Firstly, I want to appologize myself, for the fact that I don't very often provide a response that quotes sections throughout large posts such as the one you just did. The reason why is my sleep schedule is off the charts right now.. Somtimes I am just fuzzy tired like I am right now.. To be of no concern though. I am just currently not employed, and have all the free time to stay up late, or nap, or sleep in.. I've just kind of tossed routine out of the window, so I never really know when to expect that clear thinking time zone to pop up, where I can really get involved in a post.
Secondly,
Thanks for responding, and I follow quite clearly, but believe me, I will be reading it again, as I usually do read posts 2 or 3 times that are of interest to me.
Lastly, I wanted to share a bit of an interesting night last night that is related to this whole discussion. But just before I get to that, I thought I'd add a bit lead up background? I've been reading through the basic's here on these forums, checking my conclusions with what you people are presenting here. It was not even a month ago that I did not know the definition to epistemology and ontology. Nor have I really studied anything directed related to philosphy on purpose (ya know what I mean) (people calling me a philosopher before I even knew much about what that embodied, but none the less, one could just say it was my way of doing things before really being shown). Anyway, I came to many these conclusions, related to the epi and ontol concepts we are talking about here, just from my own observation and analysis and logical deduction. Thankfully however, these topics just recently opened me up to a language I could put these ideas into, and it really cleared up my ability to comprehend and see contrasts. Anyway, that is jut me blabbing a bit about myself and how I got to this stage. It has been a 3 or 4 year journey, just grasping it on my own time and methods..
Okay, so to the interesting night. As I have been browsing through, I was trying to start with the basics before moving into other things. So last night I decided it was about time to learn a bit bout, what exactly this "Schrödinger equation" was all about. So I got reading about it, and I know I've scanned through these things before on wiki, but I really dug in this time and tried to grasp it as much as possible.. but this time I really got a good hold on the material and I opened more link to learn a bit more about specific parts and terms, and I suddenly so much clicked in a way that it never had before.. all these things I've heard a thousand times, just as words describing something, but I had no sense of the essence of exactly all those words meant..
I thought about starting a topic sharing these topics that really interested me, but it mostly relates to here and what we are doing so I'll just post a brief summary of what material I was getting into.
Schrödinger equation - Wikipedia, the free encyclopedia
In physics, especially quantum mechanics, the Schrödinger equation is an equation that describes how the quantum state of a physical system changes in time. It is as central to quantum mechanics as Newton's laws are to classical mechanics.
In the standard interpretation of quantum mechanics, the quantum state, also called a wavefunction or state vector, is the most complete description that can be given to a physical system. Solutions to Schrödinger's equation describe not only atomic and subatomic systems, electrons and atoms, but also macroscopic systems, possibly even the whole universe. The equation is named after Erwin Schrödinger, who discovered it in 1926.[1]
Schrödinger's equation can be mathematically transformed into Heisenberg's matrix mechanics, and into Feynman's path integral formulation. The Schrödinger equation describes time in a way that is inconvenient for relativistic theories, a problem which is not as severe in Heisenberg's formulation and completely absent in the path integral.
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Main article: Theoretical and experimental justification for the Schrödinger equation
Einstein interpreted Planck's quanta as photons, particles of light, and proposed that the energy of a photon is proportional to its frequency, a mysterious wave-particle duality. Since energy and momentum are related in the same way as frequency and wavenumber in relativity, it followed that the momentum of a photon is proportional to its wavenumber.
DeBroglie hypothesized that this is true for all particles, for electrons as well as photons, that the energy and momentum of an electron are the frequency and wavenumber of a wave. Assuming that the waves travel roughly along classical paths, he showed that they form standing waves only for certain discrete frequencies, discrete energy levels which reproduced the old quantum condition.
Following up on these ideas, Schrödinger decided to find a proper wave equation for the electron. He was guided by Hamilton's analogy between mechanics and optics, encoded in the observation that the zero-wavelength limit of optics resembles a mechanical system--- the trajectories of light rays become sharp tracks which obey an analog of the principle of least action. Hamilton believed that mechanics was the zero-wavelength limit of wave propagation, but did not formulate an equation for those waves. This is what Schrödinger did, and a modern version of his reasoning is reproduced in the next section. The equation he found is (in natural units):
Using this equation, Schrödinger computed the spectral lines for hydrogen by treating a hydrogen atom's single negatively charged electron as a wave, \Psi(x,\,t)\;, moving in a potential well, V, created by the positively charged proton. This computation reproduced the energy levels of the Bohr model.
But this was not enough, since Sommerfeld had already seemingly correctly reproduced relativistic corrections. Schrödinger used the relativistic energy momentum relation to find what is now known as the Klein-Gordon equation in a Coulomb potential:
He found the standing-waves of this relativistic equation, but the relativistic corrections disagreed with Sommerfeld's formula. Discouraged, he put away his calculations and secluded himself in an isolated mountain cabin with a lover.[citation needed]
While there, Schrödinger decided that the earlier nonrelativistic calculations were novel enough to publish, and decided to leave off the problem of relativistic corrections for the future. He put together his wave equation and the spectral analysis of hydrogen in a paper in 1926.[2] The paper was enthusiastically endorsed by Einstein, who saw the matter-waves as the visualizable antidote to what he considered to be the overly formal matrix mechanics.
The Schrödinger equation tells you the behaviour of ψ, but does not say what ψ is. Schrödinger tried unsuccessfully, in his fourth paper, to interpret it as a charge density.[3] In 1926 Max Born, just a few days after Schrödinger's fourth and final paper was published, successfully interpreted ψ as a probability amplitude[4]. Schrödinger, though, always opposed a statistical or probabilistic approach, with its associated discontinuities; like Einstein, who believed that quantum mechanics was a statistical approximation to an underlying deterministic theory, Schrödinger was never reconciled to the Copenhagen interpretation.[5]
Wave function - Wikipedia, the free encyclopedia
A wave function or wavefunction is a mathematical tool used in quantum mechanics to describe any physical system. It is a function from a space that maps the possible states of the system into the complex numbers. The laws of quantum mechanics (i.e. the Schrödinger equation) describe how the wave function evolves over time. The values of the wave function are probability amplitudes — complex numbers — the squares of the absolute values of which give the probability distribution that the system will be in any of the possible states.
It is commonly applied as a property of particles relating to their wave-particle duality, where it is denoted ψ(position,time) and where | ψ | 2 is equal to the chance of finding the subject at a certain time and position.[1] For example, in an atom with a single electron, such as hydrogen or ionized helium, the wave function of the electron provides a complete description of how the electron behaves. It can be decomposed into a series of atomic orbitals which form a basis for the possible wave functions. For atoms with more than one electron (or any system with multiple particles), the underlying space is the possible configurations of all the electrons and the wave function describes the probabilities of those configurations.
Potential well - Wikipedia, the free encyclopedia
A potential well is the region surrounding a local minimum of potential energy. Energy captured in a potential well is unable to convert to another type of energy (kinetic energy in the case of a gravitational potential well) because it is captured in the local minimum of a potential well. Therefore, a body may not proceed to the global minimum of potential energy, as it would naturally tend to due to entropy.
Energy may be released from a potential well if sufficient energy is added to the system such that the local minimum is surmounted. In quantum physics, potential energy may escape a potential well without added energy due to the probabilistic characteristics of quantum particles; in these cases a particle may be imagined to tunnel through the walls of a potential well.
The graph of a 2D potential energy function is a potential energy surface that can be imagined as the Earth's surface in a landscape of hills and valleys. Then a potential well would be a valley surrounded on all sides with higher terrain, which thus could be filled with water (i.e., be a lake) without any water flowing away toward another, lower minimum (i.e. sea level).
In the case of gravity, the region around a mass is a gravitational potential well, unless the density of the mass is so low that tidal forces from other masses are greater than the gravity of the body itself.
A potential hill is the opposite of a potential well, and is the region surrounding a local maximum.
