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New grammar version of Schrodinger equation
Uda's School / Yuuichi Uda
About a system with one degree of freedom, I want to find an equation obeyed by Φ in the new grammar, which represents a quantum history whose quantum state at time t is represented by wave function ψ(□,t) by the following functional Φ.
Φ[χ]=exp[α∫dt φ(χ(t),t)]; ψ(x,t)= exp φ(x,t)・・・※
As a clue, it seems plausible course to select the equation for Φ as it reduces to the ordinary schrodinger equation for ψ in the special case that ※ holds.
Along this course, I tried making the following equation.
(ih／2πα) lim (1／ε)(Φ[χ']－Φ[χ])
I have not yet confirmed whether this equation always reduces to Schrodinger equation in the special case ※.
Besides, I got some undesirable result when I investigated a solution corresponding to the ground state of a harmonic oscillator.
It is the result that
is a solution when
Adding an arbitrary constant to potential energy does not change what physical system we treat, and so it may not be undesirable, but the appearance of the value of δ-function at zero point might mean failure of the theory in the worst case.
In order that V(x)＝kx^2, perhaps we have only to formally select
using a real constant E.
However, in this case, too, perhaps E includes δ(0), and we should understand that integral ∫dt[－iEt] is not well defined or is zero, and so room for criticism is left.
It might be a way for settling these problems that renew χ'(t)＝χ(t－ε).
For example, how about using an arbitrary function f such that f(t)＝t if t＜a or t＜b, selecting χ'(t)＝χ(f(t)), and altering the Hamiltonian part after it?
In order to avoid being criticized for the point that ψ(x,t) has a dimension of (length)^(-3/2) despite that φ(x,t) is dimensionless, we have only to reselect ψ(x,t)＝βexpφ(x,t) using a dimensionful factor β.
Last edited at 2010/05/01/15:33JST
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