The Hamiltonian and Quantum Mechanics

Feed: Dr. Myron Evans
Posted on: Monday, March 28, 2011 11:32 PM
Author: metric345
Subject: The Hamiltonian and Quantum Mechanics

Quantum mechanics derives directly from the hamiltonian of Rowan Hamilton:

H = T + V = E

This gives

H hat psi = E psi

which is Schroedinger’s equation with the axiom

p hat psi = – i h bar partial psi / partial x

where psi is a function operated upon by p hat. So

T hat psi = – h bar squared / (2m) partial sup 2 psi / partial t squared

so

H hat psi = (T hat + V) psi = E psi

where T hat is a second partial differential and V and E simply multiply psi.

I am not sure how many students really understand this. My Ph. D. supervisor admitted that he did not, after many years of lecturing on it. So how could his students have really understood it? The new quantum Hamilton equations and new force equation go much deeper than Schroedinger and Heisenberg ever did. So there is plenty of scope for grant applications just on UFT 176 and 177 alone, never mind all the other work going back to 1973. That is if you are minded in that way. There is nothing wrong with making grant applications, but I am basically a problem solver and it is essential to work almost full time on that in order to make real progress. So grant money to AIAS (as it fully deserves) must come in via organization of fund applications by other members of AIAS. Grant money in science usually comes in for fashionable trends – just as in any walk of life.

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177(2): Zero’th Force Eigenvalue of the Harmonic Oscillator

Feed: Dr. Myron Evans
Posted on: Wednesday, March 23, 2011 4:27 AM
Author: metric345
Subject: 177(2): Zero’th Force Eigenvalue of the Harmonic Oscillator

The new force equation of quantum mechanics is eq, (1), and the harmonic oscillator gives a zero point force eigenvalue:

F sub 0 = – k x

This is the classical result of Hooke’s law, meaning that the well know zero point energy:

E sub 0 = h bar omega / 2

is accompanied by a hitherto unknown zero point force which happens to have the classical value of Hooke’s law. Eq. (1) is a new fundamental equation of quantum mechanics and can be applied to any problem. I suggest that Horst and I apply it in UFT 177 to the first few wavefunctions of the harmonic oscillator and the first few radial functions of the H atom to make the first exploration of force eigenvalues. Eq. (1) has an unlimited number of applications in quantum mechanics and derivative subject areas of QM such as quantum optics and quantum field theory. My hand calculations of this note can also be checked as usual by computer algebra. The Casimir force originates in F sub 0.

a177thpapernotes2.pdf

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176(1): Quantum Hamilton Equations for Planar Rotation

Feed: Dr. Myron Evans
Posted on: Thursday, March 10, 2011 8:12 AM
Author: metric345
Subject: 176(1): Quantum Hamilton Equations for Planar Rotation

These are the quantum Hamilton equations for planar rotation. Once the hamiltonian is defined the equations of motion follow in such a way that x and p or phi and p sub phi are independent variables:

dx / dp = dp / dx = 0

In lagrangian dynamics, q and q dot are not independent variables as is well known.The classical Hamilton dynamics have well known advantages and all of these advantages hold for the newly discovered quantum Hamilton equations. So I will devote some time to their systematic development. Note carefully that the first QHE (eq, (4)) of the attached, and the second QHE, hold for any operator with property

A hat psi = A psi

an important special case is the hamiltonian operator:

H hat psi = H psi

i.e. the equations hold for any operator A hat of quantum mechanics and are therefore completely general in applicability.

a176thpapernotes1.pdf

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note 175(15)

Feed: Dr. Myron Evans
Posted on: Thursday, March 10, 2011 3:18 AM
Author: metric345
Subject: note 175(15)

To Dr. Horst Eckardt:

Thanks for going through the calculations. As can be seen from page 2 of note 175(14), left hand column, I followed Peter Atkins, “Molecular Quantum Mechanics”, in doing this calculation, applying the Leibniz Theorem to the product of wavefunctions psi * psi, and not to the operator. The tautology comes from the fact that:

<x> = x; x hat psi = x psi

in the position representation. In the momentum representation:

<p> = p, p hat psi = p psi

In fact one does not need the integrals to derive the quantum Hamilton equations. They follow immediately from the tautologies:

d<x> / dx = 1 ; d<p> / dp = 1.

and
[x hat, p hat] psi = i h bar psi

<[x hat, p hat]> = i h bar

So

d<x> / dx = i h bar / (i h bar) = 1 = <[x hat, p hat]> / (i h bar)

Q.E.D. Finally generalize x hat to any operator A hat to get the new equations of motion.

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Plans for Opening Chapter of “The Fermion Equation”

Feed: Dr. Myron Evans
Posted on: Monday, March 07, 2011 6:47 AM
Author: metric345
Subject: Plans for Opening Chapter of “The Fermion Equation”

To open the book a detailed derivation will be given of the Lorentz transform, based on previous notes, because this transform defines the relativistic momentum which is the Einstein energy equation. The realtivistic momentum p = gamma m v is needed for conservation of momentum in special relativity. Schroedinger’s axiom will be used to derive the wave format of the fermion equation from the Einstein energy equation, which is the relativistic momentum in another guise. After this groundwork the wave fermion equation will be derived from the tetrad postulate of geometry, and its wavefunction recognized as a 2 x 2 tetrad. Then form chapter two onwards the fermion equation will be developed as in UFT 172 ff. A lot of important progress has been made recently in ECE theory and all of it is available pen source on www.aias.us in comprehensive detail. There is essentially nothing left of the physics standard model from about nineteen thirty onwards for any serious intellectual. The physics that really stands up to scrutiny is that up to about nineteen thirty.

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The Quantum Hamilton Equations

Feed: Dr. Myron Evans
Posted on: Monday, March 07, 2011 2:41 AM
Author: metric345
Subject: The Quantum Hamilton Equations

To Dr. Horst Eckardt:

Thanks as usual for going through this note. I will give the final proof with more detail in the next note. I followed Atkins in the definition of hermiticity, his eqs. (5.2.1) to (5.2.6) of the second edition. The result is right because it is self checking. I will go through the Atkins definition and expand it. Atkins writes that his eq. (5.2.2) is obtained from the basic definition of hermiticity, his eq. (5.2.1), by taking the complex conjugate of each term (page 88 of the second edition of “Molecular Quantum Mechanics”, it should also be in your edition). It is right because he uses it in deriving the time evolution equation of quantum mechanics, his eq. (5.5.2), second edition. The other quantum Hamilton equation is obtained using the momentum representation instead of the position representation, and the tautology:

d<p> / dp = 1

The Hamilton equations are the expectation values of the quantum Hamilton equations. So this result shows that x and p can be observable simultaneously in quantum mechanics and that the whole of Copenhagen is incorrect. This is because x and p are simultaneously observable in the classical Hamilton equations, derivable from the quantum Hamilton equations. I am not quite satisfied with the proof as yet, but the result is right, self checking in at least two ways. The final version of the proof will be sent in the next note.

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The Second Hamilton Equation of Motion in Quantum Mechanics

Feed: Dr. Myron Evans
Posted on: Sunday, March 06, 2011 5:46 AM
Author: metric345
Subject: The Second Hamilton Equation of Motion in Quantum Mechanics

This is derived using the tautology d<p> / dp = 0, and the proof will be given in the next note before finally discussing the effect of relativity on the Heisenberg uncertainty principle. The latter is irrational and unscientific, so in a sense, logic cannot cure irrationality. This is why the tedious arguments about the interpretation of quantum mechanics have dragged on endlessly. However some details will be given for the final note of UFT 175. These notes will also be used for “The Fermion Equation” with tables of expectation values by Horst Eckardt’s computer algebra, tables that refute the Heisenberg uncertainty principle completely, thoroughly, and for the first time. The refutation of the uncertainty principle makes no difference at all to science because it is in fact unusable, being non-Baconian. Experiments that claim to verify the uncertainty principle will fail if studied in enough depth and with enough care. The product of the root mean square deviations from the mean of x and p, denoted delta x delta p, is a statistical property. This was Schroedinger’s original interpretation. The Langevin equation for example is a statistical equation describing the Brownian motion, no one has ever claimed that Brownian motion is “unknowable”. It was discovered by Robert Brown, a botanist and another of my Civil List predecessors. It was first explained by Einstein in 1905, and shortly thereafter by Langevin.

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175(14): A New Equation of Motion of Quantum Mechanics

Feed: Dr. Myron Evans
Posted on: Sunday, March 06, 2011 5:30 AM
Author: metric345
Subject: 175(14): A New Equation of Motion of Quantum Mechanics

This is Eq. (9) of the attached note, and produces the quantum equivalent of one of Hamilton’s equations of motion, Eq. (22). The Schroedinger postulate is derived for the first time from a tautology:

d<x> / dx = 1 , where <x> = x

given the definition of expectation value in quantum mechanics. Therefore [x, p] psi = i h bar psi is also a tautology which should not be elevated incorrectly into a principle in the Copenhagen manner. A tautology is a statement of the self-evident, in this case dx / dx = 1. Sir William Rowan Hamilton was one of my predecessors on the Civil List, and Astronomer Royal for Ireland, professor and fellow of Trinity College, Dublin, where I am sometime visiting academic.

a175thpapernotes14.pdf

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