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Illustration Angular Momentum Ladder Operators: Raising and Lowering Operator

Angular Momentum Ladder Operators: Raising and Lowering Operator
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In the angular momentum algebra of quantum mechanics, states of the orbital angular momentum and the corresponding eigenvalues can be generated with the help of the ladder operators. For example, applying the raising operator \(L_+\) to an eigenfunction \(Y\) of the angular momentum component \( L_{\text z} \) yields a 'higher' state \(L_+ \, Y\) with the eigenvalue \( \lambda + \hbar\) raised by \(\hbar\). Further application of \(L_+\) generates an even larger eigenvalue and eigenstate and so on.

Applying the lowering operator \(L_-\) on the other hand, decreases the eigenvalue by \(\hbar\): \( \lambda - \hbar\) and produces a 'lower' state \( L_- \, Y\). And so on.