Linear Harmonic Oscillator and Probable Influence (PI) via Riccati
OsherD
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Linear Harmonic Oscillator and Probable Influence (PI) via Riccati
Copyright By Owner Osher Doctorow Ph.D.
First Publishjed 2005.
Marek Nowakowski and Haret C. Rosu, in "Newton's laws of motion in
form of Riccati equation," arXiv:physics/0110066 v2 18 Jan 2002
point out that the Riccati equation:
1) dy/dx = f(x)y^2 + g(x)y + h(x)
not only have many applications to supersymmetric QM, variational
calculus, nonlinear physics, thermodynamics, renormalization group
equations, etc., but that the change of function:
2) y = -(1/f)[d(log z)/dx - g/2]
converts (1) into:
3) Dxx(z) - (Dx(log f))Dx(z) - [g^2/4 - (1/2)Dx(g) + h - Dx(log f)]
z = 0
which is a common feature of many physics applications. (For those
impatient to see the Schrodinger equation one-dimensional conver-
sion to Riccati, see Y. Neurice (U. Iowa), "Arbitrarily accurate
eignevalues for one-dimensional polynomial potentials," arXiv:
quant-ph/0202047 v3 6 Sep 2002.)
Choosing f(x) = 1 identically in (1), we get the equation of the
quantum linear harmonic oscillator in (3) rather easily by properly
selecting g, h.
I have established on sci.stat.math and math-history-list (latter
of Math Forum) and elsewhere that the Riccati differential equation
is the equation of expansion/contraction type (growth) simultaneous
motion in many directions as opposed to the usual physics motion
of one direction at a time, and also that the Riccati differential
equation is the equation of Probable Influence (PI) or Probable
Causation:
4) PI = P(A-->B) = P{(AB')'} = P(A' U B) = 1 + P(AB) - P(A)
which some readers may notice is the subtractive "analog" of con-
ditional probability (Radon derivative) P(B|A) = P(AB)/P(A)
division when P(A) is not 0. However, PI is the more fundamental
"analog" since the set/event (A-->B) is a direct analog of the
logical conditional (not related to conditional probability)
(a-->b) = ~(a ^ ~b) = ~a V b. In other words, conditional
probability (used especially by Bayesians but adopted by quantum
physicists and information/AI theorists without being aware of
the probability issues) is less fundamental than PI (probable
influence).
Osher Doctorow