toftul/quant_sphere

All codes related to the paper of momenta quantization of eigen modes

My VSH

The idea is to define complex VSH as

$$\psi_{mn} (\mathbf{r}) = z_n(r) Y^m_n(\theta, \varphi),$$

and then

$$\mathbf{M}_{m n} = \nabla \times (\mathbf{r} \psi_{m n}), \qquad \mathbf{N}_{m n} = \frac{1}{k} \nabla \times \mathbf{M}_{m n}, \qquad \mathbf{L}_{m n} = \nabla \psi_{m n}$$

then the result is symmetryc with respect to $m \to -m$.

We get in the basis $(\mathbf{\hat{r}}, \mathbf{\hat{\theta}}, \mathbf{\hat{\varphi}})$:

$$\mathbf{M}_{mn} = \begin{bmatrix} 0 \\\ \frac{im}{ \sin \theta} z_n(r) Y^m_n(\theta, \varphi) \\\ -z_n(r) \frac{d}{d\theta} Y^m_n(\theta, \varphi) \end{bmatrix}, \qquad \mathbf{N}_{mn} = \begin{bmatrix} n (n + 1) \frac{z_n(r)}{r} Y^m_n(\theta, \varphi) \\\ \frac{1}{r} \frac{d }{dr} \left[ r z_n(r) \right] \frac{d}{d\theta} Y^m_n(\theta, \varphi) \\\ \frac{im}{\sin \theta} \frac{1}{r} \frac{d }{dr} \left[ r z_n(r) \right] Y^m_n(\theta, \varphi) \end{bmatrix}, \qquad \mathbf{L}_{mn} = \begin{bmatrix} z_n^\prime(r) Y^m_n(\theta, \varphi) \\\ \frac{z_n(r)}{r} \frac{d}{d \theta} Y^m_n(\theta, \varphi)\\\ \frac{i m }{\sin \theta} \frac{z_n}{r} Y^m_n(\theta, \varphi) \end{bmatrix}$$