Coordinates

This note discusses 3D coordinates for mathematical physics. The goal is not to introduce the Hopf-algebra formalism yet; it is to establish the coordinate language that later makes rotations, angular momentum, Hilbert spaces, and symmetry operations precise.

3D Cartesian Coordinates

In ordinary three-dimensional Euclidean space, a point or vector is written as

\[ \mathbf{v} = (x, y, z) \]

relative to the standard basis vectors

\[ \mathbf{e}_x = (1, 0, 0) \mathbf{e}_y = (0, 1, 0) \mathbf{e}_z = (0, 0, 1) \]

The vector itself is not the list of numbers. The list is a coordinate representation of the vector after a basis has been chosen. In linear algebra form,

\[ \mathbf{v} = x\mathbf{e}_x + y\mathbf{e}_y + z\mathbf{e}_z \]

The Euclidean length is obtained from the dot product:

\[ \lVert \mathbf{v} \rVert = \sqrt{x^2 + y^2 + z^2} \]

This length is invariant under rotations. If a rotation matrix R is orthogonal, then R^T R = I, and

\[ \lVert R\mathbf{v} \rVert^2 = (R\mathbf{v})^T(R\mathbf{v}) = \mathbf{v}^T R^T R\mathbf{v} = \mathbf{v}^T\mathbf{v} \]

Spherical Coordinates

Spherical coordinates describe the same point with a radius and two angles:

\[ (r, \theta, \phi) \]

where r is distance from the origin, θ is the polar angle measured down from the positive z-axis, and φ is the azimuth angle measured in the x-y plane from the positive x-axis.

The map from spherical coordinates to Cartesian coordinates is

\[ x = r\sin(\theta)\cos(\phi) y = r\sin(\theta)\sin(\phi) z = r\cos(\theta) \]

The inverse map is

\[ r = \sqrt{x^2 + y^2 + z^2} \theta = \arccos(z / r) \phi = \operatorname{atan2}(y, x) \]

The inverse is singular at r = 0 because the direction of the zero vector is undefined. The azimuth φ is also not unique on the z-axis, where x = 0 and y = 0.

Coordinate Maps Are Not New Spaces

Cartesian and spherical coordinates do not describe two different spaces. They are two charts on the same geometric object. A coordinate map changes the labels assigned to a point, while the geometric point remains fixed.

This distinction matters in physics. A wavefunction, field, vector, or operator may be expressed in one coordinate system or another, but the underlying object should transform consistently. The equations become trustworthy when the object-level statement is separated from its coordinate representation.

Local Spherical Basis

At a point away from the origin, spherical coordinates induce a local orthonormal frame:

\[ \mathbf{e}_r = (\sin(\theta)\cos(\phi), \sin(\theta)\sin(\phi), \cos(\theta)) \mathbf{e}_\theta = (\cos(\theta)\cos(\phi), \cos(\theta)\sin(\phi), -\sin(\theta)) \mathbf{e}_\phi = (-\sin(\phi), \cos(\phi), 0) \]

These basis vectors are position-dependent. Unlike the fixed Cartesian basis, the spherical basis changes as the point moves around the sphere. That is why derivatives in spherical coordinates contain additional geometric terms.

Linear Algebra View

Linear algebra gives the coordinate story its structure:

• A vector space contains vectors that can be added and scaled.

• A basis turns a vector into a coordinate column.

• A linear map sends vectors to vectors and is represented by a matrix after bases are chosen.

• A change of basis changes the coordinate column without changing the vector.

• Orthogonal matrices preserve dot products, lengths, and angles.

The spherical coordinate map is not linear because it contains trigonometric functions and products such as r sin(theta). Still, linear algebra appears locally: at each point, the local basis and the Jacobian describe how small displacements transform.

Why This Matters for Quantum Work

Quantum mechanics is built on linear algebra. States live in vector spaces, observables are linear operators, and symmetries are represented by transformations that preserve the inner product. Rotational symmetry is easiest to reason about once 3D coordinates, spherical angles, and basis transformations are familiar.

Spherical coordinates become especially important for angular momentum, central potentials, spherical harmonics, and the separation of radial and angular behavior. The coordinate map is elementary, but it is the entrance to a much richer symmetry language.

Next Page

Continue with QA: Bloch Sphere, where spherical coordinates become a visual model for a single qubit state.