I should check if ∫dt ln ψ(χ(t),t) is an action when ψ
is a solution of the existing Schrodinger equation at the limit
h→0.
If so, exp of an action multiplied by α is a solution of the
new grammar version of Schrodinger equation at the limit
α→∞.
The new grammar version of quantum mechanics should
become the existing quantum mechanics at the limit α→∞.
However the
new grammar version of quantum mechanics includes α only through the
factor h／α, and so we can not distinguish the limit α→∞ from the limit
h→0.
I will check if Φ defined as Φ[χ]＝exp[α∫dt ψ(χ(t),t)] is a
solution of the new grammar version of Schrodinger equation at the limit
α→∞, where ψ is a solution of the existing Schrodinger equation at the
limit h→0.
Fortunately, the ordinary Schrodinger equation has a
certain form at the limit h→0.
Yesterday, I thought that an action functional may be a
solution of the new grammar version of Schrodinger equation in the limit
α→∞ because an action functional does not include α.
Today, I tried
to prove it and failed.
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Last edited at
2010/06/21/16:28JST
Yesterday, I seemed to succeed in solving an equation
used to solve the new grammar version of Schrodinger equation by using
fourier expansion and polar coordinates in momentum
space.
Equation:
Solution:
where f is an arbitrary
function.
I am not confident of if this is correct because I have not
yet checked this solution well.
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Last edited at
2010/05/11/15:32JST
We might be able to prove that there must be some
physical law in any world by contradiction.
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Last edited at
2010/05/03/10:30JST