Just want to know the mathematics involved behind this

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abhinavkumar_iitr05

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While going through the wave particle duality I came across the <font color="orange">UNCERTINITY PRINCIPLE</font>Now I just want to know the mathematics involved behind this. <br />
 
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yevaud

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http://www.aip.org/history/heisenberg/p08b.htm<br /><br />Start with this - it's a good if basic background of Werner Heisenberg, and discusses some of the underlying mathematics behind the Uncertainty Principle. <div class="Discussion_UserSignature"> <p><em>Differential Diagnosis:  </em>"<strong><em>I am both amused and annoyed that you think I should be less stubborn than you are</em></strong>."<br /> </p> </div>
 
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newtonian

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Yevaud - Thank you for the link. However, after considering the math of Heisenberg's postulated model, the link goes on:<br /><br />"The true quantum interaction, and the true uncertainty associated with it, cannot be demonstrated with any kind of picture that looks like everyday colliding objects. To get the actual result you must work through the formal mathematics that calculates probabilities for abstract quantum states. Clever experiments on such interactions are still being done today. So far the experiments all confirm Heisenberg's conviction that there is no "real" microscopic classical collision at the bottom." <br /><br />OK, you get the real picture by working out abstract states? That is confusing!<br /><br />OK, the link says what the current experiments do not show.<br /><br />But what actual math do the experiments show as to what is really going on in atomic physics and the probability of positions of, for example, electrons?<br /><br />For example, are they true orbitals or not actually in orbit? What is the angular momentum for example?<br /><br />I know it is not the same as planetary orbits, not at all.<br /><br />But what is it and how are the orbitals determined mathematically?<br /><br />I also know the shapes of orbitals get really complex with the more complex orbitals.
 
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newtonian

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Yevaud - Here is some background for my extension of the originial question.<br /><br />First, on angular momentum:<br /><br />Neils Bohr worked out a simple model for quantum mechanics in electron orbits.<br /><br />He quantized the angular momentum of electrons, submitting that electrons can only have angular momentums equal to Planck's constant multiplied by an integer (a whole number, not a fraction) and then divided by the geometrical constant (= 2 x pi).<br /><br />This worked out to only specific allowed radii for the electron in orbitals around the nucleus. And these correlated with specific energy levels corresponding to the angular momentum of the electrons in the orbitals.<br /><br />However, the math was overly simplistic and works out well only for the simple hydrogen atom!<br /><br />But then come Broglie waves and the particle wave phenomenon - for my next post as I try to fathom the math involved....
 
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newtonian

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abhinavkumar_iitr05 - I'm gonna try to answer your, and now my, question. First this simple start.<br /><br />If particles like electrons were indeed simply particles, the math would be relatively easy, as Bohr demonstrated.<br /><br />But particles are not simply particles - they are also waves.<br /><br />In 1927 Clinton Davison and Lester Germer directed a beam of electrons at the suface of a crystal of Nickel and noted diffraction effects explained only if electrons were waves rather than particles. <br /><br />Specifically, there was a diffraction pattern centered (peaking) at an angle of 50 degrees, but scattered at various other angles.<br /><br />Now similar diffraction effects of neutron beams are used in study of structure and properties of matter.<br /><br />And then comes "wave mechanics." Please pardon me while I study more before responding further.<br /><br />However, it is obvious some uncertainty would exist as to exactly where any specific electron would be due to the wave nature of particles.<br />
 
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newtonian

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abhinavkumar_iitr05 - OK, part of the math would involve the wave function.<br /><br />I'll let someone else post on that - except to say that the electron would have a set of probable locations mathematically proportional to the wave function of said electron wavelength.<br /><br />And, of course, the wavelength of an electron corresponds to its energy state and angular momentum.<br /><br />Actually, each electron orbital corresponds to a specific set of quantum numbers.<br /><br />Added to this set based on angular momentum, there is also the added complexity of spin.<br /><br />Electrons (and protons, etc.) have spin - electron and proton spin are 1/2. Spin plus charge produces magnetic fields, and now we have the added complexity of numerous interacting magnetic fields - notably those of electrons and protons which both involve charge in motion and therefore resulting magnetic fields.<br /><br />The spin of an electron describes what is called "fine structure." <br /><br />And then there is "superfine structure." <br /><br />Please excuse while I study further....
 
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newtonian

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abhinavkumar_iitr05 - The math is getting more complex - beyond me. <br /><br />I can at least post why it is beyond me (so far):<br /><br />Obviously, the location of the electrons will govern the location of magnetic field lines. For a simple hydrogen atom with one electron - not so hard.<br /><br />But with 10, 20, or higher individual electron magnetic fields interacting with 10, 20, or higher proton magnetic fields, plus influence of neutrons (they are made up of quarks with spin whose charges cancel out overall, but....) you can see the math would get really complex!<br /><br />Of course, one can simplify by assuming a nucleus has a simple net spin and charge and magnetic field - but it is likely far more complex than that.<br /><br />It involves superfine structure. <br /><br />"The electron is not unique in possessing spin; the protons and neutrons in the nucleus also each have a spin of 1/2. This means that the nucleus itself can have a net spin angular momentum, depending on how the individual contributions from the protons and neutrons add together. The effect of the quantization of nuclear spin can be detected in atomic spectra as hyperfine structure, or spittling on scales a thousand times smaller than the fine structure splitting due to the electron's spin." - "The World of Science," by Andromeda Oxford Ltd., 1991, Volume 13, p. 93.<br /><br />Obviously, electron location, and therefore heisenberg's uncertainty principle, would involve both fine structure and superfine structure caused by spin angular momentum of all the collective particle-waves in the specific atom studied.<br /><br />This is in addition to the uncertainty due to wave function due to particle wave duality. <br /><br />Please note that I have gone way beyond Heisenberg's uncertainty principle as first promulgated by Heisenberg (compare Schrodinger's theory which leads to the same math result). <br /><br />As the first link noted, the Heisenberg uncertainly principle is simply caused by the means o
 
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newtonian

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abhinavkumar_iitr05 - Ok, the uncertainty has limits - it is an limited set of nearly infinite possibitlities (geometrikcally finite, compare different types of infinities). <br /><br />One reason is that electron wavelengths have limits, albeit the exact position on said wavelength can have incredible variety within these limits - compare genetic variation within a kind or genetic limits)<br /><br />Erwin Schrodinger originally submitted a wave equation for particles which can then be used to map out the limits of a volume of space - and this volume, with specific mathematical (geometrical) shape, is the specific electron orbital. <br /><br />With, as noted in above posts, more complex variables.<br /><br />I will consider in this post 4 types of orbitals: <br /><br />s (sharp) - shape: spherical<br /><br />p (principle) - shape: twin lobes or dumb-bell. In a hydrogen atom there are 3 possible p orbitals which lie on a cartesian graph perpendicular to each other in the x, y, z axes coordinates.<br /><br />Note, btw, that two electrons can occupy the same orbital volume limits if, and apparently only if, their spins are 1/2 in opposite rotational angular momentum!<br /><br />Why this is is beyond me, btw.<br /><br />d orbitals: diffuse, four leaf clover shape OR hour-glass and ring. There are 5 d-orbitals calculated (as of 1991) and they are strongly directional. <br /><br />f orbitals: there are 7 f orbitals, but I'll let someone else describe their shape - it is beyond me (so far)!<br /><br />A simple note to this very complex mathematical pursuit: there are limits to the uncertainty. This is why the properties of the elements and compounds studied in chemistry have specific properties that are not very variable at all.<br /><br />To understand this you need to understand the laws of probability and statistics.<br /><br />Especially, the law of large numbers. Any sample of an element involves a large number of atoms, and any reaction medium in chemical reactions involves a large num
 
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