The energy EN of a system of N identical pairwise-interacting particles of mass M is shown to satisfy the inequalities where N > n ⩾ 2. These inequalities generalize the Stenschke result E 3(M) ⩾ 3E 2(3M/2). They hold for Bose or Fermi systems in any number of dimensions, and are shown to be the strongest possible of this type. Further extensions are made to the case where the potential is varied instead of the mass, or where both are varied together.
The energy EN of a system of N identical pairwise-interacting particles of mass M is shown to satisfy the inequalities where N > n ⩾ 2. These inequalities generalize the Stenschke result E 3(M) ⩾ 3E 2(3M/2). They hold for Bose or Fermi systems in any number of dimensions, and are shown to be the strongest possible of this type. Further extensions are made to the case where the potential is varied instead of the mass, or where both are varied together.
Lower bounds to the ground-state energy of a system composed of N identical particles interacting by two-body forces (and possibly also with an external potential) are derived. They are expressed in terms of the ground-state energies of (simpler) systems composed of a smaller number of particles having masses and/or interactions different from the original ones. This reduction process may be continued all the way down to systems involving one and two particles only. The generalization to systems containing some identical and some distinguishable particles is also discussed.
We derive a set of inequalities which relate the translation invariant problem of N identical particles with pairwise interactions to an independent particle problem. These inequalities apply to attractive power law potentials V(r)=rq, 1≤q≤∞, and superpositions of such potentials; they become identities in the harmonic oscillator case q=2. We use the inequalities to derive new upper and lower bounds for the ground state energies of fermion systems, which interact through these potentials. These bounds improve all previous results in the range 1≤q≤∞; they reduce to the exact answer in the harmonic oscillator case.
The quantum-mechanical problems of N 1-dimensional equal particles of mass m interacting pairwise via quadratic (``harmonical'') and/or inversely quadratic (``centrifugal'') potentials is solved. In the first case, characterized by the pair potential ¼mω2(xi − xj)2 + g(xi − xj)−2, g > −ℏ2/(4m), the complete energy spectrum (in the center-of-mass frame) is given by the formula E=ℏω(12N)12[12(N−1)+12N(N−1)(a+12)+ ∑ l=2Nlnl],with a = ½(1 + 4mgℏ−2)½. The N − 1 quantum numbers nl are nonnegative integers; each set {nl; l = 2, 3, ⋯, N} characterizes uniquely one eigenstate. This energy spectrum can also be written in the form Es = ℏω(½N)½ [½(N − 1) + ½N(N − 1)(a + ½) + s], s = 0, 2, 3, 4, ⋯, the multiplicity of the sth level being then given by the number of different sets of N − 1 nonnegative integers nl that are consistent with the condition s=∑l=2Nlnl. These equations are valid independently of the statistics that the particles satisfy, if g ≠ 0; for g = 0, the equations remain valid with a = ½ for Fermi statistics, a = −½ for Bose statistics. The eigenfunctions corresponding to these energy levels are not obtained explicitly, but they are rather fully characterized. A more general model is similarly solved, in which the N particles are divided in families, with the same quadratic interaction acting between all pairs, but with the inversely quadratic interaction acting only between particles belonging to the same family, with a strength that may be different for different families. The second model, characterized by the pair potential g(xi − xj)−2, g > −ℏ2/(4m), contains only scattering states. It is proved that an initial scattering configuration, characterized (in the phase space sector defined by the inequalities xi ≥ xi.1, i = 1, 2, ⋯, N = 1, to which attention may be restricted without loss of generality) by (initial) momenta pi, i = 1, 2, ⋯, N, goes over into a final configuration characterized uniquely by the (final) momenta pi′, with pi′=pN+1−i. This remarkably simple outcome is a peculiarity of the case with equal particles (i.e., equal masses and equal strengths of all pair potentials).
A system of N identical particles with masses m1 interact with each other via a pair potential V and with an additional particle of mass m0 by the pair potential U, and the system obeys nonrelativistic quantum mechanics. A general lower bound formula is derived for the ground-state energy, which reduces to Coleman’s result in the special case m0=∞. The lower bound is compared with the exact energy for the harmonic oscillator problem V (x) =k21x2 and U (x) =k02x2, which has recently been solved exactly (Hall).
We study the connections among particle statistics, frustration, and groundstate energy, in many-particle systems.Statistics of identical particles is one of the most fundamental concepts, which pervades all of quantum mechanics.A consequence of the particle statistics appears in the ground-state energy.We compare the ground-state energies of bosons and fermions with the same Hamiltonian.In noninteracting systems, the influence of particle statistics on the ground-state energy is trivial: the ground-state energy of noninteracting bosons is lower than that of free fermions because of Bose-Einstein condensation (BEC) and Pauli exclusion principle.The relation that bosons have a lower ground-state energy than fermions is described as natural inequality in this thesis.However, the comparison of the ground-state energies of bosons and fermions is not trivial in the presence of interaction, because the simple argument based on the perfect BEC breaks down.In a system of interacting bosons, it is in fact already a nontrivial question whether the BEC actually takes place.In strongly correlated systems, the influence of particle statistics on the ground-state energy is still a relatively unexplored area.We have found a sufficient condition for the natural inequality to hold for spinless and spinful cases respectively, without relying on the occurrence of BEC.That is, if all the hopping amplitudes are nonnegative, the ground-state energy of hard-core bosons is still lower than that of fermions.The same argument implies that, once we relax the condition of nonnegative hopping amplitudes, it is possible to reverse the inequality so that the ground-state energy of bosons could be higher than that of fermions.By relaxing the condition, we indeed have found several concrete examples in which the ground-state energy of hard-core bosons is proved to be higher than that of fermions.Many of the examples are even proved rigorously in the thermodynamic limit.Our study leads to a novel physical understanding of the effects of particle statistics, in terms of frustration in quantal phase.This is more general than the picture based on the perfect BEC, and is indeed applicable to systems with interaction.We can map a quantum many-particle problem to a single-particle problem on a fictitious lattice in higher dimensions.When all the hopping amplitudes are nonnegative and the particles are bosons, the corresponding single-particle problem also has only nonnegative hopping amplitudes.In such a case, there is no frustration in the quantal phase of the wavefunction.On the other hand, Fermi statistics of the original particles gives an effective magnetic flux in the corre-sponding single-particle problem.This implies a frustration in the phase of the wavefunction, which can be regarded as "statistical frustration": it is induced by the Fermi statistics and it leads to a destructive interference among propagation along different paths.In the presence of a non-vanishing flux in the original many-body problem, we observe that there is also a magnetic flux in the corresponding single-particle problem, inducing a frustration among quantal phases, which we name hopping frustration.In the original many-particle problem, the statistical frustration appears rather differently from the hopping frustration.However, upon mapping to the single-particle problem on the fictitious lattice, both hopping frustration and statistical frustration are represented by non-vanishing flux in the fictitious lattice.This provides a unified understanding of frustration.Based on the unified understanding, we find the mechanism as to why the ground-state energy of hard-core bosons can be higher than that of fermions.For many-body bosonic systems, in which there is no statistical frustration, introduction of hopping frustration will not decrease the ground-state energy.This is known as Simon's universal diamagnetism of bosons.On the other hand, in many-body fermionic systems, where statistical frustration already exists, hopping frustration if introduced, is expected to compete with statistical frustration and sometimes partially cancel with each other, resulting in the reversed inequality between the ground-state energies of the hard-core bosons and fermions.