A group-theoretical description of N identical harmonic oscillators on a line is presented. It provides a scheme for labeling the energy eigenstates that are invariant under the permutation group SN and for obtaining the symmetric operators that transform these degenerate eigenfunctions among themselves. The symmetry algebra that these generators form is in general polynomial. The 2- and 3-particle cases are considered in detail. For the simple 2-body problem the invariance algebra is found to be the cubic SU(2) algebra: [J0,J±]=±J±, [J+,J−]=2J0−αJ03. In the 3-body case, the permutational invariant states are characterized with the help of the subgroup chain U(3)⊃U(2)⊃O(2). The labeling and step operators are obtained from determining an integrity basis for the S3 scalar in U(U(3)). Generating functions techniques are used to that end; an eight-dimensional basis is found whose elements span the symmetry algebra of the three identical oscillator problem. These constants of motion are seen to generate a nonlinear algebra whose representation on the symmetric states is provided.
A group-theoretical description of N identical harmonic oscillators on a line is presented. It provides a scheme for labeling the energy eigenstates that are invariant under the permutation group SN and for obtaining the symmetric operators that transform these degenerate eigenfunctions among themselves. The symmetry algebra that these generators form is in general polynomial. The 2- and 3-particle cases are considered in detail. For the simple 2-body problem the invariance algebra is found to be the cubic SU(2) algebra: [J0,J±]=±J±, [J+,J−]=2J0−αJ03. In the 3-body case, the permutational invariant states are characterized with the help of the subgroup chain U(3)⊃U(2)⊃O(2). The labeling and step operators are obtained from determining an integrity basis for the S3 scalar in U(U(3)). Generating functions techniques are used to that end; an eight-dimensional basis is found whose elements span the symmetry algebra of the three identical oscillator problem. These constants of motion are seen to generate a nonlinear algebra whose representation on the symmetric states is provided.
It is well known that the allowed wavefunctions for an N-electron system should be antisymmetric with respect to the permutation of any pair of electron labels. On the other hand, the Hamiltonian for such system is invariant under any permutation of electron labels and, consequently, its eigenfunctions are basis for the irreducible representations of the symmetric group SN.Here, we investigate which symmetry species of the SN group are compatible with the antisymmetry principle. We illustrate the conclusions by means of simple N-particle one-dimensional models with harmonic interactions.
Abstract It is well known that the allowed wavefunctions for an N -electron system should be antisymmetric with respect to the permutation of any pair of electron labels. On the other hand, the Hamiltonian for such a system is invariant under any permutation of electron labels and, consequently, its eigenfunctions are the basis for the irreducible representations of the symmetric group S N . Here, we investigate which symmetry species of the S N group are compatible with the antisymmetry principle. We illustrate the conclusions by means of simple N -particle one-dimensional models with harmonic interactions.
We describe the implications of permutation symmetry for the state space and dynamics of quantum mechanical systems of matrices of general size N. We solve the general 11-parameter permutation invariant quantum matrix harmonic oscillator Hamiltonian and calculate the canonical partition function. The permutation invariant sector of the Hilbert space, for general Hamiltonians, can be described using partition algebra diagrams forming the bases of a tower of partition algebras Pk(N). The integer k is interpreted as the degree of matrix oscillator polynomials in the quantum mechanics. Families of interacting Hamiltonians are described which are diagonalized by a representation theoretic basis for the permutation invariant subspace which we construct for $N\ensuremath{\ge}2k$. These include Hamiltonians for which the low-energy states are permutation invariant and can give rise to large ground state degeneracies related to the dimensions of partition algebras. A symmetry-based mechanism for quantum many body scars discussed in the literature can be realized in these matrix systems with permutation symmetry. A mapping of the matrix index values to lattice sites allows a realization of the mechanism in the context of modified Bose-Hubbard models. Extremal correlators analogous to those studied in AdS/CFT are shown to obey selection rules based on Clebsch-Gordan multiplicites (Kronecker coefficients) of symmetric groups.
We study the quantum mechanical harmonic oscillator in two and three dimensions, with particular attention to the solutions as basis states for representing their respective symmetry groups — O(2), O(1,1), O(3), and O(2,1). The goal of this study is to establish a correspondence between Hilbert space descriptions found by solving the Schrodinger equation in polar coordinates, and Fock space descriptions constructed by expressing the symmetry operators in terms of creation/annihilation operators. We obtain wavefunctions characterized by a principal quantum number, the group Casimir eigenvalue, and one group generator whose eigenvalue is m + s , for integer m and real constant parameter s . For the three groups that contain O(2), the solutions split into two inequivalent representations, one associated with s = 0, from which we recover the familiar description of the oscillator as a product of one-dimensional solutions, and the other with s > 0 (in three dimensions, solutions are found for s = 0 and s = 1/2) whose solutions are non-separable in Cartesian coordinates, and are hence overlooked by the standard Fock space approach. The O(1,1) solutions are singlet states, restricted to zero eigenvalue of the symmetry operator, which represents the boost, not angular momentum. For O(2), a single set of creation and annihilation operators forms a ladder representation for the allowed oscillator states for any s, and the degeneracy of energy states is always finite. However, in three dimensions, the integer and half-integer eigenstates are qualitatively different: the former can be expressed as finite dimensional irreducible tensors under O(3) or O(2,1) while the latter exhibit infinite degeneracy. Creation operators that produce the allowed integer states by acting on the non-degenerate ground state are constructed as irreducible tensor products of the fundamental vector representation. However, the half-integer eigenstates are infinite-dimensional, as expected for the non-compact O(2,1), and no Fock space description is found for this case. For all s ≠ 0 solutions, the SU(N) symmetry of the harmonic oscillator Hamiltonian recently discovered by Bars is spontaneously broken by the ground state. The connection of this symmetry breaking to the non-separability into one-dimensional Cartesian solutions is demonstrated. We conclude with a discussion of the O(2,1) solutions, and the problem of timelike ghost excitations.
We construct representations of the symmetry algebra and derive the exact spectrum of a three-dimensional oscillator with two spin-21 degrees of freedom. The Hamiltonian and its first-order scalar and pseudo-scalar integrals were obtained in earlier classifications. Here we determine their action on the complete spectral basis. At fixed total angular momentum, a quadratic algebra acts on the three triplet orbital channels and admits a three-dimensional deformed-oscillator representation, while an overlapping two-state exchange algebra joins the middle triplet to the singlet. These finite modules determine angular--spin multiplicities but do not quantize the radial energy. A separate positive-discrete-series su(1,1) representation of the common radial Hamiltonian generates the energy sequence. We give the finite matrices, Casimir and structure functions, normalized eigenfunctions, radial ladder actions, the exceptional j=0 sector, complete energy-shell multiplicities, and the effect of particle interchange.
This study presents a complete algebraic formulation of the quantum harmonic oscillator, demonstrating that the Hermite polynomials, H n ( x ) , emerge as the fundamental solution when the power-series method is applied to the Schrödinger equation. The physical requirement of square integrability enforces the truncation of the series, leading precisely to these polynomials, which constitute the polynomial part of the energy eigenfunctions, ψ n ( x )= N n H n ( αx ) e − α 2 x 2 /2 , where n is the quantum number. This quantum number directly determines the quantized energy levels through E n =ℏω( n+ 1 2 ) . The analysis based on creation and annihilation operators further reveals that the action of the creation operator on the ground state generates the excited states along with their corresponding Hermite functions. The orthogonality of these polynomials ensures the orthonormality of the complete set of eigenfunctions. Additionally, the geometric Berry phase emerges as a natural extension when the system undergoes adiabatic cyclic changes, revealing a deeper topological structure that complements the conventional algebraic and analytic descriptions. Therefore, Hermite polynomials encapsulate both the mathematical structure and the physical essence of quantization in this fundamental system.