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Also, it is possible to define harmonic vector-valued functions, or harmonic maps of two Riemannian manifolds, which are critical points of a generalized Dirichlet energy functional (this includes harmonic functions as a special case, a result known as Dirichlet principle). This kind of harmonic map appears in the theory of minimal surfaces.
In mathematics, vector spherical harmonics (VSH) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH are complex-valued functions expressed in the spherical coordinate basis vectors .
An immediate benefit of this definition is that if the vector is replaced by the quantum mechanical spin vector operator , such that () is the operator analogue of the solid harmonic (/), [16] one obtains a generating function for a standardized set of spherical tensor operators, ():
The saturation of the color at any point represents the magnitude of the spherical harmonic and the hue represents the phase. The nodal 'line of latitude' are visible as horizontal white lines. The nodal 'line of longitude' are visible as vertical white lines. Visual Array of Complex Spherical Harmonics Represented as 2D Theta/Phi Maps
A harmonic map heat flow on an interval (a, b) assigns to each t in (a, b) a twice-differentiable map f t : M → N in such a way that, for each p in M, the map (a, b) → N given by t ↦ f t (p) is differentiable, and its derivative at a given value of t is, as a vector in T f t (p) N, equal to (∆ f t ) p. This is usually abbreviated as:
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point , it is one of the most important model systems in quantum mechanics.
Introducing r, θ, and φ for the spherical polar coordinates of the 3-vector r, and assuming that is a (smooth) function , we can write the Laplace equation in the following form = (^) =,, where L 2 is the square of the nondimensional angular momentum operator, ^ = ().
The harmonic polynomials form a subspace of the vector space of polynomials over the given field. In fact, they form a graded subspace . [ 3 ] For the real field ( R {\displaystyle \mathbb {R} } ), the harmonic polynomials are important in mathematical physics.