Eliezer: I thought your analogy to Conway’s Game of Life and the glider were brilliant. And in fact, you don’t take it far enough.
It’s a very intuitive example of position implying momentum, just as happens in QM. So you should do more examples of that.
For example: Let’s say there’s just a single photon, in 1D-land. It has some amplitude distribution for its position. Let’s say it’s mostly concentrated at some particular point, X. What does that amplitude distribution look like? Most people would probably naively guess some kind of bell-shaped curve around X.
But that can’t be right. Because such a curve is symmetric about X. Which wouldn’t indicate how the photon is moving. Yet photons are never found at rest. In 1D-land, they must be either traveling at c to the left, or c to the right. And, if position implies momentum, then the amplitude distribution of the photon at (“about”) X must be different, depending on whether the photon is traveling left or right.
So: you ought to do some examples like that. Here’s a sample amplitude distribution of a photon at X traveling to the right, here’s how it changes a time=t+1, etc. Meanwhile, here’s a different amplitude distribution of another photon also at X at time t=0, but traveling left. At a macroscopic level, we would interpret both to be identical in position (“there’s a photon near X”), but the actual amplitude modulations must be different in detail, in order to imply the different time evolution going forward.
In any case, I suggest you do more examples with that kind of comparison: diagrams of a glider in (2D) life evolving over time as the discrete case, alongside similar diagrams of a 1D photon evolving over time in the QM case.
Eliezer: I thought your analogy to Conway’s Game of Life and the glider were brilliant. And in fact, you don’t take it far enough.
It’s a very intuitive example of position implying momentum, just as happens in QM. So you should do more examples of that.
For example: Let’s say there’s just a single photon, in 1D-land. It has some amplitude distribution for its position. Let’s say it’s mostly concentrated at some particular point, X. What does that amplitude distribution look like? Most people would probably naively guess some kind of bell-shaped curve around X.
But that can’t be right. Because such a curve is symmetric about X. Which wouldn’t indicate how the photon is moving. Yet photons are never found at rest. In 1D-land, they must be either traveling at c to the left, or c to the right. And, if position implies momentum, then the amplitude distribution of the photon at (“about”) X must be different, depending on whether the photon is traveling left or right.
So: you ought to do some examples like that. Here’s a sample amplitude distribution of a photon at X traveling to the right, here’s how it changes a time=t+1, etc. Meanwhile, here’s a different amplitude distribution of another photon also at X at time t=0, but traveling left. At a macroscopic level, we would interpret both to be identical in position (“there’s a photon near X”), but the actual amplitude modulations must be different in detail, in order to imply the different time evolution going forward.
In any case, I suggest you do more examples with that kind of comparison: diagrams of a glider in (2D) life evolving over time as the discrete case, alongside similar diagrams of a 1D photon evolving over time in the QM case.