Archive for November, 2013

The Equations of Motion

Wednesday, November 27th, 2013

The equations of motion for a volume element of fluid, written in Exterior Calculus form, using a system of units which unitizes the distracting arbitrary constants such as the speed of sound, can be written (and translated)  as :

-"eqns

molecular displacement acceleration = opposite of the gradient of pressure change

pressure variation over time is opposite to the divergence of the molecular displacement  velocity

density variation over time is opposite to the divergence of molecular displacement velocity

Now you may well wonder why the mathematical form of these expression seems so opaque when it describes things so easy to say in English, such as the fact that when you crowd air molecules together, the density and pressure goes up, and vice versa.

One answer might be that to those who wish to formulate mathematical descriptions, such descriptions are only acceptable if they are sufficiently detailed to describe and account for all possible situations.  In other words, while “crowding raises the pressure” seems so obvious as to be trite, making this expression into a mathematical equation — involving vector expressions such as gradients and divergences expressed in some form of calculus or differential equation — makes the expression more useful in practical applications where it can be applied to unusual situations and unique geometries.

— MRK

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Modal Logic and Quantum Mechanics

Wednesday, November 27th, 2013

meAnd here I am again.  Yes this is me today. It’s too late to hide now.

Today I’d like to talk about modal logic. My introduction to symbolic logic and the predicate calculus began in a philosopy & logic class in the late Seventies. But that was merely true-false logic.

Modal logic is a generalization of simple logic, going beyond whether statements are true or not, but whether it is possible for them to be true, or necessary that they be true.

modal1

The combination of these operators yields some interesting theorems:

modal2

If this sounds more like philosophy (epistemology) as well as Quantum Mechanics, then perhaps it is. For example, one way to state the principle of “mixed states” such as we find in the “uncollapsed state vectors” might be the following:

modal3

Conversely, if one passes a beam of spinning particles such as electrons through a magnetic field so as to cause the “spin up” particles to deflect one way and the “spin-down” particles to deflect the other way, then the states are no longer mixed. In this case, looking at either one of the diverged result beams, we find that it is no longer possible to observe both “up” and “down” states:

modal4

More on this later –MRK

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