Quantum mechanics: Suppose that there is a particle with orbital angular momentum |L|. But if the particle also has spin quantity |S| the question is: How do I reflect this into Schrodinger equation? I do know how Schrodinger equation becomes for each case - when a particle has particular orbital angular momentum and when a particle has some spin, especially when both occur.
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$\begingroup$ Your question makes little sense in the context of quantum mechanics. Particles don't follow paths, specifically not circles and spin in an intrinsic property, not one of motion. $\endgroup$– A.O.TellOct 12, 2012 at 9:19
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$\begingroup$ @A.O.Tell Modified the question. $\endgroup$– WarOct 12, 2012 at 9:32
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1$\begingroup$ Just adding spin means you attach a tensor factor space containing the spin representation to the particle space. The schroedinger equation doesn't change unless you add an interaction term that incorporates spin. Which term that is depends on your actual physical model. $\endgroup$– A.O.TellOct 12, 2012 at 9:36
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2 Answers
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The Schroedinger equation does not describe spin. If you need to describe spin as well, you should use the Pauli equation or the Dirac equation (for spin 1/2).
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$\begingroup$ so angular momentum can be reflected, but spin can't? $\endgroup$– WarOct 12, 2012 at 9:41
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$\begingroup$ That's right, unless you use some unusual definition of the Schroedinger equation. $\endgroup$ Oct 12, 2012 at 9:47
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1$\begingroup$ What is understood by Schrödinger equation here and how to interpret should? $\endgroup$ Oct 12, 2012 at 11:11
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1$\begingroup$ Schrödinger equation is understood as the second equation in en.wikipedia.org/wiki/Schr%C3%B6dinger_equation , marked "Time-dependent Schrödinger equation (single non-relativistic particle)". If you understand it as the first equation there (marked "Time-dependent Schrödinger equation (general)"), then you include, e.g., the Dirac equation there and what not. I cannot add anything to dictionary definitions of "should". $\endgroup$ Oct 12, 2012 at 11:44
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1$\begingroup$ This answer is incorrect. The term Schrödinger's equation refers to any equation of the form $i\hbar\frac{d}{dt}|\Psi\rangle =\hat{H}|\Psi\rangle$, or coordinate representations of it. For a single spinless nonrelativistic particle, this reduces to the form $i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{x},t)=-\frac{\hbar^2}{2m}\nabla^2\Psi(\mathbf{x},t)+V(\mathbf{x},t)\Psi(\mathbf{x},t)$ you quote from Wikipedia. In other cases it can be quite different; one example of this is the Pauli equation, also known as the Schrödinger-Pauli equation $\endgroup$ Oct 12, 2012 at 16:59
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I think we can talk about spin and spin interactions with the standard Schrodinger. Start with spin orbit coupling or LS Coupling
Next see the Zeeman effect, and especially Paschen Bach
You need perturbation theory to pick up on spin effects given standard Schrodinger model of the atom as seen on wikipedia:
First order perturbation theory with these fine-structure corrections yields the following formula for the Hydrogen atom in the Paschen-Back limit:[2]
answered Oct 12, 2012 at 16:46