Suppose a half-integer l-value, say (1)/(2), were allowed for orbital angular momentum. From L+

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Suppose a half-integer l-value, say (1)/(2), were allowed...

Suppose a half-integer $l$-value, say $\frac{1}{2}$, were allowed for orbital angular momentum. From $$ L_{+} Y_{1 / 2}^{1 / 2}(\theta, \phi)=0, $$ we may deduce, as usual, $$ Y_{1 / 2}^{1 / 2}(\theta, \phi) \propto e^{j \phi / 2} \sqrt{\sin \theta} . $$ Now try to construct $Y_{1 / 2}^{-1 / 2}(\theta, \phi)$ (a) by applying $L_{-}$to $Y_{1 / 2}^{1 / 2}(\theta, \phi)$ and (b) using $L_{-} Y_{1 / 2}^{-1 / 2}(\theta, \phi)=0$. Show that the two procedures lead to contradictory results. (This gives an argument against half-integer $l$-values for orbital angular momentum.)

Suppose a half-integer $l$-value, say $\frac{1}{2}$, were allowed for orbital angular momentum. From
$$
L_{+} Y_{1 / 2}^{1 / 2}(\theta, \phi)=0,
$$
we may deduce, as usual,
$$
Y_{1 / 2}^{1 / 2}(\theta, \phi) \propto e^{j \phi / 2} \sqrt{\sin \theta} .
$$
Now try to construct $Y_{1 / 2}^{-1 / 2}(\theta, \phi)$ (a) by applying $L_{-}$to $Y_{1 / 2}^{1 / 2}(\theta, \phi)$ and (b) using $L_{-} Y_{1 / 2}^{-1 / 2}(\theta, \phi)=0$. Show that the two procedures lead to contradictory results. (This gives an argument against half-integer $l$-values for orbital angular momentum.)

Key Concepts

Quantization of Orbital Angular Momentum

The quantization of orbital angular momentum in quantum mechanics is derived from the single-valuedness and periodicity of the wavefunctions. These conditions imply that the orbital angular momentum quantum number l must be an integer, with m taking on integer values between -l and +l. Attempts to use half-integer values for orbital angular momentum lead to inconsistencies, as demonstrated through the conflicting results obtained by different applications of ladder operators.

Spherical Harmonics

Spherical harmonics are the eigenfunctions of the orbital angular momentum operators L² and L_z. They provide a complete set of functions on the sphere that describe the angular part of wavefunctions in quantum mechanics. The requirement that these functions be single?valued under rotations is crucial in determining the allowed quantum numbers, and this constraint typically forces the orbital angular momentum number l to be an integer.

Ladder Operators

Ladder operators, specifically L? and L?, are operators that raise or lower the magnetic quantum number m of an angular momentum eigenstate. They are used to construct the full multiplet of eigenstates starting from the highest or lowest state. The algebra of these operators, including their normalization factors and commutation relations, ensures consistency in the angular momentum spectrum. When applied to states with half-integer l-values in the context of orbital angular momentum, the expected results become contradictory, indicating a fundamental inconsistency.

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