Archive for the ‘Quantum Mechanics’ Category

On Proving Eigenvalue = 1 for Particular Surfaces

March 21st, 2016 No comments

So, as you have read here, I've been saying for quite a few years now that (bounded) smooth surfaces on \left[ 0, 1 \right]^2 \to \mathbb{R} have certain invariants when viewed from a particular perspective (eigenvalues, eigenfunctions).  There are particular surfaces which I dubbed Pasquali patches (for lack of a better word) and a particular construction, namely

 p(x,y) = f_1(x) g_1(y) + f_2(x) \frac{1-g_1(y) F_1}{F_2}

with F_2 = \int_0^1 f_2(x) \, dx \neq 0 which are very special in that they have lots of properties which are interesting and closely tied to probability theory.  I have now proven for this particular construction that it possesses two very specific eigenvalues given a particular operator "star" \star... one of which is \lambda_1 = 1, regardless of function choices for f_1(x), g_1(y) and almost arbitrary choice of f_2(x), which, by requirement needs F_2 \neq 0.  This mimics well known probability mathematics (except in the surface realm) and operator theory/linear algebra. I think of this as a very proud accomplishment.

Thusly, I have revamped the relevant sections in Compendium full of new and juicy recharacterizations in order to be able to do just this... particularly Section 2, the definition of Pasquali patches and Section 11.6, Relevant Generalizations as Applied to Pasquali Patches (where I have included such a proof).  Section 2 is now a lot tighter than it used to be, and I'm trying really hard to go over everything to close all the loopholes I've left so that Compendium isn't just notes but an actual... Volume of Mathematics or Book or something.

I am very close to a full eigenvalue theory which I will apply to a generalization of the Pasquali patch formula above, giving a multiplicity of calculable eigenvalues for, not just Pasquali patches, but any sufficiently well behaved surface on \left[ 0, 1 \right]^2 \to \mathbb{R} .

Part I v26

More On Convergent-in-Area Polynomials in the Interval [0,1]

October 9th, 2014 No comments

So remember last time I wrote about the rich algebraic structure of polynomials that have convergent area in the interval [0,1]? Turns out there is a lot that can be done, even more than I imagined.  I have already begun to sketch the properties out and have a long ways to go still, but at least I have defined operations on these polynomials that I think can be very useful.

Here's my newest version of Compendium where I have added these things.

Part I v23

On convergent-in-area polynomials in the interval [0,1]

August 13th, 2014 No comments

Take any finite or infinite polynomial which converges in area in the interval between [0,1].  We can define equivalence classes on such space by mapping them back to distinct probability distributions, such that each equivalence class with all its elements is a semigroup under a particular operation.  We can extend each semigroup into a group if we incorporate an identity element and inverses.  Now we've created fun to last for a lifetime!  I've included these ideas in version 20 of Compendium.

Part I v20

There is more to come, as we spawn mathematical objects of weirdness that can be related back or extend probability theory.

V19 of Compendium

August 12th, 2014 No comments

I have added version 19 of Compendium! Seems like we've come a long way!  I feel like up to around page 35ish it is pretty solid.  Will still work on tying it to QM more directly.

Part I v19

On Calculating Entropy

June 10th, 2014 3 comments

When talking about Pasquali patches that converge and stationary surfaces that they converge to, we discussed that such would stabilize because it wouldn't loop in any shape or form nor would it diverge.  Such Claim we called Entropy.  We didn't provide a mechanism to calculate such. We do that now.

The way we propose we calculate the entropy is by measuring the deviation from the stationary surface.  Thus for the case where we have a Pasquali patch p_t(x,y), t \in \mathbb{Z}^+ we could measure entropy by:

 S(t) = \int_{[0,1]^2} \vert p_t(x,y) - p_\infty(x) \vert \, dA

that is, by taking the absolute deviation from the stationary surface (notice we use the area element dA).  Since in this case the time element is configured at integer times, rather than on continuous time, letting  t \in \mathbb{R} implies using the Pasqualian.  Thus we would have 

 S(t) = \int_{[0,1]^2} \vert p(x,y,t) - p_\infty(x) \vert \, dA

It seems fairly clear that 0 < S(t) < \infty , but here we must agree that 0 means that the system is stable and has the most entropy.

We may however, rather than the absolute difference from the stationary surface, choose to emphasize larger changes and (heavily) discount smaller ones.  Thus we could suggest

 S^{\square}(t) = \int_{[0,1]^2} \left( p(x,y,t) - p_\infty(x) \right)^2 \, dA

for t \in \mathbb{R}.