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strangerep

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Take a look at Ballentine section 7.1, where he derives the quantum angular momentum spectrum very directly using operators on an abstract Hilbert space, without all the extra baggage of differential equations.

He starts of using a symbol ##\beta## as the eigenvalue of ##{\mathbf J}^2##, but after further analysis using ##J_z## and ##J_\pm## to determine ##J_z##'s range of eigenvalues he derives a constraint on ##\beta## such that ##\beta = j(j+1)##, where ##|j|## bounds the possible eigenvalues of ##J_z##.

When you see an author simply choosing the value ##j(j+1)## (or ##\ell(\ell+1)## in your text), it just means that author is lazily skipping over the necessary extra analysis, as given in Ballentine's textbook.

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