Quantum Mechanics with a Game Engine

The physics emulations that operate inside software game engines are designed to work as per Newtonian physics, so to anyone familiar with quantum theory, it may seem somewhat strange to suggest that a standard game engine could be used as a platform for creating working illustrations of quantum experiments.

If conventional accounts as to how quantum mechanics operates are correct, then it would seem so for good reason. Conventional approaches to explaining quantum phenomena begin by introducing the concept of a new type of mechanics that operates in a domain where Newtonian physics is suspended and replaced by a non-classical type of dynamics that is characterized by a wave-particle duality, where discrete particle interactions appear to be overridden by the non-local dynamics of (curiously, classical) waves.

In effect, when particles are seen to interact in a local discrete manner, then we are supposedly seeing ‘classical’ behaviour, and when we see groups of particles that stochastically form patterns that resemble wave interference patterns, we are supposedly seeing evidence of some mysterious, but hidden, ‘quantum’ world that is ruled by waves.

The basic conceptual connection that underpins this view is a clear-cut correspondence. A correspondence between the patterns that classical waves produce through the mechanism of interference, and the scattering patterns made by discrete particles.  This correspondence is the basis of a  widely accepted paradigm which places the mechanism of wave interference at the core of quantum mechanics.

Inside that paradigm, the idea that the correspondence means that it might be possible to understand the behaviour of quantum systems through some form of wave-particle duality, becomes elevated from a possible approach up to a position of certainty wherein some form of wave interference mechanism is seen to be only possible way, to explain how quantum systems produce scattering patterns that resemble wave interference patterns.

This fixed paradigm is the foundation basis for a varied range of what are termed ‘interpretations’ of quantum mechanics, some key examples are:

  • The Copenhagen Interpretation,  where the role of the observer is key. On observation, a ‘wave function’ which predicts all possible outcomes of a particle and the system it interacts with, is considered to be a complete representation of reality until a measurement, at which point it magically ‘collapses’ to a single outcome.  The choice of experiment determines whether you can observe wave-like or particle-like behaviour. Fringe variations involve a requirement for a conscious observer.
  • The quantum multiverse, where many universes spontaneously form in just the right way such that all possible outcome of an interaction get to exist in their own universe (how this works when the probabilities form continuous distributions is not explained), and furthermore, these universes are (somewhat vaguely) interconnected by… ‘interference’ such that the relative probabilities of different universes magically works out to match the probabilities calculated from QM.
  • Variations of the de Broglie/Bohm type of interpretation where every particle surfs some kind of ‘pilot wave’ that guides each particle along a trajectory that the density of trajectories exactly resembles a wave interference pattern.

From a simulation standpoint, all three suffer from the same essential problem, all three appear to require post hoc arbitrary mechanisms to fit elements of the behaviours that are expected from of classical waves and kludge this together with what we observe at measurement, i.e. discrete ‘classical’ particles so as to match the probability distributions that are calculated from quantum mechanics. Taking a somewhat flippant approach one might look at simulating these interpretations as:

  • In Copenhagen: Use the equations of quantum mechanics to calculate a probability distribution of possible outcomes for an interaction and keep them all in play, then when the user makes an ‘observation’ randomly ‘magic’ the system to a single configuration, on the basis of the weighted probability distribution calculated previously. Oh, then discard all other possible outcomes and call it ‘collapse’.
  • In the Multiverse: As per Copenhagen, use the equations of quantum mechanics to calculate a probability distribution of possible outcomes for an interaction, and render all of them, then choose one of the myriad outcome configurations and call it ‘your’ universe. Note that from a simulation perspective, Copenhagen and Multiverse interpretations begin to look to be equivalent. Both involve creating a map multiple alternate realities, both involve selecting one outcome, and the only difference is what is done with the leftovers.
  • In de Broglie/Bohm: Use the equations of quantum mechanics to calculate a probability distribution for all possible trajectories, then draw contour lines on the distribution, then announce that the contours are the result of a quantum potential.

I know the above may seem overly flippant, and not do justice to the work involved in quantum research, but there is a simple point that becomes apparent when one considers constructing working simulations. The attempt to inject classical wave-like behaviours into a quantum mechanics which involves discrete interactions results in ad-hoc mechanisms that would need to work ‘in just the right way’ to reproduce the predictions of quantum mechanics. These interpretations appear to be the best plausible attempts to rescue the hypothesis that classical wave interference is at the foundation of quantum theory.

Conversely, it may well be quite possible that the whole idea that we can explain quantum mechanics by supposing that it involves some variation of classical particles being influenced by some sort of wave interference or alternatively, particles swapping in and out of wave-like and particle-like forms of existence, is a  futile dead end, that is as viable as the idea that planetary orbital mechanics can be best explained by constructing an elaborate system of perfect crystal spheres, made of quintessence, and moving on geocentric circles offset by deferents.

So what does this discussion have to do with the utility of the game engine? In essence, the wave/particle paradigm would, as stated at the outset, rule out basing a simulation on a game engine running according to Newtonian mechanics, however, there is an alternative that may look more promising, and it looks worth consideration simply because it has the potential to allow construction of direct simulations without running into as many ad-hoc mechanisms (pilot wave/collapse/multiverse).

Method in the Madness

One avenue that could be worth following is to explore William Duane’s 1923 hypothesis for crystal diffraction, where quantum scattering is modelled using discrete momentum exchanges between an incident particle and a scattering object.

Duane showed how Einstein’s work on the photoelectric effect provides a model for quantised exchange of momentum that can be applied to the diffraction of X-rays by crystals. This model allows the problem to be treated as a discrete particle scattering problem.  In effect, Duane derived the same scattering relationships that were derived by Bragg’s wave model, “based on quantum ideas without reference to interference laws.” (www.pcontent/9/5/15nas.org/8).

Duane’s work was extended by Gregory Breit (www.pnas.org/content/9/7/238) to explain how diffraction through gratings and slits can also be analyzed using discrete momentum transfers.

At first appearance, Duane’s work might appear to be at variance with the mathematical representations in contemporary quantum theory, because of the notion that interference is crucial then leading to an expectation that the incident particles should somehow (at least transiently) act like classical waves in order to generate an interference pattern.

In effect, the conventional approach (where the incident particle is supplying the ‘waves’ that undergo an interference process) implicitly leads to the idea that the incident particle must the key element, both at scattering and detection.

In contrast, Duane’s analysis of crystal scattering shows that the scattering can also be explained if the scattering is governed by a set of permitted quantized reactions that are a property of the crystal structure, and furthermore, are completely independent of the type of incident particle or its velocity and energy.

By employing standard quantum theory to calculate the possible reactions of an extended object, Duane’s approach looks to provide a better fit to the mathematical formalism of contemporary quantum mechanics than the tortuous rationalisations of wave-particle duality, where the incident particle is thought to mysteriously ‘explore’ the scattering object as a classical wave, then recombine via collapse or ‘decoherence’.

Duane’s work shows how scattering by crystals and gratings (with Breit, including the notorious ‘double slit’) can be straightforwardly modeled by treating the reactions of the scattering object as being constrained to a quantized set of possible reactions while the incident particles are treated as localized entities that obey the rules of energy and momentum conservation.

Unlike fully deterministic classical mechanics, Duane’s model requires quantized momentum transfers that depend on the state and structure of the scattering object and this dependency injects the required uncertainty in outcomes to be consistent with quantum constraints on determining particle position at the time of interaction.  The major difference is that while the position cannot be measured (which requires some form of additional interaction), in Duane’s model the particles can have definite positions and trajectories, they just cannot be observed without disrupting the outcome of the experiment.

Because Duane’s model keeps the incident particle as a discrete entity throughout, a simple model can be constructed with particles following definite trajectories before and after the interaction. As can be seen in the simulation, the ability of this simple model to reproduce Laue scattering patterns is striking in both simplicity and beauty.

I am in the process of building a small Web-based game that people can use to explore the workings of this type of model. This is in the early stages of being ported to a web environment, the explanations and user interface are very much under construction. Nevertheless, feel free to explore.

Game engine simulation of quantized scattering at a grating.(An initial test of a WebGL version of these simulations is available here.)

At this point, the function of the test is to determine the feasibility of developing an educational example that shows how Duane’s model works to provide an alternative to de Broglie’s idea of wave-particle duality)

In essence, taking Duane’s approach involves shifting our view as to what the mathematical equations of quantum theory represent without throwing the ‘baby out with the bathwater’. The primary shift in viewpoint is to challenge the conventional view that the equations completely represent the behavior of incident particles that individually explore all avenues through a classically passive scattering object, and then later magically turn up as discrete localized objects on detection.

Despite the radical differences between interpretations based on interference/wave-particle duality, (e.g. ‘pilot wave’/collapse’/’multiple realities’), these interpretations have a common basis. All contain attempts to couple observed scattering patterns to the same type of interference processes that classical waves undergo. As if there is a one-to-one correspondence between the mathematical tools of quantum theory and the interference of classical waves.

In contrast, Duane showed that if one treats the possible momentum reactions that an extended object can make as being a discrete and quantized, then the set of reactions that are required to reproduce the observed crystal scattering can be tied to the crystal geometry using the same quantum rules that Niels Bohr had deduced for the angular momentum of atomic orbital