Quantum Mechanics with a Game Engine

The physics emulation components within software game engines are designed to operate as per Newtonian physics, so at first, it may seem somewhat strange to suggest that a standard game engine can provide working illustrations of quantum experiments.

It would seem so for good reason. The usual approaches to describing quantum mechanics involve the concept of a new type of domain, a domain where Newtonian physics is suspended and replaced by a non-classical type of dynamics characterised by some form of wave-particle duality, wherein discrete particle interactions appear to be governed by the non-local effects of wave interference.

This view that a new type of a ‘ mechanics’ that overrides classical dynamics at small scales leads to the prevailing background paradigm that there is somehow a ‘classical’ world and a ‘quantum’ world and furthermore, that when we see individual particles behaving in a local discrete manner we are seeing ‘classical’ behaviour, and when we see groups of particles forming patterns that match wave interference patterns we are seeing evidence of some mysterious ‘quantum’ world that is ruled by waves. Where the action of the waves is somehow veiled so we only see the effects of the waves in patterns of probability densities.

Perhaps the most famous example that characterises this background paradigm is Schrödinger’s cat, where we have a classical word, and quantum world, and the apparent link is the role of observation/observer.

The basis for this view has strong scientific credibility. It is quite clear that the principles of classical wave interference and diffraction correctly predict the statistical patterns that are observed when particle beams are scattered by periodic structures (e.g. gratings and crystals), and yet particles that make up the patterns are always detected as discrete entities.

The most obvious conclusion is that the classical (Newtonian) behaviour of each individual particle is overridden by the effects of wave dynamics, in such a way as to cause the statistical patterns formed by particle trajectories to match wave-like scattering patterns.

That effect of that conclusion is to place a constraint what we think we are looking for when trying to address how quantum mechanics works. As if what we need to do, is to divine some way for two apparently incompatible things to work together, and once we have achieved this, ‘interpretation’ we will understand how quantum mechanics works.

In 1915 father and son scientists William Henry Bragg and William Lawrence Bragg won the Nobel Prize for their experimental and theoretical work in developing a theory of X-Ray diffraction by crystals.

The approach taken by the Braggs involved treating X-Ray beams as behaving like classic electromagnetic waves being diffracted by the planes of atoms in a crystal. This analysis led them an equation that predicts the observed angles at which E-ray beams exit crystals.

The discovery established a strong theoretical link that connected the behaviour of X-rays to the theoretical work on electromagnetic waves developed by James Clerk Maxwell. This connection is backed by experimental observations of diffraction and interference across the electromagnetic spectrum, linking the behaviour of X-Rays in crystals to the behaviour of light, infrared, microwaves, and radio waves.

The Braggs’ work predated major developments in quantum mechanics (notably the work of Erwin Schrodinger and Werner Heisenberg in the mid-1920s). Even so, their discovery had laid a foundation for the later the concept of wave-particle duality by experimentally demonstrating that classical wave interference could provide a powerful explanation as to how beams of x-rays are scattered by crystals.

The Braggs’ interference model was further reinforced in 1923 when Louis de Broglie applied the momentum-wavelength relationship for discrete photons of light to matter particles to predict that beams of matter particles could also display interference. This was later experimentally confirmed by Davisson and Germer by scattering electron beams off nickel crystals.

Even without the full formalism of quantum theory, a conundrum arises if we accept two basic precursors: Firstly, that scattering of X-Rays and electrons is the result of classical wave interference processes. Secondly, that the scattered beams are composed of discrete particles that interact in a localised manner when detected.

The problem being the reconciliation of the discrete and localised nature of particle interactions, with the distributed and non-local features of the diffraction and interference of classical waves.

In essence the fact that the scattering of discrete particles interacting with crystals and gratings (including the double slit experiment) can be accurately predicted by applying the principles of classical wave theory, is so compelling that it is generally accepted that any interpretation of quantum theory needs to reconcile how a particle can act as a wave, as if an explanation that can reconcile two incompatible classical phenomena will somehow provide an explanation of quantum theory.

The viewpoint that classical wave interference is the source of the observed patterns leads to a presumption that the incident particle acts as both the ‘wave’ and the particle and the scattering object is a passive barrier around which the waves diffract. This, in turn, leads to some common assumptions that may not be true.

  • That there are distinct classical and quantum domains, the quantum domain is characterised by wave-like behaviour and interference, whereas in the classical domain particles are localised and discrete.
  • The equations of quantum theory are quite literally describing wave interference.
  • The job of an interpretation is to explain how a non-local and distributed wave interference process can be reconciled with the fact that particles are always detected as discrete local entities. The wave process must be concealed from direct observation.

Differing attempts to reconcile the continuous and non-local features of wave interference with the discrete interactions characteristic of quantum mechanics – tend to be regarded as the only possible options.

Whether it is a multiverse where all possible realities exist simultaneously or an ephemeral ‘wave function’ that crosses space and time only to ‘collapse’ or ‘decohere’ on measurement down to one single outcome from a continuous spatial distribution of possible outcomes, the job of these added mechanisms (collapse/multiple realities) is to link continuous wave-like probability distributions back to singular discrete events, by invoking a hidden, but classical, wave-like interference process.

In essence, the conundrum over apparently counter-intuitive effects, such as ‘wavefunction collapse’ and ‘multiple realities’ (which I would argue are ad-hoc mechanisms, not actually there in quantum mechanics itself), results from the need to add these ad-hoc devices to rescue the idea that classical ideas of wave-particle duality explain how quantum mechanics works.

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 photomodelledc 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