Starting Point: One Electron Does Not Require One Definite Path
Once we accept the distinction between having one electron and requiring that electron to possess one definite pre-measurement position, the double-slit state becomes easier to interpret.
The one-electron path state may be written
\[|\psi\rangle=\frac{|A\rangle+|B\rangle}{\sqrt2}.\]
This does not represent two electrons or half an electron travelling through each slit. It is one quantum state assigning amplitudes to the path alternatives represented by \(|A\rangle\) and \(|B\rangle\).
What Is Relative Phase?
Relative phase can first be understood using ordinary waves.
If two waves repeatedly reach their peaks and troughs together, their relative phase may be \(0^\circ\). If one repeatedly reaches a peak when the other reaches a trough, their relative phase may be \(180^\circ\).
Both situations can be coherent.
The important feature is not that the waves must line up, but that their relative positioning remains stable and predictable.
Thus:
\[\boxed{\text{coherence}=\text{a definite and preserved relative phase relationship}}.\]
Coherence in the One-Electron State
For
\[|\psi\rangle=\alpha|A\rangle+\beta|B\rangle,\]
coherence means that the relative phase between the amplitudes \(\alpha\) and \(\beta\) remains available.
This phase relationship allows the amplitudes associated with A and B to interfere when they contribute to the same later measurement outcome.
For example, at some screen position \(X\), the two amplitudes may reinforce each other. At another position \(Y\), their phases may be opposite and they may cancel.
The electron still produces only one localized detection in an individual experiment. Coherence determines the probability distribution from which those individual detections accumulate.
Now Add the Environment
Suppose the electron interacts with another physical system, denoted by \(E\).
Initially:
\[\frac{|A\rangle+|B\rangle}{\sqrt2}|E_0\rangle.\]
After interaction, quantum evolution may produce
\[\boxed{|\Psi\rangle=\frac{|A\rangle|E_A\rangle+|B\rangle|E_B\rangle}{\sqrt2}}.\]
The A alternative has become correlated with environmental state \(|E_A\rangle\), while B has become correlated with \(|E_B\rangle\).
The electron and environment are now entangled.
Why Different Environmental States Matter
Consider a simplified example at a screen position \(X\).
Before environmental interaction, suppose the path amplitudes are
\[\psi_A(X)=0.5\]
and
\[\psi_B(X)=-0.5.\]
They cancel:
\[0.5-0.5=0.\]
After becoming correlated with the environment, the corresponding expression has the form
\[0.5|E_A\rangle-0.5|E_B\rangle.\]
If \(|E_A\rangle=|E_B\rangle=|E\rangle\), then
\[(0.5-0.5)|E\rangle=0.\]
Cancellation remains possible.
But if \(|E_A\rangle\) and \(|E_B\rangle\) are different vectors, they cannot simply be treated as the same quantity and cancelled.
Orthogonal Environmental States
The strongest case occurs when
\[\boxed{\langle E_A|E_B\rangle=0}.\]
This means the two environmental states are orthogonal: perpendicular in Hilbert space.
A simple two-dimensional analogy is
\[|E_A\rangle=\begin{bmatrix}1\\0\end{bmatrix},\qquad |E_B\rangle=\begin{bmatrix}0\\1\end{bmatrix}.\]
Their inner product is zero.
Orthogonal quantum states are perfectly distinguishable. Thus the environment now contains a complete physical distinction between the A and B alternatives.
What Decoherence Means
When the electron becomes entangled with sufficiently distinguishable environmental states, the electron by itself loses observable A/B interference.
This is decoherence.
A useful beginner-level statement is:
\[\boxed{\text{decoherence}=\text{loss of observable coherence in a subsystem due to entanglement with its surroundings}}.\]
Importantly, the complete state
\[\frac{|A\rangle|E_A\rangle+|B\rangle|E_B\rangle}{\sqrt2}\]
still contains both terms.
The quantum superposition has therefore not simply disappeared from the complete electron-plus-environment description.
The phase relationship has become encoded in correlations belonging to the larger combined system rather than remaining available as ordinary interference of the electron alone.
Decoherence Is Not Automatically Collapse
This creates an essential distinction.
Decoherence explains why interference between alternatives becomes inaccessible when a system becomes entangled with its environment.
But when a measurement is actually performed, we observe one definite result:
\[A\]
or
\[B.\]
Textbook quantum mechanics traditionally represents this as
\[\frac{|A\rangle+|B\rangle}{\sqrt2}\xrightarrow{\text{measurement}}|A\rangle\]
or
\[\frac{|A\rangle+|B\rangle}{\sqrt2}\xrightarrow{\text{measurement}}|B\rangle,\]
with probabilities determined by Born’s rule.
This update to one definite result is traditionally called wavefunction collapse.
The Measurement Problem Appears
The difficulty is that ordinary quantum evolution of the combined electron and detector does not obviously perform this selection.
Suppose the detector initially has state \(|D_0\rangle\), and its interaction satisfies
\[|A\rangle|D_0\rangle\rightarrow|A\rangle|D_A\rangle\]
and
\[|B\rangle|D_0\rangle\rightarrow|B\rangle|D_B\rangle.\]
Because ordinary quantum evolution is linear, a superposition evolves as
\[\frac{|A\rangle+|B\rangle}{\sqrt2}|D_0\rangle\]
\[\longrightarrow\]
\[\boxed{\frac{|A\rangle|D_A\rangle+|B\rangle|D_B\rangle}{\sqrt2}}.\]
The evolution preserves both branches. It does not, by itself in this description, randomly discard one and retain the other.
Yet an experiment produces a definite recorded outcome.
This creates the central question:
If ordinary quantum evolution leaves both alternatives in the combined state, why do measurements give us one definite outcome?
This is the core of the quantum measurement problem.
What Does Quantum Evolution Mean?
Quantum evolution simply means how a quantum state changes with time.
If a system begins at time zero in
\[|\psi(0)\rangle\]
and later has state
\[|\psi(t)\rangle,\]
then
\[|\psi(0)\rangle\rightarrow|\psi(t)\rangle\]
is its quantum time evolution.
In classical mechanics, Newton’s laws describe how quantities such as position and velocity evolve. In ordinary nonrelativistic quantum mechanics, the Schrödinger equation describes how an isolated system’s quantum state evolves with time.
Thus an interaction such as
\[|A\rangle|D_0\rangle\rightarrow|A\rangle|D_A\rangle\]
can be understood as part of the quantum evolution of the combined electron-detector system.
The Important Distinction to Keep
At this stage, three concepts should remain separate:
Coherence: alternatives in a quantum state retain a usable relative phase relationship and can interfere.
Decoherence: interaction and entanglement with other systems make that interference inaccessible when considering the subsystem alone.
Collapse: the traditional textbook rule that updates the state to the particular outcome obtained in a measurement.
The unresolved question of how, or whether, the last process should be understood as a physical mechanism is part of the measurement problem and leads to different interpretations of quantum mechanics.
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- Note type: learning-note
- Subject: learning-notes
- Source: Quantum Computing for Everyone