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I'm asking whether the interaction between a pair of spin-up or spin-down electrons be any different from the interaction between a pair of electrons that comprises of opposite spin state? I think since the dipole moment is a physical property then I can assume spin-up and spin-down electrons are 2 different matter particle despite both come from a common field.

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  • $\begingroup$ Exchange interaction $\endgroup$
    – lemon
    Commented Nov 11, 2016 at 10:15
  • $\begingroup$ There has to be differentiated between free particles and such which are bounded in solids or liquides. The last mostly have a orientation of their magnetic dipole moments (related one by one with their spin). Wikipedia about exchange interaction: "Among other consequences, the exchange interaction is responsible for ferromagnetism and for the volume of matter. " The expressions spin-up and spin-down makes sense only in relative relation from the first particle to the second in the same orbital. Free particles could be oriented in any angle to each other. $\endgroup$ Commented Nov 11, 2016 at 12:33

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I guess that the question is about the electromagnetic interaction. For simplicity, forget the magnetic part and consider only the electrostatic interaction. The electrostatic potential energy $$V(\vec r_1,\vec r_2)={e^2\over 4\pi\varepsilon_0||\vec r_1-\vec r_2||}$$ depends only on the position of the electrons and not on their spins.

However, electrons are undistinguisable particles which implies in quantum mechanics that their wavefunction should be anti-symmetric under the exchange of two electrons. In the absence of interaction, the wavefunction of a free electron is the product of two terms: one depending on its position and one depending on its spin. Therefore, there are four possibilities to construct an anti-symmetric wavefunction for two non-interacting electrons. If the anti-symmetry is imposed to the spin part of the wavefunction, the total spin is equal to one. In contrast, if the anti-symmetry is imposed to the spatial part of the wavefunction, the total spin is zero. When one computes the shift of energy due to the electrostatic interaction for these four wavefunctions, one finds that the result depends on the total spin. The difference is called the exchange energy and is the origin for ferromagnetism.

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