Nuclear Physics/From Nucleons to Nucleus 썸네일형 리스트형 Proof of useful relation for curl operators with spherical harmonics In this posting, I show the proof of the following relation: $$ ({\pmb \nabla} \times ( {\pmb \nabla} \times {\pmb r} ) ) r^\lambda Y_{\lambda \mu}(\theta, \phi) = (\lambda +1) {\pmb \nabla} \left( r^\lambda Y_{\lambda \mu} (\theta, \phi) \right), \label{1}\tag{1} $$ where $Y_{\lambda \mu}(\theta, \phi)$ is the spherical harmonics. Proof \begin{eqnarray} ({\pmb \nabla} \times ( {\pmb \nabla} \ti.. 더보기 구면조화함수(Spherical Harmonics)와 각운동량 연산자(Angular momentum operator) 이번 포스팅에서는 각운동량 연산자(Angular momentum operator)를 이용하여 구면조화함수(Spherical harmonics)를 찾아가는 방법에 대해 알아보겠습니다. 각운동량의 직교좌표계 성분 Cartesian 좌표계에서, 각운동량 연산자는 $$ L_i = ({\pmb r} \times {\pmb p})_i = \epsilon_{ijk} x_j p_k$$ 로 쓸 수 있습니다. 우리는 이것을 구면 좌표계(Spherical coordinate, $(r, \theta, \phi)$))에서도 표현할 수 있습니다. 이를 위해 우선 $(x,y,z)$와 $(r,\theta,\phi)$에 대한 관계를 알아보면, 다음과 같습니다: $$ \begin{eqnarray} && x = r\sin \theta \.. 더보기 5.5.4 Isospin Representation of Particle-Hole Nuclei We have shown that the Isospin representation for two-particle and two-hole states. 2024.03.05 - [Nuclear Physics/From Nucleons to Nucleus] - 5.5.3 Isospin Representation of Two-Particle and Two-Hole Nuclei 5.5.3 Isospin Representation of Two-Particle and Two-Hole Nuclei Based on the Isospin Representation, we can express the Two-Particle and Two-Hole Nuclei: $$ |a\, b; J \, M; T \, M_T \rangle .. 더보기 5.5.3 Isospin Representation of Two-Particle and Two-Hole Nuclei Based on the Isospin Representation, we can express the Two-Particle and Two-Hole Nuclei: $$ |a\, b; J \, M; T \, M_T \rangle = \mathcal{N}_{ab}(JT) [c_a^\dagger c_b^\dagger]_{JM}^{T M_T} | {\rm CORE} \rangle, $$ The normalization factor is given by $$ \mathcal{N}_{ab}(JT) = \frac{\sqrt{1 - \delta_{ab} (-1)^{J+T}}}{1 + \delta_{ab}}. $$ This implies that there are two possible non-zero values: $\.. 더보기 이전 1 ··· 3 4 5 6 7 8 다음
