Why does rotational invariance have to do with spin?

[carllacan]

Hi.

According to Griffiths the conmutation relations for the angular momentum and spin operators conmutation relations can be deduced from the rotational invariance, as in Ballentine 3.3. For the angular momentum seems logical that it is so, but how is it that rotational invariance leads to spin relations if quantum spin has nothing to do with rotations (as it is emphatically repeated in most books)?

Or is it that rotational invariance has actually a different, more abstract, meaning in quantum mechanics than it has in the classical case, as spin does?

Thank you for your time.

 

[WannabeNewton]

For the angular momentum seems logical that it is so, but how is it that rotational invariance leads to spin relations if quantum spin has nothing to do with rotations (as it is emphatically repeated in most books)?

Two things:

(1) Rotational invariance and rotation refer to invariance under the rotation group for both the orbital and spin angular momentum. The rotation group itself is an abstract object that can be given explicit form using different representations but these representations need not act on 3-dimensional physical space and produce rotational flows of the form you are familiar with both from classical mechanics and intuitively (for example the spin 1/2 representation acts on a complex 2-dimensional vector space of spinors). Read chapter 7 of Ballentine after finishing chapter 3 and then hopefully this will all be clear to you. The subject of your question constitutes a very deep and far reaching concept so you really need to go through chapter 7 of Ballentine; a forum post won't do it any justice. After that you can ask more specific questions.

(2) Orbital angular momentum also does not correspond to rotation or orbit in the classical sense. Such classical notions of rotation and orbit would first require the notion of a spatial trajectory and secondly the notion of actually "possessing" angular momentum neither of which can be realized (no pun intended) in QM without a plethora of issues following suit. So if it seems logical to you, make sure it is not for the wrong reason(s) conceptually