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As spin quantum number cannot be derived from Schrödinger's equation, it cannot predict opposite electron spin. I mean to ask that how do we obtain the information conveyed by the spin quantum number when we don't speak of it in this theory.

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    $\begingroup$ As far as I know, in nonrelativistic quantum mechanics the existence of spin is a postulate, i.e. we assume it's there (and the reason for assuming it's there is because we saw it). $\endgroup$ – orthocresol Mar 4 '17 at 18:23
  • $\begingroup$ Regarding your edit, exactly what information are you talking about? The idea is that we postulate that spin is a form of angular momentum with quantum numbers 0, 1/2, 1, ... Angular momentum is a topic that is very well-described by the Schrodinger equation, and spin is nothing but a form of angular momentum, so all the rules that apply to spin quantum numbers are simply derived from the theory of angular momentum. This includes the Pauli exclusion principle which can also be derived in the non-relativistic picture. The only postulate that needs to be made is the existence itself of spin. $\endgroup$ – orthocresol Mar 4 '17 at 19:16
  • $\begingroup$ and once we assume (for example) that an electron has spin-1/2, it automatically follows from the rules of angular momentum that it possesses "up" and "down" states, because the allowed values of $m_s$ for a particle with $s = 1/2$ are exactly $+1/2$ ("up") and $-1/2$ ("down"). For more information on angular momentum you might want to consult Atkins Molecular Quantum Mechanics chpt 4 although any decent QM textbook will have a section on it. $\endgroup$ – orthocresol Mar 4 '17 at 19:19
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Good question.

This was one indication that there had to be something more than just the Schrodinger equation. Another problem which was perceived quite quickly is that the Schrodinger equation treated time differently than the spatial dimensions which is not in the spirit of special relativity. Thus, Paul Dirac constructed what we now call the Dirac equation. When you solve the Dirac equation in a Coulomb potential, spin, and hence the fine structure of the hydrogen atom energy levels in an electric field, falls out naturally.

This proves to be fairly technical and I won't try to do any of the mathematics for this (I don't completely understand it all), but you can find solutions to the Dirac equation for some common cases here.

As orthocresol points out, we first observed the intrinsic angular momentum of the electron in the Stern-Gerlach experiment, so people guessed what this would look like at first (this is how it was used in the Schrodinger equation) and Dirac came along and explained that people had guessed correctly. This can also be seen from the fact that the Dirac equation reduces to the Schrodinger equation in the non-relativistic, low energy case.

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  • $\begingroup$ Why do we keep using the term non relativistic for schrodinger equation? $\endgroup$ – lily May 21 '17 at 9:53
  • $\begingroup$ @lily The Dirac equation can be thought of as a relativistic version of the Schrodinger equation. By relativistic, I mean relating to the theory of special relativity. That is, it is accurate when particles are moving close to the speed of light. $\endgroup$ – jheindel May 21 '17 at 18:48
  • $\begingroup$ That might seem stupid..but as far as I understand it, electrons can never get their speeds close to that of light..so why do we need to have a relativistic picture of that $\endgroup$ – lily May 23 '17 at 19:18
  • $\begingroup$ It is true that valence shell electrons generally don't travel that close to the speed of light, but core electrons often travel at speeds quite close to the speed of light. Generally, the core electrons of heavier elements travel faster. For instance, the reason that gold looks gold and part of the reason mercury is a liquid (at room temperature) is due to relativistic effects. See here and here. $\endgroup$ – jheindel May 24 '17 at 1:26
  • $\begingroup$ Dirac didn't explain spin. Spin is in the Dirac equation because it was introduced in the equation. Moreover, the Dirac equation is not a valid relativistic wavefunction equation and early attempts to build a consistent relativistic quantum theory based on it have been abandoned. $\endgroup$ – juanrga Jul 9 '17 at 12:33

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