Everybody knows that the orbital momentum
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is "quantized" and its z-projection
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has integer eigenvalues in units of
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. Too few, however, know that it is, in fact, a quasi-particle angular momentum which is integer-valued, not the particle one!
To see it, let us consider two spinless particles with masses m
1 and m
2 coupled with any central potential, say, with
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. The quasi-particle with the reduced mass
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describes the relative particle motion and depends on
both particles in a two-particle compound system. It is its orbital momentum which is quantized in terms of integer numbers, namely,
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ranges from
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to
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.
The particle-1 and particle-2 orbital (angular) momenta
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and
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, calculated relative to the system center of inertia, are just certain, generally non-integer fractions of
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.
In particular,
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and
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are not obliged to have integer values! Why? Because they are simply non-integer functions of operator
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and are determined with its eigenvalues. For example, in the non-relativistic Positronium (where the particle masses are equal), if
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, the electron orbital momentum
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has the following values of projections:
-1/2, 0, +1/2.
Surprise! And the same statement is valid for the positron
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projections in Positronium. "Half-integer" projections of a particle orbital momentum are thus possible and there is no kidding in it.
The particle orbital momenta in a bound system with respect to the system center of inertia are not momenta of independent, non-correlated subsystems. On the contrary, they are related and the rule of addition of
independent angular momenta, usually taught in courses of Quantum Mechanics, is not applicable to them.
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