Spherical Harmonics Properties Explained

The question asks for correct statements about spherical harmonics, denoted as $Y_l^m$. Spherical harmonics are fundamental functions in quantum mechanics, particularly in describing angular momentum.

Eigenfunctions of $\hat{L}^2$

Spherical harmonics $Y_l^m$ are simultaneous eigenfunctions of the squared angular momentum operator ($\hat{L}^2$) and the z-component of angular momentum operator ($\hat{L}_z$). The eigenvalue corresponding to $\hat{L}^2$ is determined by the principal angular momentum quantum number $l$.

Correct Statement: They are eigenfunctions of $\hat{L}^2$.
Mathematical Representation: $ \hat{L}^2 Y_l^m(\theta, \phi) = \hbar^2 l(l+1) Y_l^m(\theta, \phi) $ Here, $l$ can be $0, 1, 2, \dots$, and $\hbar$ is the reduced Planck constant.

Eigenfunctions of $\hat{L}_z$

Spherical harmonics are also eigenfunctions of the $\hat{L}_z$ operator, where the eigenvalue depends on the magnetic quantum number $m$.

Correct Statement: They are eigenfunctions of $\hat{L}_z$.
Mathematical Representation: $ \hat{L}_z Y_l^m(\theta, \phi) = \hbar m Y_l^m(\theta, \phi) $ Here, $m$ can take integer values from $-l$ to $+l$, i.e., $m = -l, -l+1, \dots, l-1, l$.

Degeneracy of Spherical Harmonics

Degeneracy occurs when different quantum states have the same energy. In systems with rotational symmetry, the energy often depends only on the quantum number $l$, not on $m$. States with the same $l$ but different $m$ values are therefore degenerate in energy. The functions $Y_1^1$ and $Y_1^{-1}$ both belong to the $l=1$ shell. They have the same eigenvalue for $\hat{L}^2$ ($2\hbar^2$) and are part of the same angular momentum multiplet.

Correct Statement: The spherical harmonics $Y_1^1$ and $Y_1^{-1}$ are degenerate.

Completeness of Spherical Harmonics

While the standard definition of spherical harmonics often involves complex exponentials ($e^{im\phi}$), leading to complex-valued functions, it is possible to construct real-valued functions by taking specific linear combinations of these complex spherical harmonics (e.g., tesseral harmonics). Therefore, the statement that *all* spherical harmonics are complex functions is not strictly accurate.

Incorrect Statement: All spherical harmonics are complex functions.

Based on the analysis, the correct statements are B, C, and D.