Origins: Brain and Self Organization

By Karl Pribram | Go to book overview
where δ is the Dirac delta function, then the solution of the equation L(y)=g(x) can be written in the form
y(x) = ∫ G(x′, x)*g(x′)dx′ (A.7)To illustrate the meaning of this seemingly pure formal trick we can imagine an arbitrary physical problem associated with a differential equation, e.g. eq. (2.12), where g(x) is a forcing term on the otherwise autonomous system, represented by the homogeneous L[y]= 0 equation. We can realize that
G(x′,x) *g(x′)dx′ (A.8)
represents a local concentrated influence of the forcing term. Consequently, the right hand side, the integral part, of eq. (2.21) represents the superposition of the localized/quantized influences. That is why the Green's function is often called the influence function.
Appendix B: Minimal Requirements for Quantum Mechanical Observables
Having now provided a generic quantum theoretical framework, two questions naturally emerge:
a. How can an observable be conserved (i.e., be a dynamical invariant)?
b. How can a conserved quantity be found?

To provide the fundamental ideas for answering the first question, a simplifying assumption is needed. Assume a time independent Hamiltonian H. Let F be an observable in the state ψ. If its value, 〈F〉 conserved, that is constant, then its time derivative

. (B.1)
should be equal to zero. Using the complex conjugate Schrödinger equation
. (B.2)
eq. (A.1) takes the form
. (B.3)

-92-

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