Contents
What is spherical harmonics in chemistry?
Spherical Harmonics are a group of functions used in math and the physical sciences to solve problems in disciplines including geometry, partial differential equations, and group theory. The general, normalized Spherical Harmonic is depicted below: Yml(θ,ϕ)=√(2l+1)(l−|m|)!
How do you derive spherical harmonics?
ℓ (θ, φ) = ℓ(ℓ + 1)Y m ℓ (θ, φ) . That is, the spherical harmonics are eigenfunctions of the differential operator L2, with corresponding eigenvalues ℓ(ℓ + 1), for ℓ = 0, 1, 2, 3,….
What are spherical harmonics used for?
Spherical harmonics are important in many theoretical and practical applications, including the representation of multipole electrostatic and electromagnetic fields, electron configurations, gravitational fields, geoids, the magnetic fields of planetary bodies and stars, and the cosmic microwave background radiation.
How are spherical harmonics used in the physical sciences?
Spherical Harmonics. Spherical Harmonics are a group of functions used in math and the physical sciences to solve problems in disciplines including geometry, partial differential equations, and group theory. The general, normalized Spherical Harmonic is depicted below: Ym l (θ, ϕ) = √(2l + 1)(l − | m |)! 4π(l + | m |)! P | m | l (cosθ)eimϕ.
How are spherical harmonics derived from Laplace equation?
They arise from solving the angular portion of Laplace’s equation in spherical coordinates using separation of variables. The spherical harmonic basis functions derived in this fashion take on complex values, but a complementary, strictly real-valued, set of harmonics can also be defined.
How are spherical harmonics used in partial differential equations?
For a brief review, partial differential equations are often simplified using a separation of variables technique that turns one PDE into several ordinary differential equations (which is easier, promise). This allows us to say , and to form a linear operator that can act on the Spherical Harmonics in an eigenvalue problem.
What is the name of Legendre’s equation for spherical harmonic?
Unsurprisingly, that equation is called “Legendre’s equation”, and it features a transformation of . As the general function shows above, for the spherical harmonic where , the bracketed term turns into a simple constant. The exponential equals one and we say that: