Alpha Institute for Advanced Studies

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.

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

F psi = E d psi / dx

a176thpaper.pdf

partial (H hat psi) / partial x = i h bar d (partial psi / partial x) / dt

with
H hat psi = H psi = E psi

E = H = T + V

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

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.

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.

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