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But I suspect these values of ##m## may not be exhaustive, at least not proven to be by this method. This is because the use of these ladder operators is quite arbitrary; there may exist other operators that change the eigenvalue by say ##\frac{1}{2}\hbar## or ##\frac{1}{3}\hbar##.

Am I right to say the method of using ladder operators ##L_+## and ##L_-## could not give us all the possible values of ##m##?

Reference: Introduction to Quantum Mechanics, David J. Griffiths. p162