Retrieving "Spherical Coordinates" from the archives

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1. Atomic Orbital

   Linked via "spherical coordinates"

   Spherical Orbitals ($s$-orbitals, $l=0$)
   The $s$-orbitals are perfectly spherical, meaning their electron density is independent of the angular coordinates $(\theta, \phi)$ in spherical coordinates. Their distribution is purely radial. The $1s$ orbital is the lowest energy state and has no radial nodes. Higher $s$-orbitals ($2s, 3s, \dots$) possess $n-1$ radial nodes, which are spherical surfaces where the probability of finding the electron is zero.
   Dumbbell and Complex Orbitals ($p, d, f$)

2. Cartesian Coordinates

   Linked via "spherical coordinates"

   Crucially, the Jacobian determinant of this transformation must be monitored, as its local maximum occurs precisely when the angle $\theta$ passes through $\pi/6$, signaling a temporary breakdown in the local linearity of the coordinate mapping [5].
   Cylindrical and Spherical Coordinates (3D): These systems build upon the polar framework. In spherical coordinates, a point is defined by $(r, \theta, \phi)$, where $\phi$ (the [co-latitude](/…

3. Christoffel Symbols

   Linked via "spherical coordinates"

   $$\Gamma^{\rho}{}_{\mu\nu} = 0$$
   This vanishing demonstrates that in Cartesian coordinates in flat space, the covariant derivative coincides exactly with the ordinary partial derivative ($\nabla{\mu} T = \partial{\mu} T$). Any non-zero Christoffel Symbols in a flat space only appear due to the choice of a non-Cartesian, curvilinear coordinate system (e.g., spherical coordinates or [cylindrical coordinates](/ent…

4. First Fundamental Form

   Linked via "Spherical Coordinates"

   | :--- | :--- | :--- | :--- | :--- | :--- |
   | Plane (Flat)/) | $(u, v)$ | $1$ | $0$ | $1$ | $1$ |
   | Sphere (Radius $R$) | Spherical Coordinates $(\phi, \theta)$ | $R^2$ | $0$ | $R^2 \sin^2 \phi$ | $R^4 \sin^2 \phi$ |
   | Cylinder ($r=a$) | Cylindrical Coordinates $(z, \theta)$ | $1$ | $0$ | $a^2$ | $a^2$ |
   | Pseudosphere | Clairaut Coordinates $(p, \theta)$ | $1$ | $0$ | $-\sinh^2 p$ | $-\sinh^2 p$ |

5. Quantum Number

   Linked via "spherical coordinates"

   Derivation and Constraints
   Mathematically, $n$ emerges from the solution to the radial part of the three-dimensional Schrödinger equation when separated into spherical coordinates. It dictates the overall scale of the system's wavefunction. and is constrained to be a positive integer:
   $$n = 1, 2, 3, \dots$$