A Quote by Erwin Schrodinger
In this communication I wish first to show in the simplest case of the hydrogen atom (nonrelativistic and undistorted) that the usual rates for quantization can be replaced by another requirement, in which mention of "whole numbers" no longer occurs. Instead the integers occur in the same natural way as the integers specifying the number of nodes in a vibrating string. The new conception can be generalized, and I believe it touches the deepest meaning of the quantum rules.
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Nature does not count nor do integers occur in nature. Man made them all, integers and all the rest, Kronecker to the contrary notwithstanding.
There can never be two or more equivalent electrons in an atom, for which in a strong field the values of all the quantum numbers n, k1, k2 and m are the same. If an electron is present, for which these quantum numbers (in an external field) have definite values, then this state is 'occupied.'
Arithmetic starts with the integers and proceeds by successively enlarging the number system by rational and negative numbers, irrational numbers, etc... But the next quite logical step after the reals, namely the introduction of infinitesimals, has simply been omitted. I think, in coming centuries it will be considered a great oddity in the history of mathematics that the first exact theory of infinitesimals was developed 300 years after the invention of the differential calculus.
Quantum mechanics brought an unexpected fuzziness into physics because of quantum uncertainty, the Heisenberg uncertainty principle. String theory does so again because a point particle is replaced by a string, which is more spread out.
If we apply the term revolution to what happened in North America between 1776 and 1829, it has a special meaning. Normally, the word describes the process by which man transforms himself from one kind of man, living in one kind of society, with one way of looking at the world, into another kind of man, another society, another conception of life.... The American case is different: it is not a question of the Old Man transforming himself into the New, but of the New Man becoming alive to the fact that he is new, that he has been transformed already without his having realized it.
The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules... One might therefore conjecture that these axioms and rules of inference are sufficient to decide any mathematical question that can at all be formally expressed in these systems. It will be shown below that this is not the case, that on the contrary there are in the two systems mentioned relatively simple problems in the theory of integers that cannot be decided on the basis of the axioms.
From the results so far obtained it is difficult to avoid the conclusion that the long-range atoms arising from collision of alpha particles with nitrogen are not nitrogen atoms but probably atoms of hydrogen, or atoms of mass 2. If this be the case, we must conclude that the nitrogen atom is disintegrated under the intense forces developed in a close collision with a swift alpha particle, and that the hydrogen atom which is liberated formed a constituent part of the nitrogen nucleus.
The page of my notebook was filled with many messy integrals, but all of a sudden I saw emerge a formula for counting. I had begun to calculate a quantity on the assumption that the result was a real number, but found instead that, in certain units, all the possible answers would be integers. This meant that areas and volumes cannot take any value, but come in multiples of fixed units.
The beauty of string theory is the metaphor kind of really comes very close to the reality. The strings of string theory are vibrating the particles, vibrating the forces of nature into existence, those vibrations are sort of like musical notes. So string theory, if it's correct, would be playing out the score of the universe.
The trouble with integers is that we have examined only the very small ones. Maybe all the exciting stuff happens at really big numbers, ones we can't even begin to think about in any very definite way. Our brains have evolved to get us out of the rain, find where the berries are, and keep us from getting killed. Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions.
The central idea of string theory is quite straightforward. If you examine any piece of matter ever more finely, at first you'll find molecules, atoms, sub-atomic particles. Probe the smaller particles, you'll find something else, a tiny vibrating filament of energy, a little tiny vibrating string.
Integers are the fountainhead of all mathematics.
The series of integers is obviously an invention of the human mind, a self-created tool which simplifies the ordering of certain sensory experiences.
The best theory comes from string theory, which states that dark matter is nothing but a higher vibration of the string. We are, in some sense, the lowest octave of a vibrating string.
There are three distinct kind of judges upon all new authors or productions; the first are those who know no rules, but pronounce entirely from their natural taste and feelings; the second are those who know and judge by rules; and the third are those who know, but are above the rules. These last are those you should wish to satisfy. Next to them rate the natural judges; but ever despise those opinions that are formed by the rules.
In quantum mechanics there is A causing B. The equations do not stand outside that usual paradigm of physics. The real issue is that the kinds of things you predict in quantum mechanics are different from the kinds of things you predict using general relativity. Quantum mechanics, that big, new, spectacular remarkable idea is that you only predict probabilities, the likelihood of one outcome or another. That's the new idea.
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