Two Slit Copenhagen:

Text by Chris Dewdney

In the physics of everyday objects, as described by Newton and Maxwell, waves and particles are completely different things. Particles follow paths whereas waves are spread out and can interfere. Quantum mechanics was born when Louis de Broglie proposed (in his PhD thesis) that particles also have a wave-like aspect which determines the way they behave. Today, in standard quantum mechanics, the motion of a particle is completely described by calculating the evolution of its associated wave according to the Schroedinger equation. The intensity, or brightness, of the quantum wave in a given region indicates the chance of finding a particle within that region. The probability of being found in a given region is all that standard, or Copenhagen, quantum mechanics can calculate.

This animation illustrates a quantum wave impinging, from the lower left, onto two slits, or gaps, in an otherwise impenetrable screen. Before the beam reaches the slits there is a pretty much constant chance of finding the particle at any point across the beam. As the quantum wave moves through the slits, two separated beams are created that immediately begin to spread out, whilst still very near the slits this happens in a smooth manner. The spreading beams from each slit start to overlap but, instead of the intensities of the beams just adding together to produce a smooth overlapping pattern, a series of bright and dark interference bands form diverging from a point between the slits towards the screen. Such interference patterns have long been familiar in wave optics. But here we are not dealing with a simple continuous wave effect. The bright points of light appearing on the screen, one at a time, show where the individual quantum particles arrive. The particles appear at the screen one-by-one, but only in those regions where the beam is bright, and they arrive there in an apparently random sequence that cannot be predicted by quantum calculations. The quantum particles’ particle-like nature is revealed in the localised bright spots, whereas their wave-like nature is demonstrated by their distribution in a wave-like interference pattern. The behaviour seen in this animation is a demonstration of the wave-particle duality of quantum mechanics.

Were the quantum particles to behave simply as particles, with no wave-like aspects, then they would just pass through the slits and head straight towards the screen without spreading or interfering. In quantum mechanics, when interference is suppressed, the beams still spread out and overlap but the intensity remains smoothly bright. The particles more likely arrive at the screen where the beam is bright and the tale-tale points of light are spread out smoothly, with no gaps and no fringes appearing. This we see in the last section of the animation.

Interestingly, the suppression of the interference fringes can be brought about by introducing a detector that can record through which slit each individual particle passes. Once the detector has determined the specific slit through which an individual particle has passed, the wave from the slit not taken has no effect on its motion. It is as if the wave from the other slit has collapsed. The detection of the particle’s location at the slits appears to force the quantum particle to show only its particle-like aspect thereafter.


Two Slit: Bohm – Quantum Potential

Text by Chris Dewdney

The Copenhagen Interpretation forbids any conception of how the particles move from the source to the screen. In stark contrast, Bohm’s theory provides a clear and distinct description of the way in which particles move along trajectories nonetheless accounting for all of the observations predicted by quantum theory. Bohm’s approach dispels the so-called quantum paradoxes, replacing conundrums with clear and intuitive explanations. In the two-slit case, each particle passes through one or other of the slits but its motion is guided by the interfering waves from both slits, through the medium of the quantum potential. The quantum potential becomes significant where quantum effects are observed. Here, we see the particles moving over the quantum potential surface (represented in ghostly white), dashing across the valleys and lingering on the flat plateaux to arrive at the screen in the brighter areas of the interference pattern. Between the two slits and the screen no previously known, or classical forces are acting, the space is empty yet the particles do not follow straight trajectories as Newton’s laws demand. Instead, the trajectories “are bent several times backwards and forwards, with a motion like that of an eel”.

Information concerning the whole experimental environment is encoded in the form of the quantum potential (itself determined by the particle’s quantum wave) which “informs” the particle how to move.

The behaviour seen in these animations illustrates that, according to quantum mechanics, the behaviour of individual particles depends very sensitively on their whole environmental context.

To find out how particles move it is not sufficient to simply specify the configuration of a system and any forces acting at a given time, we need to also specify the quantum mechanical wave function encoding information about the whole system and its environment. This is the origin of quantum wholeness.


Configuration Space

Text by Chris Dewdney

A detector is placed between the slits, it can tell through which slit each of the particles passes.

When it is switched on the interference disappears. The particle paths no longer wriggle into the bright fringes; the fringes have disappeared. Now, the beams emerging from the slits each act as if the other beam were not there. Each path from a particular slit is unaffected by the presence of the other.

Richard Feynman claimed that no “hidden variable” theory could ever be devised to account for this loss of interference. But, Bohm’s theory has a perfectly acceptable explanation of this loss of interference. The quantum mechanical wave function tells the probability amplitudes of all possible motions, but this wave function must  be described using a multidimensional, abstract mathematical space, known as configuration space.  In principle, within this space there is one quantum wave function that describes the evolution of the possible configurations of everything in the universe, including ourselves. The relationship between objects in our everyday world is determined within this abstract space and proximity in physical space is not necessary in order for objects to be correlated. Nonlocality in physical space emerges from the (local) behaviour in configuration space.

To explain quantum behaviour it is necessary to transcend description in everyday space and time. This is the most staggering implication of quantum theory, there is no possibility to explain our world whilst we remain bound by it.

It is not possible to visualise spaces with more dimensions than three, and as stated previously, the space in which the wave function of the universe evolves has many, many dimensions, but if we can factor out from the universal wave function, the wave function describing single particles passing through the slits (in two physical dimensions) and we represent the detector by another single particle moving in just one dimension then the space we need has three dimensions which can be visualised. This is what is illustrated here.

The box which appears on the right of the animation provides an illustration of this more general three dimensional configuration space. What we see there is not physical space. The horizontal plane of the box represents the two dimensions in which the particle moves through the two slits. The vertical dimension in the box  is not the real space vertical dimension, it is the detector particle’s dimension. Each single path in this three dimensional configuration space describes at once both the motion of the two slit particles and the motion of the particle representing the detector. A single point in this space tells where the two-slit particle is and also where the detector particle is.

If the detector records the particle moving through the left slit, the detector particle moves upwards (shown as a glowing triangle in the animation) taking the two slit particle trajectory quickly upwards onto the top plane of the box. A detection at the right slit produces a similar downward motion. The action of the detector pulls the beams apart along the detector dimension, they no longer overlap in configuration space and the interference disappears. The set of trajectories from the left slit spread across the top plane whilst those from the right slit spread across the lower plane.

The real space motion of the two-slit particle can be found by projecting out of the configuration space into real space. In this case the projection can be achieved by collapsing the detector dimension, or simply viewing the box from above. Doing so, shows the trajectories from the slits crossing in real space, but in the quantum mechanical configuration space they do not cross at all.

Our perception reveals events happening in physical space, but it is quite wrong to assume that all that can happen is determined solely within that space in which proximity and local interaction appear fundamental. The actual arena in which the whole universe evolves is the abstract quantum space, and within this space objects are bound together and act together as a one whole. Distant objects, such as the detector and the particle in this simple animation, are not independent, but are irreducibly entangled, their correlated motions appearing as a “spooky” action at a distance”.

Everything that can possibly happen is represented by a different branch of the wave function. In the many worlds interpretation all of the possibilities are real and exist somehow together. In Bohm’s theory just one of the possibilities is the actual one.


Spin Measurement

Text by Chris Dewdney

The animation shows two possible motions for a single spinning particle moving through a Stern-Gerlach (SG) measuring device, as calculated in Bohm’s theory.

The SG device measures a quantum property known as “spin”. For particles such as electrons, quantum mechanics correctly predicts the probabilities that a given particle passing through the device will appear in one of two separated spots on the detecting screen, one in the upper region and one in the lower region.  In any given case, quantum mechanics cannot predict in which spot the particle will land. Classical physics, counter to observations, predicts a smeared out distribution of spots between the two extremes. The behaviour of the SG measuring device is similar for any orientation of the magnets, only a maximum of two distinct spots ever appear on the screen.

The operation of the  SG device constitutes a measurement, as by looking at where an individual particle lands on the screen it is possible to infer the value of its spin (along the direction of the axis of the magnets). Each particle that passes through the device appears in one of the two spots and in this way its spin is measured. The results are often referred to as “spin up” or “spin down” along the axis of the magnets.

Whilst passing through the device the particles have no location, according to ordinary quantum mechanics, there are no trajectories. Instead, in the quantum calculation the wave function develops two branches corresponding with the two outcomes. Quantum mechanics can predict the probabilities associated with each branch, the probability that a given particle will end up in one or other of the spots, but it cannot predict which one. Quantum mechanics cannot account for the fact that, for each particle, we see only one of the two possibilities. Quantum mechanics predicts possibilities, but we perceive a definite world. Quantum mechanics cannot account for our perception of a definite world. This is the measurement problem of quantum mechanics. Attempting to account for our experience of a definite world has led to many extraordinary ideas. Some argue that all of the possibilities are real simultaneously (the many worlds theory), whilst others invoke the power of human consciousness to collapse the many branches of the wave function to just one, in this case making the particle appear in one of the spots.

Bohm’s theory has no measurement problem, it gives a rather more prosaic account of our definite perceptions. In Bohm’s theory, the fact that a specific particle ends up in a particular location is accounted for by the fact that it always has a definite location; it follows a trajectory from beginning to end, just as classical particles do. Similarly, the actual world is always in a definite state notwithstanding the fact that its wave function encompasses many possibilities. The role of the wave function in Bohm’s theory is to determine how the definite state of the world evolves.

In the SG case it turns out that the explanation for arrival in just one spot is particularly simple, especially if the particle always enters the magnetic field with its spin axis lying horizontally. In this case, if the particle starts out in the upper part of the beam then it moves upwards whilst its spin axis rotates finally arriving in the upper spot with its axis pointing upwards. Similarly, if it starts out in the lower part of the beam it lands in the lower spot with its axis pointing downwards. The animation shows both possibilities simultaneously, but there is in fact only one particle in the SG device at a time. All we need to make the world definite is a definite trajectory, an intuitive concept eschewed in the Copenhagen orthodoxy.



Text by Chris Dewdney

The statistics of the behaviour of a quantum system are calculated, not by using functions in real three dimensional space, but by solving Schroedinger’s equation for the evolution of multidimensional wave function in an abstract mathematical space known as configuration space. A two particle quantum system is described by a six dimensional wave function. Under some circumstances the six dimensional function consists in a simple multiplication of two functions, one for each particle, and in this factorisable case the two particles behave independently. In other circumstances the wave function cannot be factored, it is then described as entangled and the behaviour of the two entangled particles shows correlations. These correlations cannot be thought of as arising from influences that pass from one particle to the other through our three dimensional space.  Nether, as J.S. Bell showed, can the correlations be thought of as arising from some cooking up of some local variables associated with each particle at some time in the past. From the point of view within our three dimensional space there appears to be a ”spooky action at a distance”.

Einstein, Rosen and Podolsky proposed a thought experiment in 1935 (now known as the EPR experiment) in which entanglement played a central role. EPR’s experiment became a laboratory possibility when Bohm reformulated it in terms of the entanglement of particle spins. Many experiments have now verified that the quantum correlations in the spins of two entangled  particles persist even when the particles are so widely separated that it is not possible for any signal to pass between them.

Bohm’s theory can account for the observed correlations as the variables describing the particles are nonlocal variables. When the particles’ state is entangled, the motion of one of the particles can depend on the motion of the other even though they are widely separated and incapable of signalling each other. This is probably the deepest mystery of quantum mechanics. We see this nonlocality clearly and explicitly in Bohm’s theory.

The animation shows the motion of two spin-entangled particles. There are two Stern-Gerlach (SG) devices widely separated (one on Earth and one on Andromeda, for example) and the two entangled particles travel one to each device where their spins are measured. When the two SG’s are aligned along the same direction, if one particle is measured to be spin up – travelling upwards when leaving the SG, then the other will be measured to be spin down and travel downwards at its SG. How is this correlation of the motion of the particles established across vast distances of space when no signal can possibly pass between the particles?

As the animation shows, in Bohm’s theory whichever particle is uppermost as it enters its SG device travels upwards whilst the other consequently travels downwards. In the opening moments of the animation we see the particle travelling to the left (say, towards Earth) at a lower point in its beam compared to the particle travelling to the right (say, towards Andromeda). Consequently the Earth particle is measured as “spin down” whilst the Andromeda particle is measured as “spin up”. If the Earth particle keeps the same position, but the Andromeda particle’s position is different so that it is now relatively below its earthbound twin, then the earthbound particle is now found to be spin up instead. This is the case even though nothing at all has changed on the Earth.

It is not just the relative position of the particles that counts. As the animation shows towards the end, if the operator on Andromeda chooses to switch off her SG device (something that cannot be known on Earth for 2.5 million years) the path of the particle on Earth can change immediately.

The trajectories predicted by Bohm’s theory, when particles are entangled, show that what happens to the particle on Earth depends, nonlocality, on the circumstances pertaining where the distant particle is on Andromeda. Flicking a switch to turn off the magnets in the SG device on Andromeda can change what happens on Earth. This type of behaviour, which is explicit within Bohm’s theory, explains the origin of the correlations predicted by quantum mechanics.

We see that proximity in space does not necessarily determine whether particles can be correlated. In the quantum mechanical multidimensional configuration space that lies beyond our everyday space, there is a single entangled wave function that develops different branches according to the measurements performed. A single point in configuration space represents both particles and in Bohm’s theory this point moves to enter just one of the branches that develop as the measurements take place. The local behaviour in configuration space appears nonlocal in our three dimensional everyday space.

Nonlocality, the inextricable binding of distant systems, is perhaps the most staggering consequence of quantum theory. We can neither formulate nor understand quantum behaviour bound by the confines of everyday space and time.