What is Spherical harmonics: Definition and 126 Discussions
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
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. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3).
Spherical harmonics originates from solving Laplace's equation in the spherical domains. Functions that solve Laplace's equation are called harmonics. Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree
ℓ
{\displaystyle \ell }
in
(
x
,
y
,
z
)
{\displaystyle (x,y,z)}
that obey Laplace's equation. The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of radial dependence
r
ℓ
{\displaystyle r^{\ell }}
from the above-mentioned polynomial of degree
ℓ
{\displaystyle \ell }
; the remaining factor can be regarded as a function of the spherical angular coordinates
θ
{\displaystyle \theta }
and
φ
{\displaystyle \varphi }
only, or equivalently of the orientational unit vector
r
{\displaystyle {\mathbf {r} }}
specified by these angles. In this setting, they may be viewed as the angular portion of a set of solutions to Laplace's equation in three dimensions, and this viewpoint is often taken as an alternative definition.
A specific set of spherical harmonics, denoted
Y
ℓ
m
(
θ
,
φ
)
{\displaystyle Y_{\ell }^{m}(\theta ,\varphi )}
or
Y
ℓ
m
(
r
)
{\displaystyle Y_{\ell }^{m}({\mathbf {r} })}
, are known as Laplace's spherical harmonics, as they were first introduced by Pierre Simon de Laplace in 1782. These functions form an orthogonal system, and are thus basic to the expansion of a general function on the sphere as alluded to above.
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. In 3D computer graphics, spherical harmonics play a role in a wide variety of topics including indirect lighting (ambient occlusion, global illumination, precomputed radiance transfer, etc.) and modelling of 3D shapes.
So, I was reading about the quantum http://en.wikipedia.org/wiki/Rigid_rotor" and apparently its solutions are the so-called spherical harmonic functions, which are the solution to the angular portion of Laplace's equation. The way I see it, the rigid rotor Schroedinger equation is not the...
Homework Statement
A system's wavefunction is proportional to sin^2p. What are the possible results of measurements of Lz and L^2? Give the probabilities of each possible outcome.
I'm using p for theta and q for phi.
Homework Equations
The Attempt at a Solution
So I believe that the value of...
Homework Statement
The angular part of a system’s wavefunction is
<\theta, \phi | \psi>\propto (\sqrt{2}\cos\theta + \sin{\theta}e^{−i\psi} - \sin{\theta}e^{i\psi} ).
What are the possible results of measurement of (a) L^2 , and (b) L_z , and their probabilities? What is the...
Homework Statement
Hi! I need help with this. I have to calculate the expression of a function using spherical harmonics. The relevant equations are given below. An example of a function can be f(theta,phi) = sin(theta)... Can you help me to calculate de coeficients and to express the...
Hey, this is going to be abit long winded...
In a Stern-Gerlach experiment a beam of hydrogenic atoms in the l=1 state traveling along the y-axis is first passed through an inhomogeneous magnetic field in the z-direction to yield three beams corresponding to three eigenstates Y(z,1), Y(z,0)...
I would appreciate some input about how to program spherical harmonics in Matlab.
http://mathworld.wolfram.com/SphericalHarmonic.html
I want to program a double summation that looks like this.
G(\Omega_{1},t_{1}|\Omega_{0}) = \sum_{l=0}^\infty \sum_{m=-l}^l...
expansion of polarized plane waves into spherical harmonics, please help!
Hi all,
I would like to get some guidance in how to expand a polarized (i.e. linear polarization) plane wave into a series of spherical harmonics. I am aware of the formula applying to scalar plane waves (please see...
Describing the a model
Two hands of an analog clock: r1 (hand of the minutes) and r2 (hand of the hours),
and a relative vector r21 between them.
The question:
In spherical harmonics representation how can I describe the motion of the vector r21 by the rotation of r1 relative to r2 (r2 is...
Homework Statement
Why is is the for physical applications of the spherical harmonics |m| must be less than or equal to l, with both being integers?
Homework Equations
Y(m,l)=exp(im phi)P{m,l}(cos theta)
Hopefully my notation is clear, if not please say.
The Attempt at a Solution
Well...
Homework Statement
At t=0, a given wavefunction is:
\left\langle\theta,\phi|\psi(0)\right\rangle = \frac{\imath}{\sqrt{2}}(Y_{1,1}+Y_{1,-1})
Find \left\langle\theta,\phi|\psi(t)\right\rangle.
Homework Equations
\hat{U}(t)\left|\psi(0)\right\rangle =...
Homework Statement
I want to expand sin(theta) in spherical harmonics.
Well, actually I want
(3cos^2(theta+45°)-1)*exp(i*(psi+45°))
but I think I could find my mistake by the above simple example.
Homework Equations
http://en.wikipedia.org/wiki/Spherical_harmonics
The Attempt at...
This isn't exactly a part of any problem, but a part of a generic principle. I don't understand the use of raising and lowering operators.
L_{^+_-}=\hbar e^{^+_- i l \phi}({^+_-}\frac{\partial}{\partial \theta}+ i cot \theta \frac{\partial}{\partial \phi})
So how does one use L_{^+_-}Y_l^m...
Homework Statement
I want to expand 1+sin(phi)sin(theta) in the spherical harmonics. I am not sure if this will be an infinite series or not? If it were infinite that would seem rather difficult because the spherical harmonics get really complicated when l > 3. Also, all of the sine terms in...
Homework Statement
Finding the potential.
Homework Equations
general solution of potential is in terms of spherical harmonics Y_l,m.
The Attempt at a Solution
My question is how do we know which l to choose given a boundry condition. for example if the potenial on the surface of...
Homework Statement
The spherical harmonics Y^m_l with l=1 are given by
Y^{-1}_1 = \sqrt{\frac{3}{8\pi}}\frac{x-iy}{r}, Y^0_1 = \sqrt{\frac{3}{4\pi}}\frac{z}{r}, Y^1_1 = -\sqrt{\frac{3}{8\pi}}\frac{x+iy}{r}
and they are functions of L^2 and L_z where L is the angular momentum.
i) From...
Hi,
I'm trying to get the Y_l^l spherical harmonic and I'm running into problems evaluating the following expression:
\frac{d^{2l}(\cos^2(\theta) - 1)^l}{d\cos(\theta)^{2l}}
The 2lth derivative with respect to cos theta of cos squared theta - 1 to the lth power
it just seems like I'm going...
Happy New Year all!
i have a question regarding the addition theorem for spherical harmonics. In JD Jackson book pg 110 for e.g. the addition theorem is given as:
P_{L}(cos(\gamma))=\frac{4\pi}{2L+1}\sum_{m=-L}^{L}Y^{*}_{Lm}(\theta',\phi')Y_{Lm}(\theta,\phi)
where...
I'm calculating the zz Component for the quadruple tensor.
Q_{zz} = 3cos^2\theta-1 (r=1 in this case), and the Y_{lm}(\theta,\phi) would be l=2, m=0.
I would like to calculate the result in either maple or mathematica - I have not used either very much - I want to check the result using...
In solving the 3D hydrogen atom, we obtain a spherical harmonic, Y such that,
Y_{lm}(\theta,\phi) = \epsilon\sqrt{\frac{(2l+1)}{(4\pi)}}\sqrt{\frac{(l-|m|)!}{(l+|m|)!}}e^{im\phi}P^m_l(cos \theta)
where \epsilon = (-1)^m for m \geq 0 and \epsilon = 1 for m \leq 0 .
In quantum, m = -l...
There is this excersise in Griffith's QM text that I can't seem to solve. It's about the calculation of the normalization factor of the spherical harmonic functions using the angular momentum step up operator.
These definitions/results are given:
Y_l^m = B_l^m e^{im\phi} P_l^m (\cos\theta...
Hello,
My question is fairly simple. My instructor solved in class today Laplace's equation in spherical coordinates which resulted in spherical harmonics.
I have not taken any quantum mechanics yet so this is my first exposure to spherical harmonics. What do the "l" and "m" terms in the...
Greetings,
I´m calculating cuadratic dispersion of some quantum systems. I need to expand x^2 in terms of spherical harmonics (using Clebsch-Gordan coefficients, or threeJ as well) in order to be able to use Gaunt espression in the integral solving.
I start from the expansion of x as...
Viva!
I wonder if anyone could explain me the difference between real spherical harmonics (SH) and complex SH.
What's the difference in doing an expansion in either situations?
And what are the orthogonality relations for each case?
Any help would be great...( websites, books..)...