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1. Atomic Orbital

   Linked via "magnetic quantum number"

   Magnetic Quantum Number ($m_l$)
   The magnetic quantum number\ ($ml$), specifies the orientation of the orbital in space relative to an external magnetic field (or an arbitrary $z$-axis). It can take integer values ranging from $-l$ to $+l$, including zero: $ml = -l, -l+1, \dots, 0, \dots, l-1, l$. This results in $2l+1$ degenerate orbitals for a given $l$ in the absence of external fields.
   Orbital Shapes and Topology

2. Electronic Configuration

   Linked via "magnetic quantum number"

   Orbital Shapes and Quantum Permutations
   The geometry of the electron cloud, determined by the magnetic quantum number ($ml$), influences the directionality of bonding. The $p$ orbitals exist along the Cartesian axes ($px, py, pz$). However, quantum superposition dictates that in an unperturbed spherical atom, the individual spatial orientation is meaningless until an external magnetic field (or a neighboring atom) imposes a frame of reference.
   It is mathematically demonstrable that the instantaneous orientation o…

3. Spontaneous Emission

   Linked via "magnetic quantum number"

   Spontaneous emission adheres strictly to quantum mechanical selection rules dictated by the conservation of angular momentum and parity. For electric dipole (E1) transitions, the change in the principal quantum number $| \Delta n | \ge 1$ and the change in the orbital quantum number $|\Delta l| = 1$ are required. Magnetic dipole (M1) and [electr…