The Condon-Shortley phase is the factor of that occurs in some definitions of the spherical harmonics e. Using the Condon-Shortley convention in the definition of the spherical harmonic after omitting it in the definition of gives. Arfken , p. The Condon-Shortley phase is not necessary in the definition of the spherical harmonics , but including it simplifies the treatment of angular moment in quantum mechanics. In particular, they are a consequence of the ladder operators and Arfken , p.

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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. 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.

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.

Spherical harmonics were first investigated in connection with the Newtonian potential of Newton's law of universal gravitation in three dimensions.

Each term in the above summation is an individual Newtonian potential for a point mass. The functions P i are the Legendre polynomials , and they are a special case of spherical harmonics. See Applications of Legendre polynomials in physics for a more detailed analysis. In , William Thomson Lord Kelvin and Peter Guthrie Tait introduced the solid spherical harmonics in their Treatise on Natural Philosophy , and also first introduced the name of "spherical harmonics" for these functions.

The solid harmonics were homogeneous polynomial solutions of Laplace's equation. By examining Laplace's equation in spherical coordinates, Thomson and Tait recovered Laplace's spherical harmonics. See the section below, "Harmonic polynomial representation".

The term "Laplace's coefficients" was employed by William Whewell to describe the particular system of solutions introduced along these lines, whereas others reserved this designation for the zonal spherical harmonics that had properly been introduced by Laplace and Legendre. The 19th century development of Fourier series made possible the solution of a wide variety of physical problems in rectangular domains, such as the solution of the heat equation and wave equation.

This could be achieved by expansion of functions in series of trigonometric functions. Whereas the trigonometric functions in a Fourier series represent the fundamental modes of vibration in a string , the spherical harmonics represent the fundamental modes of vibration of a sphere in much the same way. Many aspects of the theory of Fourier series could be generalized by taking expansions in spherical harmonics rather than trigonometric functions.

This was a boon for problems possessing spherical symmetry , such as those of celestial mechanics originally studied by Laplace and Legendre. The prevalence of spherical harmonics already in physics set the stage for their later importance in the 20th century birth of quantum mechanics. The spherical harmonics are eigenfunctions of the square of the orbital angular momentum operator. Laplace's equation imposes that the Laplacian of a scalar field f is zero. In spherical coordinates this is: [2].

By separation of variables , two differential equations result by imposing Laplace's equation:. Applying separation of variables again to the second equation gives way to the pair of differential equations.

These angular solutions are a product of trigonometric functions , here represented as a complex exponential , and associated Legendre polynomials:.

Such an expansion is valid in the ball. In quantum mechanics, Laplace's spherical harmonics are understood in terms of the orbital angular momentum [4]. The spherical harmonics are eigenfunctions of the square of the orbital angular momentum. Laplace's spherical harmonics are the joint eigenfunctions of the square of the orbital angular momentum and the generator of rotations about the azimuthal axis:. These operators commute, and are densely defined self-adjoint operators on the weighted Hilbert space of functions f square-integrable with respect to the normal distribution as the weight function on R 3 :.

Furthermore, L 2 is a positive operator. If Y is a joint eigenfunction of L 2 and L z , then by definition. Furthermore, since. This polynomial is easily seen to be harmonic. Several different normalizations are in common use for the Laplace spherical harmonic functions.

In acoustics , [7] the Laplace spherical harmonics are generally defined as this is the convention used in this article. This normalization is used in quantum mechanics because it ensures that probability is normalized, i.

The disciplines of geodesy [10] and spectral analysis use. The magnetics [10] community, in contrast, uses Schmidt semi-normalized harmonics. In quantum mechanics this normalization is sometimes used as well, and is named Racah's normalization after Giulio Racah. Alternatively, this equation follows from the relation of the spherical harmonic functions with the Wigner D-matrix.

In the quantum mechanics community, it is common practice to either include this phase factor in the definition of the associated Legendre polynomials , or to append it to the definition of the spherical harmonic functions.

There is no requirement to use the Condon—Shortley phase in the definition of the spherical harmonic functions, but including it can simplify some quantum mechanical operations, especially the application of raising and lowering operators. The geodesy [11] and magnetics communities never include the Condon—Shortley phase factor in their definitions of the spherical harmonic functions nor in the ones of the associated Legendre polynomials.

A real basis of spherical harmonics can be defined in terms of their complex analogues by setting. The Condon-Shortley phase convention is used here for consistency. The corresponding inverse equations are. The real spherical harmonics are sometimes known as tesseral spherical harmonics. The reason for this can be seen by writing the functions in terms of the Legendre polynomials as.

The same sine and cosine factors can be also seen in the following subsection that deals with the cartesian representation. As is known from the analytic solutions for the hydrogen atom, the eigenfunctions of the angular part of the wave function are spherical harmonics.

This is why the real forms are extensively used in basis functions for quantum chemistry, as the programs don't then need to use complex algebra. Here, it is important to note that the real functions span the same space as the complex ones would. Essentially all the properties of the spherical harmonics can be derived from this generating function.

They are, moreover, a standardized set with a fixed scale or normalization. It may be verified that this agrees with the function listed here and here. The spherical harmonics have deep and consequential properties under the operations of spatial inversion parity and rotation. The spherical harmonics have definite parity. That is, they are either even or odd with respect to inversion about the origin. The statement of the parity of spherical harmonics is then. That is,.

However, this is not the standard way of expressing this property. In the standard way one writes,. The rotational behavior of the spherical harmonics is perhaps their quintessential feature from the viewpoint of group theory.

Many facts about spherical harmonics such as the addition theorem that are proved laboriously using the methods of analysis acquire simpler proofs and deeper significance using the methods of symmetry. The Laplace spherical harmonics form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions.

On the unit sphere, any square-integrable function can thus be expanded as a linear combination of these:. This expansion holds in the sense of mean-square convergence — convergence in L 2 of the sphere — which is to say that.

This is justified rigorously by basic Hilbert space theory. For the case of orthonormalized harmonics, this gives:. The convergence of the series holds again in the same sense, but the benefit of the real expansion is that for real functions f the expansion coefficients become real. As a rule, harmonic functions are useful in theoretical physics to consider fields in far-zone when distance from charges is much further than size of their location.

At the same time, it is important to get invariant form of solutions relatively to rotation of space or generally speaking, relatively to group transformations. It is easy to verify that they are the harmonical functions. James Clerk Maxwell used similar considerations without tensors naturally. Hobson analysed Maxwell's method as well. Formula for harmonical invariant tensor was found in paper. The last relation is Euler formula for homogeneous polynomials actually. The two relations allows to substitute found tensor into Laplace equation and to check straightly that tensor is the harmonical function:.

The last property is important for theoretical physics for the following reason. Potential of charges outside of their location is integral to be equal to the sum of multipole potentials:. The convolution is applied to tensors in the formula naturally. Integrals in the sum are called in physics as multipole moments.

Three of them are used actively while others applied less often as their structure or that of spherical functions is more complicated. Therefore, simplified moments give the same result and there is no need to restrict calculations for dipole, quadrupole and octupole potentials only.

It is the advantage of the tensor point of view and not the only that. Spherical functions have a few recurrent formulas. The property is occurred due to symmetry group of considered system.

The vector ladder operator for the invariant harmonical states found in paper [20] and detailed in. The ladder operator is analogous for that in problem of the quantum oscillator. It generates Glauber states those are created in the quantum theory of electromagnetic radiation fields. Spherical harmonics accord with the system of coordinates.


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Spherical polar coordinates are convenient for the description of 3-dimensional physical systems that posses spherical or near-spherical symmetry; for such systems they are preferred over other coordinate systems such as Cartesian or cylinder coordinates. Spherical harmonics are ubiquitous in atomic and molecular physics. In quantum mechanics they appear as eigenfunctions of squared orbital angular momentum. Further, they are important in the representation of the gravitational and magnetic fields of planetary bodies, the characterization of the cosmic microwave background radiation, the rotation-invariant description of 3D shapes in computer graphics, the description of electrical potentials due to charge distributions, and in certain types of fluid motion. Completeness implies that this expansion converges to an exact result for sufficient terms. In an approximate non-converged expansion, the expansion coefficients may be used as linear regression parameters, meaning that they may be chosen such that the expanded function gives a best fit to the original function, which means that the two functions will "resemble" each other as closely as possible.


Condon-Shortley Phase

By using our site, you acknowledge that you have read and understand our Cookie Policy , Privacy Policy , and our Terms of Service. Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It only takes a minute to sign up. And, also they claim this is useful in quantum mechanical operations, such as raising and lowering. You don't need it: it's a sign convention and the only thing you need to do with it is to be consistent.


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This is a classic, which should be in the library of anyone who works in the field of atomic spectroscopy. The Theory of Atomic Spectra. Edward Uhler Condon , E. Condon , G. Condon and Shortley has become the standard comprehensive work on the theory of atomic spectra. The first two chapters contain a brief historical introduction and an exposition of quantum mechanics along the lines formulated by Dirac.

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