The double slit experiment

Having discussed the issues with quantum measurement in general, and shown that standard interpretation of quantum mechanics is incomplete, Young's double slit experiment with electrons will be discussed in two variants (for background see Chapter 1 of Hall [1]):

• The standard configuration, to be described below, Figure (1).
• A configuration with a pointer that acts behind the slits to point in the direction of the passing electron, Figure (2). 

Figure (1) The setup for the double slit experiment is shown. An electron source sends one particle at a time toward the screen with the slits. The slits are marked by \(\delta_1\) and \(\delta_2\) and a sample region \(\Delta\) is shown on the detector screen.

Standard configuration

The experiment, Figure (1), is as follows:

• There is a source of electrons that move towards a screen with two slits.
• The intensity of the beam is low and only one electron is moving towards the detector at any time.
• The slits are marked $\delta_1$ and $\delta_2$
• \(\Delta\) be some arbitrary region on the electron detector.

Let $Y_1$ and $Y_2$ be the projection operators for position in the regions of the two slits $\delta_1$ and $\delta_2$. Then $Y_1 + Y_2$ is the projection of position for the union $\delta_1 \cup \delta_2.$ Let $X$ be the operator for position in a local region $\Delta$ on the detection screen. Assume the electron is only constrained to pass through the slits without being constrained as to which, then under those conditions the conditional probability is given by the Law of Alternatives:

\[\begin{eqnarray}
p(X|Y_1 + Y_2) &=& p(X|Y_1)p(Y_1|Y_1 + Y_2) + p(X|Y_2)p(Y_2|Y_1 + Y_2)\nonumber \\
& &+ [\textbf{tr}(Y_1 \rho Y_2 X)+\textbf{tr}(Y_2 \rho Y_1 X)]/ \textbf{tr}(\rho (Y_1+ Y_2)).
\end{eqnarray} \, \, \, \, \, \, \, (1)\]

This can be written more compactly as
\[
p(X|Y_1 + Y_2) = p(X|Y_1)p(Y_1|Y_1 + Y_2) + p(X|Y_2)p(Y_2|Y_1 + Y_2) + p(X| Y_1+Y_2 )_I  \, \, \, \, (2)
\]
where \( p(X| Y_1+Y_2 )_I\) is the interference term. 

Note that if $X$ commutes with either $Y_1$ or $Y_2$, this interference term vanishes, because \(Y_i Y_j = 0\) for \( i \ne j \) . This can be observed if the detector is right next to the two-slit screen because \(X\) then coincides with either \(Y_1\) or \(Y_2\). If the detector is a distance from the two-slit screen (e.g. \(X= \Delta\)), then the state of the electron evolves unitarily via  $\alpha_t,$ so $\alpha_t(Y_i)=u_t Y_i u^{-1}_t$ no longer commutes with $\alpha(X)$, giving rise to the non-zero interference term.

 

The standard explanation of the interference effect is that the state of the particle is, or acts as, a coherent pair of waves emanating from the slits, which exhibit constructive and destructive interference effects. This was, of course, the explanation for Young's original experiment with light. For individual quantum particles, however, there is the unexplained local event observed at the detection screen.  This is a problem for the theory. While the Born interpretation of the wavefunction provides a probability distribution for the particle position it requires the detection screen, operating outside what is described by the mathematical theory, to act as a sampling mechanism for that distribution.

The explanation given in this blog is that the two-slit screen functions as a preparation of the state for the particle, by which the state is conditioned, or reduced, to pass through the region $\delta_1\cup\delta_2 $. This reduction is not a position measurement, since $\delta_1\cup\delta_2$ is not a localised region (as it would be for a single-slit screen). Once the particle reaches a detection screen then, in interaction with the screen, it appears in a random local region $\Delta$ and its position takes a value. Just as in the standard Born interpretation of the wavefunction, it is not explained in the theory how the electron takes the value that the detector detects other than invoking random sampling of the possible values.

So, the interference pattern on the detector is built up over time as more electrons arrive and are sampled by the detector. 

The introduction of an interaction with a pointer

This section is adapted from Bricmont [2], Appendix 5.A and Maudlin [3].

The experiment, illustrated in Figure (2), is now as follows

• There is again a source of electrons that move towards a screen with two slits. 
• The intensity of the beam is low and only one electron is moving towards the detector at any time.
• The slits are marked $\delta_1$ and $\delta_2$.
• \(\Delta\) be some arbitrary region on the electron detector.
• A pointer \(P\) is introduced. It is a quantum object with three states neutral, \(P_0\), points to slit \(1\), \(P_1\) and points to slit \(2\), \(P_2\). The interaction with the electron causes the pointer to move towards it.

Figure (2) The setup for the double slit experiment is as in Figure (1) but for the addition of a three state pointer that interacts with the electron as it passes through slit \(\delta_1\) or \(\delta_2\).

Again let $Y_1$ and $Y_2$ be the projection operators for position in the regions of the two slits $\delta_1$ and $\delta_2$. Then $Y_1 + Y_2$ is the projection of position for the union $\delta_1 \cup \delta_2.$ Let $X$ be the operator for position in a local region $\Delta$ on the detection screen. The operator representing the pointer has three eigenstates and therefore a three-dimensional Hilbert space \(\mathcal{H}_P\). Without the pointer the Hilbert space is \(\mathcal{H}_0\). The Hilbert space of the total system is \(\mathcal{H}_P \otimes \mathcal{H}_0\). The total system consists of a single electron and a pointer constrained by the screen with the two slits, and the detector.

The possible constituent states are: 

• \(\phi_1\) be the state of the pointer pointing towards the slit \(\delta_1\)
• \(\phi_2\) be the state of the pointer pointing towards the slit \(\delta_2\)
• \(\phi_0\) be the state of the pointer pointing in the neutral direction \(P_0\)
• \(\psi_1\) be the state of the electron passing through slit \(\delta_1\)
• \(\psi_2\) be the state of the electron passing through slit \(\delta_2\)
• \(\Psi_0\) be the state of the electron with the pointer in the neutral position \(P_0\).

where \(\phi_0\), \(\phi_1\) and \(\phi_2\) are eigenstates and therefore orthogonal. This is not the case for \(\psi_1\) and \(\psi_2\). 

Assuming the pointer starts in its neutral state, the initial wave function is
\[\begin{eqnarray}
\Psi_0 &=& \phi_0 \otimes (\psi_1 +\psi_2) \nonumber\\
&=&\phi_0 \otimes \psi_1 + \phi_0 \otimes \psi_2
\end{eqnarray}\]
Two treatments of the situation will now be discussed. In the first, the electron carries its charge through either the \(\delta_1\) or \(\delta_2\) and the pointer reacts and in the second the charge is not constrained to pass through only one slit at a time. The first treatment would be consistent with the ontology of Bohmian mechanics or stochastic mechanics. The second would be consistent with the electron with its charge passing through both slits or not physically existing at all at that point in the experiment. This is consistent with the ontology proposed by Bell [4] for the formulation of quantum mechanics proposed by GRW [5]. In their ontology there can be a local event only with extremely low probability in a run of the experiment. 

Treatment I: The pointer reacts to which slit the electron passes through

Here the situation is idealised to assume that the pointer reacts perfectly to the electron going through either slit 1 or slit 2. This is not a measurement because the reaction is neither registered nor signalled. At no point does anyone know which slit the electron has passed through.

Time unitary evolution in quantum mechanics is linear, therefore \(\Psi_0\) evolves to

\[
\Psi = \phi_1 \otimes \psi_1 + \phi_2 \otimes \psi_2.
\]
Inserting this for the state into equation (1), and using the notation for the interference term in equation (2), gives
\[
p(X| Y_1+Y_2 )_I=\frac{\textbf{tr}(Y_1 \rho_\Psi Y_2 X)+\textbf{tr} (Y_2 \rho_\Psi Y_1 X)}{\textbf{tr} (\rho_\Psi (Y_1+ Y_2 ))} 
\]
\[
p(X| Y_1+Y_2 )_I= \frac{\mathfrak{N}}{\mathfrak{D}},
\]
where
\[
\mathfrak{N} =(\phi_2 \otimes \psi_2 ,P \otimes X \phi_1 \otimes \psi_1) +(\phi_1 \otimes \psi_1, P \otimes X \phi_2 \otimes \psi_2 )
\]
\[
\mathfrak{D}=(\phi_1 \otimes \psi_1 , \phi_1 \otimes \psi_1)+(\phi_2 \otimes \psi_2, \phi_1 \otimes \psi_1)
\]
\[
+(\phi_1 \otimes \psi_1, \phi_2 \otimes \psi_2) +
(\phi_2 \otimes \psi_2, \phi_2 \otimes \psi_2)
\]
Using that the states of pointer are orthogonal
\[
p(X| Y_1+Y_2 )_I= 0
\]

The quantum interference term disappears. \(p(X| Y_1+Y_2 )\) is just a combination of the pattern for each slit on its own. So, even though no measurement is registered the presence of the pointer and its interaction with the electron is enough to eliminate the interference pattern. This is often explained (by Feynman [6] for example) by the electron being watched to determine which slit the electron passes through. The pointer is reacting to but not determining the outcome. The interference pattern disappears due to what is known as entanglement, not measurement.

Treatment II: The pointer does not react to which slit the electron passes through

In this treatment the assumption that the total charge is carried through only one of the two slits is not made or if it does the pointer cannot unambiguously react to it. This leads to a more general linear combination of the possibilities. Generally, the \(\Psi\) evolves to
\[\Psi = \sum_{i \in \{0,1,2\}} a_i \phi_i \otimes \psi_1 + \sum_{i \in \{0,1,2\}}b_i \phi_i \otimes \psi_2.
\]
\[
a_0=b_0, a_1 = b_2, a_2=b_1.
\]
The pattern to be observed on the detection screen in this treatment would now be 
\[
(\Psi, P \otimes X \Psi) = (\sum_{i \in \{0,1,2\}} a_i \phi_i \otimes \psi_1 + \sum_{i \in \{0,1,2\}}b_i \phi_i \otimes \psi_2, \]

\[
 P \otimes X [\sum_{i \in \{0,1,2\}} a_i \phi_i \otimes \psi_1 + \sum_{i \in \{0,1,2\}}b_i \phi_i \otimes \psi_2]).
\]
Using the orthogonality of the pointer states,
\[
(\Psi, P \otimes X \Psi) =\sum_{i \in \{0,1,2\}}|a_i|^2 (\psi_1, X \psi_1) + \sum_{i \in \{0,1,2\}}|a_i|^2 (\psi_2, X \psi_2)+\]
\[\sum_{i \in \{0,1,2\}} a^*_1 a_2 (\psi_2, X \psi_1) + \sum_{i \in \{0,1,2\}}a^*_2 a_1 (\psi_1, X \psi_2)
\]
where superscript \(*\) denotes the complex conjugate.
Using \(C= \sum_{i \in \{0,1,2\}}|a_i|^2\) to simplify to 
\[
(\Psi, P \otimes X \Psi) = C( (\psi_1, X \psi_1) + (\psi_2, X \psi_2))+
 \mathfrak{Re}\{2 a^*_1 a_2 (\psi_2, X \psi_1)\}.
\]

So, the interference pattern (\(\mathfrak{Re}\{2 a^*_1 a_2 (\psi_2, X \psi_1\}\)) persists. This behaviour is consistent with a physical situation where no charged particle exists in the region of the slits, as in the Bell ontology for the GRW collapse theory.

Experimental tests and ontological comparisons

The setup with the pointer, as described above, is an idealisation. This pointer is a quantum object that will react reliably to a passing charge particle but with no registration of the direction pointed. If there is no passing charged particle, then there would be nothing to react to. 

It is conceivable that the pointer could be realised by a molecule with an appropriate electrical dipole moment that can be fixed in position immediately behind the screen, between the two slits, but free to rotate. Maudlin [3] discusses the setup with a reacting proton trapped between the slits. Any practical experiment would implement the pointer in a way that would inevitably deviate for the ideal. This could lead to a situation where the interference pattern is weakened but not destroyed.

If quantum theories are constructed to be empirically equivalent but with distinctly different ontological models, then a discussion of how credible these ontological models are within different scenarios can provide a valid critical comparison. The result in Treatment I is consistent with an ontology in which the electron carries its charge on one continuous trajectory, such as in Bohmian mechanics or Nelson's stochastic mechanics. That is, each electron exists in the region of only one of the slits. Then the presence of a pointer reacting to the charge but not measuring it would be sufficient to destroy the interference pattern. This would give support to

• A Bohm or Nelson type theory in which the electron follows a continuous trajectory through the experimental setup. The trajectory is deterministic in the case of Bohm but stochastic in the case of Nelson.
• A quantum chance theory. The local appearance of the electron as a dispositional property that appears as a value locally in the region of only one of the slits due to the interaction of the electron with the pointer. However, the theory does not as it stands describe how this appearance is made actual. It would be an assumption that the pointer acts to sample the distribution.

By contrast, if pointer shows no reaction, as in Treatment II, then that would undermine the explanatory force of the Bohm or Nelson ontologies and indicate a quantum world in which one or more of the following is the case:

• A registering measurement is needed to destroy the interference pattern. This could be called the Copenhagen point of view.
• The charge is spread across possible positions (although this would have to be split equally across the two slits to give no pointer reaction)
• The charged particle may not actually exist in the region of pointer. Although the Bell ontology for GRW could be thought of as a mechanism for locally actualising the charge, the mechanism that they propose does not occur frequently enough to produce the effect in this experiment.
• The proposal for a theory of quantum chance in which dispositional property of the electron to appear at a locality does not entail the actual appearance due to the interaction with the pointer.

Treatments I and II both assume a behaviour of the pointer. There is a full mathematical formalism that would, in principle, provide the answer to whether the theory predicts that the interference pattern persists or disappears once the electron point interaction is included in the Hamiltonian of the system shown in Figure (2). 

The theory put forward in this blog is open to either the outcome where the pointer reacts and to that in which it does not. This is because it provides no quasi-classical insight into what the pointer or the electron may do. The quantum chance theory provides transition probabilities that must be calculated from first principles. They correspond to dispositional powers that do not appear as such in any Field of Sense. This is because the dispositional properties, although existing in the real sphere, only provide an effect when an interaction and context affords a Field of Sense, and the form of the appearance depends on the details of the interaction. What is clear is that there is no local appearance of the electron to which the pointer can react. The physical details of the pointer interaction with the quantum state my give rise to an actual local appearance of the electron but it may not. A treatment using the total system Hamiltonian will have no mechanism to break the symmetry between the two slits and so cannot be expected to eliminate the interference pattern.

[1] Hall, B. C., Quantum Theory for Mathematicians, Springer, 2013

[2] Bricmont, J., Quantum Sense and Nonsense, Springer Nature, 2017

[3] Maudlin, T., (2019) Philosophy of Physics: Quantum Theory, Princeton University Press., 2019

[4] Bell, J. S., Are there quantum jumps?, Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy, Cambridge University Press, 2004, 201-212

[5] Ghirardi, G. C., Rimini, A. & Weber, T., Unified dynamics for microscopic and macroscopic systems, Phys. Rev. D, American Physical Society, 1986, 34, 470-491

[6] Feynman, R., The Feynman Lectures on Physics, Volume III Quantum Mechanics, 

California Institute of Technology, 2013