Senin, 27 Maret 2017

Heisenberg's Matrix Theory of QM

Heisenberg's Matrix Theory of Quantum Mechanics

1. Heisenberg was unhappy about the quantum mechanics of his time which
was based upon the measurements of frequency and location which are not
measurable. So, he launched the formalism of QM which is based upon
measurable quantities. This move was consistent with the trend of
positivism and instrumentalism among his contemporaries.

2. Heisenberg looked for an answer to this problem in the failed Bohr's
correspondence principle. According to this principle of Bohr, there is a
close connection between the quantum transition (in old QM) of an
oscillating system and the Fourier components of the oscillating motion in
the sense of classical mechanics. However, this correspondence gives the
right answer only when the quantum number is large.

3. As a solution to this shortfall of Bohr's correspondence principle,
Heisenberg tried to find a new correspondence principle which yields
correct predictions for all quantum numbers.

4. In the end, Heisenberg came up with desirable infinite matrix
representation of physical quantity and coordinate using Fourier
components with which he converted Hamiltonian equation of motion into
infinite matrix version of canonical equation of motion:

dp/dt = [(2(pi)i)/h](pH-Hp)
dx/dt = [-2(pi)i/h](xH-Hx).

The first equation describes the quantum systems as Hamiltonian operator.
The second equation describes the canonical equation of motion. With them
we calculate the absolute values of the amplitude of p and x. Also we
calculate the frequency of the phase of p and x.

5. Furthermore, Heisenberg showed "empirical equivalence" of his QM and
Schrodinger's QM. Physicists argue that, these two major theories one
coming from the Hamilton's classical equation plus Bohr's correspondence
Principle and the other coming from Hamiltonian classical equation plus de
Broglie's relation are empirically equivalent, establishes that some thing
at least empirically intrinsic is achieved

6. Again, after all, all of this is invalid as to begin with the classical
energy concept is invalid to begin with. Moreover, Schrodinger's wave
mechanics is invalid as discussed in [1] and [2] of this Part. One may
question if Correspondence Principle and Pilot Wave Theory are equivalent.
Considering that the Pilot wave theory is logically inconsistent, we must
conclude that this question is meaningless as by virtue of inconsistent
theory this theory can imply any theory.

7. What about the question of Heisenberg's QM itself? Putting the issue of
energy concept aside, it did not use de Broglie's relation. So, may it be
the case that it means something as the only problem with it seems to be
that of correspondence principle? To my recollection, Bohr used the
photon/(em wave) duality for his early quantum mechanics. Without this
false assumption, QM does not exist.

[5] Photon/(em wave) Duality Revisited

We may add more problems to the claimed duality between photons and
em waves. According to the Uncertainty Principle, a particle can not
be localized twice. This makes it impossible to do Michelson Morley
Experiment. The photon is localized at the mirror and then at the
emitter of the photon.


Akira

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Victor Christianto
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