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Fundamental problem with the Schrodinger's wave mechanics


For those who are interested in mathematical foundations of QM, here is an explanation by Prof. Akira Kanda on Fundamental problem with the Schrodinger's wave mechanics:

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So, why did Schrodinger put de Broglie's relation and Hamiltonian together to
achieve "particle-wave duality"? I can not find convincing scientific
reason for this.

1. Einstein and Planck showed a "successful" wave-particle duality of
light and photon. Einstein used relativistic energy-momentum to do this
[not knowing the contradiction coming from e=hf=0/0 which is
e=sqrt((cp)^2+(m0)^2c^4)=cp=m0cv/sqrt(1-(v/c)^2=(0/0)cv=c^2hhf].
This apparently "solved" the mystery of the double slit experiment for
light. Indeed, Schrodinger managed to "explicate" this using the
uncertainty principle for light/photon. [By the way, Suntora showed that
the blackbody radiation problem is resolved by considering the light wave
used to be monochromatic waves rather than harmonic waves. So, Planck's
quantization is not needed at all.]

2. De Broglie wanted to have the same duality for any particle to
explicate the double slit experiment. Following the steps of Planck who
quantized mathematical em waves, he tried to quantize mathematical plane
waves. In stead of  Einstein's relativization 0/0 of Planck's photon, he
relativized (transformed) wave "functions" not wave equations in terms of
wavenumber and frequency. [BTW this was a correct thing to do. As I have
been arguing for many years, wave functions and wave equations are not the
same thing. For example, Galilean transformation of wave functions are
wave functions but it is not the case for wave equations. But, I do not
see why it has to be Lorentz transformation rather than Galilean
transformation. The only reason seems to be because Einstein relativised
Plank's photon using Lorentz transformation.  0/0 comes from the gamma
factor which was the wrong thing to do.]

3. It turned out to be that the Lorentz transformation of plane wave
equation as "(wave number)-(wave frequency) relation" and that of
"energy-momentum relation" are mathematically the "same form. From this
"symbolic analogy", de Broglie concluded that "if there was a connection
between wave and particle, the following relation must be true:
                             p=(h/2pi)k and E=(h/2pi)f
where k is the wave number and f is the wave frequency." What is important
here is that de Broglie never said that there "is" a connection between
wave and particle. He said that "if" there is then it must be expressed as
the equations above. This is not to say that wave and particle are always
connected through the equations (de Broglie relation) above.

4. Schrodinger was looking for a mathematical formalism to connect
particles of Newtonian mechanics (not just a particle as de Broglie
presented) and desired wave representation of it. He represented classical
particles as Hamiltonian energy equations. Then using the hypothetical de
broglie relation namely p=(h/2pi)k, he converted Hamilton's energy
equation to the so called Schrodinger wave equation.

So, after all, Schrodinger's wave mechanics was developed upon "not the
fact but the assumption" that the wave particle duality exists in the form
of de Broglie relation. So, even today it is not clear if Schrodinger
really established a physical fact that wave-particle duality exists. He
just assumed the wave-particle duality and presented a mathematical
equation which describes such assumed duality.

This fundamental problem was never understood nor noticed. Physics
community was too busy with "harvesting " the fruits grown in the
forbidden garden. I do not see that Feynman-Landau-Lifschitz Lagrangian
formalism of QM resolved this very essential problem. Nobody really
understood what Schrodinger-Feynman-Landau-Lifschitz did including
themeselves. Worshipping of mathematical formulas is sticking its ugly
head as usual.

*********************************

Akira

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