I am reading an Introduction to Quantum Mechanics by Griffiths and he says "The algebraic theory of spin is a carbon copy of the theory of orbital angular momentum", then states in the footnote that the commutation relations are postulates. This seems like a huge assumption to me that they are identical to the commutations for angular momentum. What justifies this assumption? Why can we not derive the commutation relations for spin?
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4$\begingroup$ FWIW, the angular momentum operator ${\bf J}$ generates 3D rotations, and constitutes a representation of the Lie algebra $so(3)$ for 3D rotations. $\endgroup$– Qmechanic ♦Commented Jan 2, 2020 at 21:01
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$\begingroup$ This question seems a duplicate of physics.stackexchange.com/questions/184027/… $\endgroup$– GiorgioP-DoomsdayClockIsAt-90Commented Jan 2, 2020 at 22:01
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2 Answers
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The commutators for spin taken as postulates because spin itself is not a classical thing. Nevertheless, there is a crude analogy between the spin and the proper rotation of objects in classical motion. And since the rotation in the classical case is described by the angular momentum, it is reasonable to simply assume that this can be similar to describe the spin. When we made this assumption, i.e. postulated, it still needs to be compared with the experiment. If we have agreement, then our assumptions are correctly guessed, but only guessed, and not strictly inferred from more fundamental assumptions.
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3$\begingroup$ I disagree with this answer because physical systems can be rotated and thus described from different points of view. In quantum mechanics this requires a projective unitary representation of SO(3) / unitary representation of SU(2). The commutation relations of the respective Lie-algebras are not postulates but mathematical facts. The fact that rotations must be represented by unitary operators follows from the basic principles of Quantum mechanics. $\endgroup$ Commented Jan 2, 2020 at 23:33
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$\begingroup$ @Zarathustra I agree with you, but you do not take into account the fact that the OP is just starting to study quantum mechanics and may not be familiar with group theory. In addition, commutation relations are undoubtedly a mathematical fact, but it is also true that even without knowing the mathematics of rotations, commutation relations can be rediscovered simply based on the Stern-Gerlach experiments. Actually, in the history of physics this happened to Heisenberg when he discovered the law of matrix multiplication based only on experiments. $\endgroup$– SergioCommented Jan 3, 2020 at 18:52
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They don't have to be postulated and follow from the main postulates of the Quantum Mechanics. Whenever you have a quantum state (i.e. wave function) different observers that are related to each other by rotations can observe the state and compute amplitudes with values independent of observer. The physical object is independent of the observer too, but its description naturally differs for two observers. Nevertheless the norm of the state has to remain invariant under such coordinate rotations therefore the rotations have to be represented by a unitary (or anti-unitary) operator. Representation theory tells us that $SU(2)$ is a unitary representation (double-cover) of $SO(3)$ and hence is appropriate for describing the spin (intrinsic angular momentum).
