An integral transformation generalizing the Bargmann transform is established among two sets of wavefunctions associated with the configuration and Bargmann-Fock-Segal-like representations of a relativistic harmonic oscillator. The time-independent configuration-space wavefunctions are also studied and the lack of unitarity occurring when factorizing the time dependence of the wavefunction is solved by modifying the scalar product.
An integral transformation generalizing the Bargmann transform is established among two sets of wavefunctions associated with the configuration and Bargmann-Fock-Segal-like representations of a relativistic harmonic oscillator. The time-independent configuration-space wavefunctions are also studied and the lack of unitarity occurring when factorizing the time dependence of the wavefunction is solved by modifying the scalar product.
Recently we have obtained, on the basis of a group approach to quantization, a Bargmann-Fock-like realization of the Relativistic Harmonic Oscillator as well as a generalized Bargmann transform relating fock wave functions and a set of relativistic Hermite polynomials. Nevertheless, the relativistic creation and annihilation operators satisfy typical relativistic commutation relations of the Lie product (vector-z, vector-z(sup dagger)) approximately equals Energy (an SL(2,R) algebra). Here we find higher-order polarization operators on the SL(2,R) group, providing canonical creation and annihilation operators satisfying the Lie product (vector-a, vector-a(sup dagger)) = identity vector 1, the eigenstates of which are 'true' coherent states.
We investigate the dynamics of the spin-less relativistic particle subject to an external eld of a harmonic oscillator potential.The KleinGordon equation with one-and three-dimensional vector and scalar parabolic potentials is solved using the expansion of the wavefunction in properly selected basis-sets.The resonance states are determined using the complex coordinate rotation method.The analytic expressions for the rst-and secondorder relativistic energy corrections are derived perturbatively.The relativistic model of the harmonic oscillator in the momentum representation, originally proposed by Znojil, is also discussed.
Continuing the line of development of Paper I [J. Math. Phys. 15, 1295 (1974)], we enlarge the concept of canonical transformations in quantum mechanics in two directions: first, by allowing the definition of a canonical transformation to be made through the preservation of an so(2,1) algebra, rather than the usual Heisenberg algebra, and providing the bridge between the classical and quantum mechanical descriptions, and, second, through the complexification of the transformation group. In this paper we study specifically the transformations which can be interpreted as the radial part of n-dimensional complex linear transformations in Paper I. We show that we can build Hilbert spaces of analytic functions with a scalar product defined through integration over half the complex plane of a variable which has the meaning of a complex radius. A unitary mapping to the ordinary Hilbert space Lrn−12(0,∞) is provided with a kernel involving a Bessel function. Special cases of this are shown to be the Barut-Girardello, one-dimensional Bargmann and Hankel transforms. The transform kernels provide a series of representations of a subsemigroup of SL(2,C) and allow the construction of coherent states for the harmonic oscillator with an extra centrifugal force. We present a hyperdifferential operator realization of these transforms which yields new Baker-Campbell-Hausdorff and special function relations.
The Segal-Bargmann transform A is a unitary isomorphism of the quantum mechanical configuration space L2(Rn,ν) onto the Segal-Bargmann, or phase, space HL2(Cn,μ) of entire functions square integrable with respect to μ. Here μ and ν are Gaussian measures, and for z∈Cn and measurable f:Rn→CAf(z) :=∫Rn dν(t)f(t)exp(−z2/2+√z⋅t) if the integral converges. Af is defined for all f∈Lp(Rn,ν) if 1<p⩽∞, but A does not preserve Lp structure if p≠2. Specifically, it is proven here that A is a compact map from Lp(Rn,ν) to Lq(Cn,μ) if 1⩽q<2 and p>1+q/2 (which implies p>q) and in this case ‖A‖p→q⩽((1+q/2−p′q/2)(1−q/2))−n/2q where p′ is the dual of p. Moreover, a calculation with test functions shows that ‖A‖p→q=∞ if q>2 or p<1+q/2 (which is implied by p<q). Using an improved operator norm estimate (analogous to the Hausdorff-Young inequality) we prove Hirschman type entropy inequalities for f∈L2(Rn,ν), (p−1−2−1)S(f )⩽(q−1−2−1)S(Af )+nCp,q‖f‖22, where 1⩽q<2, p>1+q/2, Cp,q is given explicitly as a function of p and q and S(g) is the entropy of g. Examples given here show A is not entropy preserving. So these inequalities provide constraints on the change in entropy. Similarly, we get energy-entropy, or log-Sobolev, inequalities a1S(Af )+a2S(f )⩽a3‖f‖22+a4〈f,Nf〉, for f in a dense subspace of L2(Rn,ν). Here 〈f,Nf〉 is the Dirichlet form of the energy operator N in the configuration space and the ai can have various signs and are also given explicitly. The principal results of this article are generalized Heisenberg uncertainty relations, namely they are constraints on Af, the phase space representation, given information about f, the configuration space representation.
Abstract The three-dimensional harmonic oscillator is solved in Bargmann space. The treatment is pedagogically more transparent than the standard ones, at the price of introducing the Bargmann transform in the context of the one-dimensional oscillator. The standard solid harmonics are similarly derived with minimal technical effort, amounting to a complete self-contained exposition suitable for introductory courses in quantum mechanics or mathematical methods of physics. It provides an early exposure to wavelets, with important contemporary applications in signal analysis and quantum optics.