Research Article | Open Access
H. Fakhri, "Spherical Harmonics : Positive and Negative Integer Representations of for and ", Advances in High Energy Physics, vol. 2016, Article ID 3732657, 7 pages, 2016. https://doi.org/10.1155/2016/3732657
Spherical Harmonics : Positive and Negative Integer Representations of for and
The azimuthal and magnetic quantum numbers of spherical harmonics describe quantization corresponding to the magnitude and -component of angular momentum operator in the framework of realization of Lie algebra symmetry. The azimuthal quantum number allocates to itself an additional ladder symmetry by the operators which are written in terms of . Here, it is shown that simultaneous realization of both symmetries inherits the positive and negative - and -integer discrete irreducible representations for Lie algebra via the spherical harmonics on the sphere as a compact manifold. So, in addition to realizing the unitary irreducible representation of compact Lie algebra via the ’s for a given , we can also represent noncompact Lie algebra by spherical harmonics for given values of and .
The set of principal, azimuthal, magnetic, and spin quantum numbers describe the unique quantum state of a single electron for any system in which the potential depends only on the radial coordinate. The labels and of the usual complex spherical harmonics are the second and the third numbers of this set. Spherical surface harmonics are an orthonormal set of vibration solutions for eigenvalue equation of the Laplace-Beltrami operator on the sphere as a compact Riemannian manifold. They also form the wave functions which represent the orbital angular momentum operator () and have a wide range of applications in theoretical and applied physics [1–4]. The three components of the angular momentum operator, that is , , and , are Killing vector fields that generate the rotations about -, -, and -axes, respectively. All spherical harmonics with the given quantum number form a unitary irreducible representation of Lie algebra. In fact, they can be seen as representations of the symmetry group of rotations about a point and its double-cover . It can also be noted that the spherical harmonics are just the independent components of symmetric traceless tensors of rank . The properties of the spherical harmonics are well known and may be found in many texts and papers (e.g., see [5–9]).
In spite of the fact that the problem of quantization of particle motion on a sphere is 80 years old, there still exist some open questions concerning the symmetry properties of the bound states. The aim of this work is to introduce new symmetries based on the quantization of both azimuthal and magnetic numbers and of the usual spherical harmonics . In order to provide the necessary background and also to attribute a quantization relation for azimuthal quantum number , here, we present some basic facts about the spherical harmonics [2, 3]. In Section 2, with the application of angular momentum operator, we review realization of the unitary irreducible representations of Lie algebra on the sphere in terms of spherical harmonics by shifting only. In Section 3, the representations of the ladder symmetry with respect to azimuthal quantum number are constructed in terms of a pair of ladder operators, and its corresponding quantization relation is also expressed as an operator identity originated from solubility in the framework of supersymmetry and shape invariance theories. In Section 4, these results are applied to show that Lie algebra can also be represented irreducibly by using spherical harmonics. Finally, in Section 5 we discuss the results and make some final comments.
2. The Unitary Irreducible Representations of Lie Algebra via Orbital Angular Momentum Operator
This section covers the standard and the well-known formalism of commutation relations in order to encounter spherical harmonics. In what follows, we describe points on using the parametrization (, , ) where is the polar (or colatitude) angle and is the azimuthal (or longitude) angle. For a given with the lower bound , we define the -dimensional Hilbert space with the spherical harmonics as bases: Also, the infinite dimensional Hilbert space is defined as a direct sum of finite dimensional subspaces: . We must emphasize that the bases of are independent spherical harmonics with different values for both indices and . The spherical harmonics as the bases of constitute an orthonormal set with respect to the following inner product over the sphere :Therefore, similar to the Fourier expansion, they can be used to expand any arbitrary square integrable function of latitude and longitude angles. is the natural invariant measure (area) on the sphere . The following proposition is an immediate consequence of the raising and lowering relations of the index of the associated Legendre functions [2, 4, 10].
Proposition 1. Let one introduce three differential generators , , and on the sphere , corresponding to the orbital angular momentum operator as They satisfy the commutation relations of Lie algebra as follows: is a self-adjoint operator, and two operators and are Hermitian conjugate of each other with respect to the inner product (3). Each of the Hilbert subspaces realizes an -integer unitary irreducible representation of Lie algebra as The Hilbert subspace contains the lowest and highest bases with the lowest and highest weights and , respectively. They are annihilated by the operators and : and . Meanwhile, an arbitrary basis belonging to each of the Hilbert subspaces can be calculated by an algebraic method as follows: Also, the Casimir operator corresponding to the generators (4), that is, is a self-adjoint operator and has a -fold degeneracy on as
A given unitary irreducible representation is characterized by the index . The spherical harmonics , via their index, describe quantization corresponding to commutation relations of the three components of orbital angular momentum operator. is always a Killing vector field which corresponds to an angular momentum about the body-fixed -axis. The Casimir operator along with the Cartan subalgebra generator describes the Hamiltonian of a free particle on the sphere with dynamical symmetry group and -fold degeneracy for the energy spectrum. It must be emphasized that the spherical harmonics and their mathematical structure, as given by Proposition 1, are playing a more visible and important role in different branches of physics. Proposition 1 implies that the spherical harmonics are created by orbital angular momentum operator. Schwinger has developed the realization of this proposition in the framework of creation and annihilation operators of two-dimensional isotropic oscillator .
3. Ladder Symmetry for the Azimuthal Quantum Number
It is evident that simultaneous realization of laddering relations with respect to two different parameters and of the associated Legendre functions gives us the possibility to represent laddering relations with respect to the azimuthal quantum number of spherical harmonics. Representation of such ladder symmetry by the spherical harmonics with the same but different induces a new splitting on the Hilbert space : The following proposition provides an alternative characterization of the mathematical structure of spherical harmonics.
Proposition 2. Let one define two first-order differential operators on the sphere : They satisfy the following operator identity in the framework of shape invariance theory: are the adjoint of the operators with respect to the inner product (3); that is, one has . Each of the Hilbert subspaces realizes the semi-infinite raising and lowering relations with respect to as The lowest bases, that is, belonging to the Hilbert subspaces with and , are, respectively, annihilated by and as and . Meanwhile, an arbitrary basis belonging to each of the Hilbert subspaces with and can be calculated by the algebraic method:
Proof. The proof follows immediately from the raising and lowering relations of the index of the associated Legendre functions .
According to the limitation obtained from the commutation relations of , must be an odd and even nonnegative integer for the orbital and spin angular momenta, respectively. Although the relation (13) is identically satisfied for any constant number , however, it is represented only via the nonnegative integers (odd positive integer values for ) of spherical harmonics . This is an essential difference with respect to the spin angular momentum. In fact, the relation (13) distinguishes the orbital angular momentum from the spin one. It also implies that the number of independent components of spherical harmonics of a given irreducible representation of Lie algebra, that is, , is derived by the shift operators corresponding to the azimuthal quantum number . If we take the adjoint of (13), we obtain , which is identically satisfied. Thus, Proposition 2 presents a symmetry structure, called ladder symmetry with respect to the azimuthal quantum number of spherical harmonics. Note that, indeed, the identical equality (13) has been originated from a brilliant theory in connection with geometry and physics named supersymmetry. In other words, although contrary to and the two operators and do not contribute in a set of closed commutation relations, however, the operator identity (13) for them can be interpreted as a quantization relation in the framework of shape invariance symmetry (for reviews about supersymmetric quantum mechanics and shape invariance, see [12–17]). Thus, the operators and describe quantization of the azimuthal quantum number which, in turn, lead to the presentation of a different algebraic technique from (8), in order to create the spherical harmonics , according to (16). Furthermore, spherical harmonics belonging to the Hilbert subspaces have parity , since commutes with the parity operator. Thus, the operators and can be interpreted as the interchange operators of parity: and .
4. Positive and Negative Integer Irreducible Representations of for
The laddering equations (6a) and (6b) as well as (14a) and (14b), which describe shifting the indices and separately, lead to the derivation of two new types of simultaneous ladder symmetries with respect to both azimuthal and magnetic quantum numbers of spherical harmonics. Our proposed ladder operators for simultaneous shift of and are of first-order differential type, contrary to . They lead to a new perspective on the two quantum numbers and in connection with realization of (consequently, ) Lie algebra which in turn is accomplished by all spherical harmonics with constant values for and , separately. First, it should be pointed out that the Hilbert space can be split into the infinite direct sums of infinite dimensional Hilbert subspaces in two different ways as follows: The constant values for the expressions and of spherical harmonics have been labeled by and , respectively.
Proposition 3. Let one define two new first-order differential operators on the sphere : They, together with the generators and 1, satisfy the commutation relations of Lie algebra are the adjoint of the operators with respect to the inner product (3); that is, one has . Each of the Hilbert subspaces realizes separately -integer irreducible positive representations of Lie algebra as (It must be pointed out that, by defining and , the Lie algebra (19) can be considered as commutation relations corresponding to the Lie algebra: and . This means that 1 is a trivial center for the semisimple Lie algebra . In , a short review on the three different real forms , , and of Lie algebra has been presented. There, their differences in connection with the structure constants and their representation spaces have also been pointed out.) Also, the Casimir operator corresponding to the generators , , and , has an infinite-fold degeneracy on the Hilbert subspace as The Hilbert subspaces and with and as nonnegative and positive integers contain, respectively, the following lowest bases: They are annihilated as and and also have the lowest weights and . Meanwhile, the arbitrary bases of the Hilbert subspaces and can be, respectively, calculated by the algebraic methods as
Proof. The relations (18), (20a), and (20b) can be followed from the realization of laddering relations with respect to both azimuthal and magnetic quantum numbers and , simultaneously and agreeably. It is sufficient to consider that two new differential operators satisfy the simultaneous laddering relations with respect to and as The relations (26a) and (26b) are obtained from (6a), (6b), (14a), and (14b). The relations (19) and (20c) are directly followed. The adjoint relation between the operators can be easily checked by means of the inner product (3). The commutativity of operators , , and with results from (19). The eigenequation (22) follows immediately from the representation relations (20a)–(20c). The relation (20b) implies that and are the lowest bases for the Hilbert subspaces and , respectively. Then, with repeated application of the raising relation (20a), one may obtain the arbitrary representation bases of Lie algebra as (24a) and (24b).
Although the commutation relations (19) are not closed with respect to taking the adjoint, however, their adjoint relations and are identically satisfied.
Proposition 4. Let one define two new first-order differential operators on the sphere as They, together with the generators and 1, satisfy the commutation relations of Lie algebra as are the adjoint of the operators with respect to the inner product (3); that is, one has . Each of the Hilbert subspaces realizes separately -integer irreducible positive representations of Lie algebra as Also, the Casimir operator corresponding to the generators , , and , has an infinite-fold degeneracy on the Hilbert subspace as The Hilbert subspaces and with and as nonnegative and positive integers contain, respectively, the following highest bases: They are annihilated as and and also have the highest weights and . Meanwhile, the arbitrary bases of the Hilbert subspaces and can be, respectively, calculated by the algebraic methods as
Proof. The proof is quite similar to the proof of Proposition 3. So, we have to take into account that the two new differential operators are represented by spherical harmonics whose corresponding laddering equations shift both the azimuthal and magnetic quantum numbers and simultaneously and inversely:
Here, again the adjoint of commutation relations (28) becomes and , which are identically satisfied.
Thus, all unitary and irreducible representations of of dimensions with the nonnegative integers can carry the new kind of irreducible representations for . The new symmetry structures presented in the two recent propositions, the so-called positive and negative discrete representations of , in turn, describe the simultaneous quantization of the azimuthal and magnetic quantum numbers. Therefore, the Hilbert spaces of all spherical harmonics not only represent compact Lie algebra by ladder operators shifting for a given , but also represent the noncompact Lie algebra by simultaneous shift operators of both quantum labels and for given values and .
5. Concluding Remarks
For a given azimuthal quantum number , quantization of the magnetic number is customarily accomplished by representing the operators , , and on the sphere with the commutation relations compact Lie algebra, in a -dimensional Hilbert subspace . Furthermore, for a given magnetic quantum number , quantization of the azimuthal number is accomplished by representing the operators and on the sphere with the identity relation (13), in an infinite-dimensional Hilbert subspace .
Dealing with these issues together, simultaneous quantization of both azimuthal and magnetic numbers and is accomplished by representing two bunches of operators and on the sphere with their corresponding commutation relations of noncompact Lie algebra, in the infinite-dimensional Hilbert subspaces and , respectively. For given values and , they are independent of each other, the so-called positive and negative - and -integer irreducible representations, respectively. As the spherical harmonics are generated from by the operators , they are also generated from and by and , as well as from and by and , respectively. Therefore, not only ’s with the given value for represent Lie algebra, but also ’s with the given values for subtraction and summation of the both quantum numbers and represent separately (hence, ) Lie algebra as well. In other words, two different real forms of Lie algebra, that is, and , are represented by the space of all spherical harmonics . This happens because the quantization of both quantum numbers and are considered jointly. Indeed, we have
We point out that the idea of this paper may find interesting applications in quantum devices. For instance, coherent states of the noncompact Lie group have been defined by Barut and Girardello as eigenstates of the ladder operators , and by Perelomov as the action of the displacement operator on the lowest and highest bases [20, 21]. So, our approach to the representation of noncompact Lie algebra provides the possibility of constructing two different types of coherent states of on compact manifold . Also, realization of the additional symmetry named Lie algebra for Landau levels and bound states of a free particle on noncompact manifold can be found based on the above considerations.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
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Copyright © 2016 H. Fakhri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.