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

---

Victor Christianto

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