Quantum systems of three identical particles on a plane are analyzed from the viewpoint of symmetry. Upon reduction by rotation, such systems are described in the space of sections of a line bundle over a three-dimensional shape space whose origin represents triple collision. It is shown that if the total angular momentum is nonzero, then the wave section must vanish at the origin, while if it is zero, then the wave section can be finite at the origin. Since the particles are assumed to be identical, the quantum system admits the action of the symmetric group S3 as well, which stands for the group of particle exchanges and is commutative with rotation. Hence the reduced system still admits the S3 action, so that Bose and Fermi states can be discussed in the space of sections of the line bundle. A detailed analysis of a system of three free particles on a plane is presented in the latter part of the article.
Quantum systems of three identical particles on a plane are analyzed from the viewpoint of symmetry. Upon reduction by rotation, such systems are described in the space of sections of a line bundle over a three-dimensional shape space whose origin represents triple collision. It is shown that if the total angular momentum is nonzero, then the wave section must vanish at the origin, while if it is zero, then the wave section can be finite at the origin. Since the particles are assumed to be identical, the quantum system admits the action of the symmetric group S3 as well, which stands for the group of particle exchanges and is commutative with rotation. Hence the reduced system still admits the S3 action, so that Bose and Fermi states can be discussed in the space of sections of the line bundle. A detailed analysis of a system of three free particles on a plane is presented in the latter part of the article.
According to symmetrization postulate for a system of identical particles, wave function has to be completely symmetric or completely anti-symmetric. In this paper we want to mathematically justify this postulate ignoring the spin part of wave function in three dimension. For a system of N identical particles, if the solution to the governing Schrodinger equation meets these criteria: a) the probability density remains invariant when any two particle positions are exchanged over time, b) the wave function is continuous and has a continuous gradient, and the system exhibits the following characteristics: c) the configuration space, which is 3N dimensional, is connected, and d) the potential term in the Hamiltonian is invariant under the exchange of any two particle positions, then the wave function must be either totally symmetric or totally antisymmetric over time.
The unitary group acting on the Hilbert space H:=(C2)⊗3 of three quantum bits admits a Lie subgroup, US3(8), of elements which permute with the symmetric group of permutations of three objects. Under the action of such a Lie subgroup, the Hilbert space H splits into three invariant subspaces of dimensions 4, 2 and 2 respectively, each corresponding to an irreducible representation of su(2). The subspace of dimension 4 is uniquely determined and corresponds to states that are themselves invariant under the action of the symmetric group. This is the so called {\it symmetric sector.} The subspaces of dimension two are not uniquely determined and we parametrize them all. We provide an analysis of pure states that are in the subspaces invariant under US3(8). This concerns their entanglement properties, separability criteria and dynamics under the Lie subgroup US3(8). As a physical motivation for the states and dynamics we study, we propose a physical set-up which consists of a symmetric network of three spin 21 particles under a common driving electro-magnetic field. {For such system, we solve the control theoretic problem of driving a separable state to a state with maximal distributed entanglement.
In both classical and quantum mechanics, a dynamical system with symmetry is reduced to a system which has a smaller number of degrees of freedom. In this paper we formulate path integrals for the quantum system reduced by symmetry. The original system is defined in terms of a Riemannian manifold M, on which a Lie group G acts by isometric transformations. Then we show that the path integral on M is reduced to a family of path integrals on a quotient space Q = M/G and that the reduced path integrals are classified by irreducible unitary representations of G. Stratification geometry of the set of spaces (M, Q), which is a generalization of the concept of principal fiber bundle, is used to describe the reduced path integral and consequently the path integral is expressed as a product of three factors; the rotational energy amplitude, which represents integration over the fiber directions, the vibrational energy amplitude, which represents integration over the directions perpendicular to the fiber, and the holonomy factor, which is caused by non-integrability of the vibrational directions. An attention is also paid for a singular point in Q, which arises when the group action of G on M is not free. We determine the boundary condition of the path integral kernel for a path which runs through the singularity.
Singular configurations have long been out of consideration in the study of many-particle systems. In this paper we show that a group theoretical method can provide boundary conditions on wavefunctions at singular configurations of many particles. The consequences of this method are qualitatively stated as follows. (i) If the projection of angular momentum on an axis does not vanish, the particles will not be aligned on the axis, i.e. the wavefunction vanishes at collinear configurations along the axis. (ii) If the total angular momentum does not vanish, the simultaneous collision at a point will not take place, i.e. the wavefunction vanishes at the configuration of simultaneous collision. Furthermore, on the assumption of analyticity of the wavefunction at singular configurations, the behaviour of the wavefunction around the singular configurations shows that (a) the larger the projection of angular momentum on an axis becomes, the less the particles are likely to be aligned on the line, and that (b) the more the total angular momentum grows, the less likely the particles are to collide at a point. The proof is carried out for three-body systems by transforming the power series expansion of a wavefunction into a Fourier series expansion in terms of the angular momentum eigenfunctions, in both cases of collinear configurations and of the triple collision, without reference to the Hamiltonian operator. These results are described quantitatively in terms of the angular momentum quantum numbers in propositions 1 and 3 in the text.
The Hamiltonian for a system of n particles that are identical, that is, particles that can be substituted for each other with no physical change, must be completely symmetric under any interchange of its arguments. Let Ψ(1, 2, …, n) be any solution of the Schrödinger equation, depending on the coordinates (spatial and spin) of the n identical particles 1 to n, and let P be any permutation of the n numbers 1 to n. Then PΨ(1, 2, …, n) ≡ Ψ(P1, P2, …, Pn) is a function which depends on the coordinates of particle Ψ in the same manner as the original function Ψ depended on the coordinates of particle i. Then using the symmetry of H, we can show that the operator P commutes with the Hamiltonian ( 2-1 ) H ( P Ψ ) = P ( H Ψ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429493645/749bab20-4a79-4f5a-8c66-d9e9a36a87b9/content/eq89.tif"/>