From the point of view of functional analysis, wave functions are understood as vector in Hilbert’s space of quantum states. When the number of identical particles is greater than one, there appears a specific algebraic structure, where the same Hilbert space is a graded algebra over the ring of symmetric polynomials (bosonic excitations). Generators of the algebra (vacuums) are known in the Cartesian basis. In this work we translate them to the basis of good angular momentum for the specific case of two particles. Then three generators (Ψ1; Ψ2; Ψ3) make up the ground-state vector and the fourth Ψ4 = Ψ1Ψ2Ψ3 is a pseudoscalar found among doubly excited states, of which there are 28. When all those states are written in the form of bosonic excitations of the vacuums and in the basis of good angular momentum, it is revealed that Ψ4 is a component of a specific excited state of angular momentum l=3 and projection m = ±2. Therefore this state is identified as a band head of excited states with different symmetry then the band which is generated by excitations of the ground state.
From the point of view of functional analysis, wave functions are understood as vector in Hilbert’s space of quantum states. When the number of identical particles is greater than one, there appears a specific algebraic structure, where the same Hilbert space is a graded algebra over the ring of symmetric polynomials (bosonic excitations). Generators of the algebra (vacuums) are known in the Cartesian basis. In this work we translate them to the basis of good angular momentum for the specific case of two particles. Then three generators (Ψ1; Ψ2; Ψ3) make up the ground-state vector and the fourth Ψ4 = Ψ1Ψ2Ψ3 is a pseudoscalar found among doubly excited states, of which there are 28. When all those states are written in the form of bosonic excitations of the vacuums and in the basis of good angular momentum, it is revealed that Ψ4 is a component of a specific excited state of angular momentum l=3 and projection m = ±2. Therefore this state is identified as a band head of excited states with different symmetry then the band which is generated by excitations of the ground state.
Many-body Hilbert space is a functional vector space with the natural structure of an algebra, in which vector multiplication is ordinary multiplication of wave functions. This algebra is finite-dimensional, with exactly [Formula: see text] generators for N identical particles, bosons or fermions, in d dimensions. The generators are called shapes. Each shape is a possible many-body vacuum. Shapes are natural generalizations of the ground-state Slater determinant to more than one dimension. Physical states, including the ground state, are superpositions of shapes with symmetric-function coefficients, for both bosons and fermions. These symmetric functions may be interpreted as bosonic excitations of the shapes. The algebraic structure of Hilbert space described here provides qualitative insights into long-standing issues of many-body physics, including the fermion sign problem and the microscopic origin of bands in the spectra of finite systems.
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.
The conventional method of constructing wave functions of pure angular momentum for a composite system is based on the idea of coupling the angular momenta of the constituents. Instead of this synthetic method, an analytical approach is here introduced. It is pointed out that an arbitrary trial wave function for the total system in a unique way must be resolvable into orthogonal components of pure angular momentum associated with different quantum numbers. A particular component can be selected by means of certain 'projection operators', which annihilate all components except the one desired. Such a projection operator is simply a product of commuting factors, each one of which annihilates a specific eigenfunction to the angular momentum under consideration. Physically this idea is of importance, since one can now start out from a rough model wave function based essentially on qualitative arguments, for instance of the independent-particle type, and then obtain a mathematically and physically much better trial function by selecting the particular component of the original function which has the correct symmetry type desired. The projection operators are studied in some detail both as products and in expanded form. The case of degenerate subspaces of the same angular momentum quantum numbers is discussed, and the problem of constructing an orthogonal subset of functions is solved by a simple elimination procedure; the connection with the conventional seniority idea is briefly discussed.
A completely algebraic and representation-independent solution is presented of the simultaneous eigenvalue problem for H, L2, and L3, where H is the Hamiltonian operator for the three-dimensional, isotropic harmonic oscillator, and L is its angular momemtum vector. It is shown that H can be written in the form h/ω(2ν°ν+λ°⋅λ+3/2), where ν° and ν are raising and lowering (boson) operators for ν°ν, which has nonnegative integer eigenvalues k; and λ° and λ are raising and lowering operators for λ°⋅λ, which has nonnegative integer eigenvalues l, the total angular momentum quantum number. Thus the eigenvalues of H appear in the familiar form h/ω(2k+l+3/2), previously obtained only by working in the coordinating the operators ν° and λ° to a ’’vacuum’’ vector on which ν and λ vanish. The Lie algebra so(2,1) ⊕ so(3,2) is shown to be a spectrum-generating algebra for this problem. It is suggested that coherent angular momentum states can be defined for the oscillator, as the eigenvectors of the lowering operators ν and λ. A brief discussion is given of the classical counterparts of ν,ν°, λ, and λ°, in order to clarify their physical interpretation.
Abstract Harmonic wave functions for integer and half-integer angular momentum are given in terms of the Euler angles (θ, ϕ, ψ) that define a rotation in SO(3), and the Euclidean norm r in <?CDATA R3?> R 3 , keeping the usual meaning of the spherical coordinates (r, θ, ϕ). They form a Hilbert (super)-space decomposed in the form <?CDATA H=H0⊕H1?> = 0 ⊕ 1 . Following a classical work by Schwinger, 2-dimensional harmonic oscillators are used to produce raising and lowering operators that change the total angular momentum eigenvalue of the wave functions in half units. The nature of the representation space <?CDATA H?> is approached from the double covering group homomorphism SU(2) → SO(3) and the topology involved is taken care of by using the Hurwitz-Hopf map <?CDATA H:R4→R3?> H : R 4 → R 3 . It is shown how to reconsider H as a 2-to-1 group map, <?CDATA G0=R+×SU(2)→R+×SO(3)?> G 0 = R + × SU ( 2 ) → R + × SO ( 3 ) , translating it into an assignment (z 1, z 2) ↦ (r, θ, ϕ, ψ) whose domain consists of pairs (z 1, z 2) of complex variables, under the appropriate identification of <?CDATA R4?> R 4 with <?CDATA C2?> C 2 . It is shown how the Lie algebra of G 0 is coupled with two Heisenberg Lie algebras of 2-dimensional (Schwinger’s) harmonic oscillators generated by the operators <?CDATA {z1,z2,zˉ1,zˉ2}?> { z 1
Hilbert space and inner product In Ch. 2 it was noted that quantum wave functions form a linear space in the sense that multiplying a function by a complex number or adding two wave functions together produces another wave function. It was also pointed out that a particular quantum state can be represented either by a wave function ψ( x ) which depends upon the position variable x , or by an alternative function ψ( p ) of the momentum variable p . It is convenient to employ the Dirac symbol |ψ〉, known as a “ket”, to denote a quantum state without referring to the particular function used to represent it. The kets, which we shall also refer to as vectors to distinguish them from scalars , which are complex numbers, are the elements of the quantum Hilbert space H . (The real numbers form a subset of the complex numbers, so that when a scalar is referred to as a “complex number”, this includes the possibility that it might be a real number.) If α is any scalar (complex number), the ket corresponding to the wave function ɑψ( x ) is denoted by α|ψ〉, or sometimes by |ψ〉α, and the ket corresponding to ø( x )+ψ( x ) is denoted by |ø〉+|ψ〉 or |ψ〉+|ø〉, and so forth. This correspondence could equally well be expressed using momentum wave functions, because the Fourier transform, (2.15) or (2.16), is a linear relationship between ψ( x ) and ( p ), so that αø( x )+βψ( x ) and αϕ( p ) + βψ( p ) correspond to the same quantum state α|ψ〉+β|ø〉.