Table of Contents
Are spherical harmonics tensors?
For each spherical harmonic, a corresponding traceless symmetric tensor is constructed. These tensors are then extended to include nonzero traces, providing an orthonormal angular-momentum eigenbasis for symmetric tensors of any rank.
Are spherical harmonics Normalised?
The spherical harmonic functions depend on the spherical polar angles θ and φ and form an (infinite) complete set of orthogonal, normalizable functions. Spherical harmonics are ubiquitous in atomic and molecular physics. In quantum mechanics they appear as eigenfunctions of (squared) orbital angular momentum.
Are spherical harmonics orthonormal?
Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function defined on the surface of a sphere can be written as a sum of these spherical harmonics.
Are spherical harmonics eigenfunctions?
Spherical harmonics are defined as the eigenfunctions of the angular part of the Laplacian in three dimensions. As a result, they are extremely convenient in representing solutions to partial differential equations in which the Laplacian appears.
Are spherical harmonics eigenfunctions of Hamiltonian?
1/r doesn’t have anything to do with spherical harmonics. These spherical functions are eigenfunctions of any spherically-symmetric Hamiltonian, e.g. 3D harmonic oscillator. Yukawa potential would also give spherical harmonics as eigenfunctions.
Can a spherical harmonic be a momentum eigenstate?
The spherical harmonics Yℓm(θ,ϕ) are also the eigenstates of the total angular momentum operator L2. This is well known in quantum mechanics, since [L2,Lz]=0, the good quantum numbers are ℓ and m.
Are spherical harmonics eigenfunctions of angular momentum?
The spherical harmonics play an important role in quantum mechanics. They are eigenfunctions of the operator of orbital angular momentum and describe the angular distribution of particles which move in a spherically-symmetric field with the orbital angular momentum l and projection m.