The Central Confusion: There Is Only One Electron
While trying to understand coherence in the double-slit experiment, an important objection appears repeatedly:
If there is only one electron, then surely it must pass through either A or B. How can there be two phases or two path amplitudes?
This question reveals an assumption inherited from classical physics:
\[\boxed{\text{one particle}\Rightarrow\text{one definite position and one definite trajectory at every moment}}\]
If that assumption is imposed from the beginning, then the idea of coherent A and B alternatives does indeed appear impossible.
Quantum mechanics, however, does not generally make that assumption.
One Electron Is Not the Same Statement as One Definite Position
There is still exactly one electron.
If the electron eventually hits a detection screen, one localized detection is observed. It does not produce half an electron at one location and half at another.
But this does not imply that before the position measurement the electron must already have possessed one definite position represented in the quantum state.
Quantum mechanics distinguishes:
\[\boxed{\text{number of particles}}\]
from
\[\boxed{\text{whether a particular observable such as position has a definite value}}.\]
There can therefore be one electron while its quantum state does not correspond to one definite position.
The Connection With an Ordinary Qubit
This is closely related to the familiar qubit expression
\[|\psi\rangle=\alpha|0\rangle+\beta|1\rangle.\]
The presence of both \(|0\rangle\) and \(|1\rangle\) in the state does not mean there are two qubits.
There is one quantum system described by one state.
The coefficients \(\alpha\) and \(\beta\) are amplitudes associated with the possible basis-state measurement outcomes.
Similarly, for the path degree of freedom of one electron we can write
\[|\psi\rangle=\alpha|A\rangle+\beta|B\rangle.\]
This does not mean there are two electrons or half an electron in each path.
There is one electron described by one quantum state.
Amplitude Is Not Probability
It is tempting to replace the amplitudes by ordinary probabilities and say that the electron simply has a certain probability of having secretly taken A and another probability of having secretly taken B.
But amplitudes contain more information than probabilities.
For a path measurement:
\[P(A)=|\alpha|^2\]
and
\[P(B)=|\beta|^2.\]
For example, the states
\[\frac{|A\rangle+|B\rangle}{\sqrt2}\]
and
\[\frac{|A\rangle-|B\rangle}{\sqrt2}\]
both give
\[P(A)=P(B)=\frac12\]
when the path itself is measured.
Nevertheless, the two states can behave differently when their alternatives are recombined because the relative phase is different.
Thus:
\[\boxed{\text{amplitude contains magnitude and phase}}\]
while
\[\boxed{|\text{amplitude}|^2=\text{measurement probability}}.\]
What Relative Phase Means
Relative phase can first be understood using ordinary waves.
If two waves repeatedly reach their peaks together, their relative phase can be described as \(0^\circ\). If one reaches a peak whenever the other reaches a trough, their relative phase can be \(180^\circ\).
Importantly, both situations can be coherent.
Coherence does not mean that the phases must be identical.
It means that their relationship remains definite and predictable:
\[\boxed{\text{coherence}=\text{a stable, usable relative phase relationship}}.\]
What Are A and B in the Electron Experiment?
In the double-slit discussion, A and B are not two electrons.
They label two path alternatives in the state of the same electron:
\[|A\rangle\]
and
\[|B\rangle.\]
The quantum state assigns amplitudes to these alternatives:
\[|\psi\rangle=\alpha|A\rangle+\beta|B\rangle.\]
It is therefore more precise to say:
One electron has one quantum state, and that state assigns amplitudes to possible path measurement outcomes.
It is potentially misleading to picture two physical electrons or to imagine that the electron has literally been divided into pieces.
Position Is Also a Quantum Observable
The deeper difficulty appears when we insist that the one electron must nevertheless occupy one definite position before measurement.
Quantum mechanics does not generally require this.
For position, a quantum state can be represented by a wavefunction
\[\psi(x).\]
The wavefunction assigns an amplitude to different possible position outcomes \(x\).
The probability density for a position measurement is obtained through Born’s rule:
\[P(x)\propto|\psi(x)|^2.\]
This does not mean that many electrons exist at all those positions.
There remains one electron. Its quantum state simply need not correspond to one definite position before the position is measured.
When a position measurement is performed, however, the electron is detected at one location.
The Crucial Distinction
The conceptual breakthrough is therefore to separate two statements that initially seem identical:
Statement 1: There is only one electron.
Statement 2: That electron must possess one definite position and one definite trajectory at every moment.
Quantum mechanics accepts the first statement but does not generally assume the second.
Therefore:
\[\boxed{\text{one electron}\not\Rightarrow\text{one definite pre-measurement position}}.\]
This distinction is essential before trying to understand coherence and decoherence in the double-slit experiment.
Where This Leads Next
The next question is unavoidable:
If quantum mechanics does not say that the electron already has one definite position or trajectory, what should we actually imagine physically happening between the source and the final measurement?
This is where the distinction between the mathematical quantum state, measurement outcomes and claims about an underlying physical trajectory becomes especially important.
Note metadata
- Note type: learning-note
- Subject: learning-notes
- Source: Quantum Computing for Everyone