slits the electron is in the superposition

|ψ> = (1/√2)( |L> + |R> )

Let |x> represent a position eigenstate on the detection screen. Define the two complex amplitudes

ψ_L(x) := <x|L>,   ψ_R(x) := <x|R>

The amplitude to arrive at position x is then

<x|ψ> = (1/√2)( ψ_L(x) + ψ_R(x) )

So the probability density on the screen is

P(x) = |<x|ψ>|² = (1/2)|ψ_L(x) + ψ_R(x)|²

Expand it out:

P(x) = (1/2)( |ψ_L(x)|² + |ψ_R(x)|² + ψ_L(x)ψ_R^*(x) + ψ_L^*(x)ψ_R(x) )

The last two terms are the interference (cross) terms. They carry the relative phase information between the left and right alternatives. With those terms present, we get fringes.

3. Add the simplest possible which-path detector: one bit of memory

Now introduce the simplest imaginable which-path detector. Model it as a two-state system (call it a “bit” if you like) with orthonormal basis states |0> and |1>. Assume it starts in |0>. The coupling at the slits is defined as follows:

If the electron takes the left slit, flip the bit. If it takes the right slit, leave it alone.

In symbols, the interaction implements the correlations

|L>|0> → |L>|1>    and    |R>|0> → |R>|0>

This is entirely unitary: a controlled operation. Now apply linearity to the incoming superposition. The combined system (electron + detector) evolves as

(1/√2)( |L> + |R> )|0> → |Ψ> = (1/√2)( |L>|1> + |R>|0> )

This is the whole mechanism of decoherence in miniature. There is no mysterious “collapse” here. The total state |Ψ> is a perfectly good, pure state. But the electron by itself is no longer in a pure superposition. It is entangled with something that stores which-path information.

4. The exact calculation: how the interference term disappears

We now compute the probability P(x) of finding the electron at position x on the screen without conditioning on the detector. That “without conditioning” clause is crucial. In a normal experiment we do not read out every microscopic environmental degree of freedom. We just look at the screen.

First compute the environment-valued amplitude:

<x|Ψ> = (1/√2)( <x|L>|1> + <x|R>|0> )

So

<x|Ψ> = (1/√2)( ψ_L(x)|1> + ψ_R(x)|0> )

This is not a single complex number. It is a vector in the detector’s two-dimensional Hilbert space. The probability of a hit at x, ignoring the detector outcome, is the squared norm of that vector:

P(x) = ||<x|Ψ>||² = <Ψ|x><x|Ψ>

Now expand, using orthonormality

  ( <1|1> = <0|0> = 1 and <1|0> = <0|1> = 0 ):

P(x) = (1/2)( |ψ_L(x)|²<1|1> + |ψ_R(x)|²<0|0> + ψ_L(x)ψ_R^*(x)<1|0> + ψ_L^*(x)ψ_R(x)<0|1> )

But the cross terms vanish because the detector states are orthogonal:

<1|0> = <0|1> = 0

Therefore

P(x) = (1/2)( |ψ_L(x)|² + |ψ_R(x)|² )

That is the incoherent sum of the two single-slit contributions. No fringes.

The key point is almost embarrassingly simple once you see it: the two “paths” no longer contribute amplitudes that add as complex numbers. They contribute orthogonal vectors in the detector/environment space. Orthogonal vectors do not interfere; their norm squares add.

5. Partial which-path information: fringes fade rather than vanish

A real detector need not be perfectly sharp. Suppose the detector ends up in two (possibly non-orthogonal) states |d_L> and |d_R>, with overlap

γ := <d_R|d_L>

The joint state is

|Ψ> = (1/√2)( |L>|d_L> + |R>|d_R> )

Run the same calculation and the probability becomes

P(x) = (1/2)( |ψ_L(x)|² + |ψ_R(x)|² + 2 Re{ γ ψ_L(x) ψ_R^*(x) } )

The interference term is suppressed by γ. If γ ≈ 1 (detector states essentially identical) you recover full interference. If γ ≈ 0 (detector states orthogonal) the fringes disappear. This is the precise mathematical meaning of the informal claim that decoherence “washes out” interference.

6. What this does and does not explain

This little model lays bare what decoherence actually buys you:

• It shows, in purely unitary quantum mechanics, how creating a record of which path was taken eliminates interference in the local statistics on the screen.
• It explains why a macroscopic environment is so effective: overlaps like γ are driven rapidly towards zero when records are amplified into many degrees of freedom.

But it also shows what decoherence does not do by itself. It does not tell you why, in one particular run, the electron hit this pixel rather than the adjacent one. Decoherence explains the emergence of classical-looking probability distributions; it does not, on its own, settle the “single outcome” question. That is a separate interpretive problem, and it’s best kept separate if you want conceptual hygiene.

7. The minimal takeaway

The double-slit experiment doesn’t merely illustrate “wave–particle duality” in the popular-science sense. It gives you the cleanest possible laboratory for the central idea of decoherence:

Interference requires not just a superposition in the system, but the absence of distinguishable records in the environment.

Once a which-path record exists—even a single bit—the cross terms that generate fringes are multiplied by an environment overlap that becomes (effectively) zero. The interference pattern dies at the level of ensemble statistics, even though the total state vector of system plus environment remains perfectly coherent.

---