Cellular automata rules for quantum mechanics

Question

My limited understanding of quantum theory is that a quantum system is completely described by its wave function, which deterministically evolves according to Schrödinger's equation until wave function collapse.

Schrödinger's equation is essentially just a differential equation, which means (as far as I can see) that the state at every point in space changes depending only on its surroundings (i.e local rules). Correct me if I'm wrong but it's at the point of wave function collapse that the non-locality comes in, which is what cellular autonoma (CA) models can't deal with very well.

I'm not hypothesizing or suggesting that the universe IS a cellular autonoma (as Wolfram has suggested), but what would a CA model of Schrödinger's equation be like? (Ignoring wave function collapse for the minute)

Providing my understanding described above isn't wrong, I want to know, for the CA model:

1. What values describe the state at a given point? (i.e. what information would be in each CA cell?)
2. What rules are followed every discrete time tick to progress these values?

I ask because I've gained a good intuition (an "understanding" that I'm happy with at least!) of heat flow, wave propagation and the rules of fluid dynamics through interpreting their differential equations as local cellular automata rules, and writing little toy simulation programs to see how they behave and look. I'm a computer science/mathematics undergraduate and I've always said that I don't understand something until I can write a program that simulates it, so being a little too ambitious for my own good I want to write a simulation for a QM system! My background is I've read a few Penrose/Feynman books, and I've got the complete Feynman lectures on physics (not read them yet...), and physics has always been a side hobby, but I'm not too hot yet on the mathematical language that is used for QM.

Answer 1

There is nothing stopping you from interpreting the Schrodinger equation as rules for a cellular automata, in fact, the Schrodinger equation has the same form as the diffusion equation, but evolving in imaginary time. Let's write down some rules.

$$ i \hbar \dot \psi = H \psi $$ assuming we have a time independent potential $$ i \hbar \dot \psi = -\frac{\hbar^2}{2m} \psi'' + V \psi $$ Taking first order differences for the derivatives, we create

$$ i \hbar \frac{1}{dt} \left( \psi^{t+1}_x - \psi^t_x \right) = -\frac{\hbar^2}{2m} \frac{1}{dx^2} \left( \psi^t_{x-1} - 2 \psi^t_{x} + \psi^t_{x+1} \right) + V \psi^t_x $$ rearranging we obtain $$ \psi^{t+1}_x = (1 - i\bar V) \psi_x + i \alpha \left( \psi_{x-1} - 2 \psi_x + \psi_{x+1} \right) $$ where I have dropped the time index on the right, $\alpha = h dt / 2m dx^2$ and $\bar V = V dt / \hbar $, or perhaps more symbolically

$$ \psi^{t+1} = \psi + i \alpha \ \text{lap}(\psi) - i \bar V \psi $$ where $\text{lap}$ stands for the lattice Laplacian operator (so that we can easily generalize to higher dimensions).

Compare this to the standard diffusion equation, for which we have $$ \phi^{t+1} = \phi - D \ \text{lap}(\phi) $$

The Schrodinger equation appears very similar, although the field is complex and its change at each location is a $i$ rotation of its current state Laplacian and potential.

I will end with a cautionary note. I tried to implement this simple cellular automata, with hopes of adding a movie of the solution of a harmonic oscillator or something similar, but as presented, the automata is horribly ill conditioned and quickly blows up with a checkerboard mode. This is due to the fact that we have tried to describe the first order differential equation with a simple forward euler evolution, which doesn't have good convergence properties. By using RK4 (Runge Kutta 4th order), I was able to get the automata to behave nicely, but it wasn't all that inspiring of a movie, so I'll leave it as an exercise to the reader to implement this for themselves.

I will also note that this isn't really all that novel. Solving quantum systems by discretizing the Schrodinger equation is fairly common practice, as it is for most numerical solutions to differential equations generally, but people usually don't think or talk about these numerical solutions as cellular automata, although you could think of them as such if it aids in understanding.

Answer 2

Excellent question, I have a similar mindset having studied similar subjects. The cellular interpretation of differential equations really helps me develop intuition and play around with them.

You are looking for a continuous cellular automaton formulation, which can also be interpreted/implemented as a reaction diffusion-system with n-"chemicals" or vector components (one for each continous variable).

Check out the https://github.com/GollyGang/ready framework which was developed for studying exactly these types of systems - the included example Schroedinger1926/two_slit.vti is exactly what you are looking for here is a movie: https://www.youtube.com/watch?v=te_JU3RZ2eM&ab_channel=TimHutton

A fairly stable rule (I think this is probably a second order (leapfrog integration) scheme) is used: You need 4 real components (called "chemicals" in ready) a,b,c,d at each point in space and update them according to:

delta_a = -laplacian_b - c*b;
delta_b =  laplacian_a + c*a;
d = a*a + b*b + c;

The authors write

In our implementation, 'a' represents the real part of the complex number and 'b' the imaginary part. A fixed energy background is given by 'c'. The probability density is shown in 'd', with the energy barrier overlaid.

Other rules might be stable with bigger timesteps or more simulation steps - your mileage may vary. The examples under NumericalMethods give a quick overview of available low-order integration schemes, each with vastly different stability after tens of thousands of timesteps.

Aside

In case you are interested: I have asked and answered a question about a framework/software package for studying systems in the "cellular automaton" way here: https://softwarerecs.stackexchange.com/questions/77313/can-you-recommend-an-interactive-cellular-automata-2d-sparse-grid-visualization/77487#77487 I also wonder how you would best incorporate the collapse phenomenon into this kind of model. It seems it is not confirmed that collapse is really instantaneous phenomenon:

Experimental bounds on the collapse time are of order 0.1 ms to 0.1 ps https://iopscience.iop.org/article/10.1088/1742-6596/410/1/012153/pdf

so maybe we just need to propagate some extra information at one cell per step or so to propagate the collapse information (and make sure that if two collapses happen, only one of them prevails)...