Re: Quaternion problem--> quantum differeentials
- To: mathgroup at smc.vnet.net
- Subject: [mg68069] Re: Quaternion problem--> quantum differeentials
- From: Roger Bagula <rlbagula at sbcglobal.net>
- Date: Fri, 21 Jul 2006 17:36:07 -0400 (EDT)
- References: <e7ap5q$952$1@smc.vnet.net> <e7lf1d$3l9$1@smc.vnet.net> <e7no3j$97f$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Here is an effort to get a space-time polar quantum LaPlacian.
Quantum numbers in the q^4 powers result from a hyper-plate
type of vibrational equation.
The result doesn't appear normalizable in simple terms.
Dividing out :
MatrixPower[qr.qrs, -1]
would give a Klein-Gordon type factor.
Table[FullSimplify[ExpandAll[dqr[[j]].dqrs[[j]]]*MatrixPower[qr.qrs,
-1]], { j, 1, 4}]
n and o quantum number factors have a complex wave function term which
if solves as unity
in each case tends to give quantum number angular interdependence.
A radial Chladni type vector on the "plate" as: qr0 as the base
Rodrigues quaternion
psi=(Cos[l*r0]+Cos[m*r]+Cos[n*t]+Cos[o*p])*qr0
might represent standing waves for a four quantum model such as this.
Mathematica:
Clear[a, m, m1, qr, dqr, qrs, dqrs, v, w, x, y, z, t, q, i, j, k]
<< Algebra`Quaternions`
i = {{0, 1}, {-1, 0}};
j = {{0, I}, {I, 0}};
k = {{I, 0}, {0, -I}};
e = IdentityMatrix[2];
q[t_, x_, y_, z_] := e*t + x*i + j*y + k*z;
(* Rodrigues quaternion : quantum number functional
definition : r0 radial distance/ Bessel - like,
r time angle, {t, p} polar sphere angles*)
qr = Exp[l*r0]*q[Cos[m*r], Sin[m*r]*
Sin[n*p]*Cos[o*t], Sin[m*r]*Sin[n*p]*Sin[0*t], Sin[m*r]*Cos[n*p]];
qrs = Exp[l*r0]*q[Cos[m*r], -Sin[m*r]*Sin[n*p]*Cos[o*t], \
-Sin[m*r]*Sin[n*p]*Sin[0*t], -Sin[m*r]*Cos[n*p]];
(* Linear vector differential definition *)
v = {r0, r, p, t};
(* space - time polar differential factors defined: not derived directly*)
w = {1, 1/Exp[2*r0], 1/Exp[2*r0], 1/(Exp[2*r0]*Sin[t]^2)};
dqr = Table[FullSimplify[w[[i]]*D[qr, {v[[i]], 2}]], {i, 1, 4}]
dqrs = Table[FullSimplify[w[[i]]*D[qrs, {v[[i]], 2}]], {i, 1, 4}]
Table[FullSimplify[ExpandAll[dqr[[j]].dqrs[[j]]]], {j, 1, 4}]
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