We consider the problem of designing a variety of "system guided" basis sets for quantum mechanical anharmonic oscillators. Using ideas based on supersymmetric quantum mechanics, we design canonical transformations of the usual position and momentum to generate generalized "Cartesian-like positions, W and momenta, p_W" with unit Poisson brackets. These are quantized following Dirac, leading to an infinite family of potential "operator observables". The fundamental issue is that all but one of the operators are not Hermitian (formally self-adjoint) in the original position representation. We show that the resulting operators are "quasi-Hermitian" relative to the x-representation and that all are Hermitian in the W-representation. Depending on how one treats the Jacobian of the canonical transformation in the expression for the classical momentum, p_W, quantization yields a) continuous mutually unbiased bases (MUB) b) orthogonal bases (with Dirac delta normalization) c) biorthogonal bases (with Dirac delta normalization) d) new W-harmonic oscillators yielding standard orthonormal bases (as functions of W) and associated coherent states. The MUB include W-generalized Fourier transform kernels whose eigenvectors are the W-harmonic oscillator eigenstates, with the spectrum (p/m i, p/m i). The W, p_W satisfy the uncertainty product relation: Delta-W * Delta-p_W gte 1/2.
We consider the problem of designing a variety of "system guided" basis sets for quantum mechanical anharmonic oscillators. Using ideas based on supersymmetric quantum mechanics, we design canonical transformations of the usual position and momentum to generate generalized "Cartesian-like positions, W and momenta, p_W" with unit Poisson brackets. These are quantized following Dirac, leading to an infinite family of potential "operator observables". The fundamental issue is that all but one of the operators are not Hermitian (formally self-adjoint) in the original position representation. We show that the resulting operators are "quasi-Hermitian" relative to the x-representation and that all are Hermitian in the W-representation. Depending on how one treats the Jacobian of the canonical transformation in the expression for the classical momentum, p_W, quantization yields a) continuous mutually unbiased bases (MUB) b) orthogonal bases (with Dirac delta normalization) c) biorthogonal bases (with Dirac delta normalization) d) new W-harmonic oscillators yielding standard orthonormal bases (as functions of W) and associated coherent states. The MUB include W-generalized Fourier transform kernels whose eigenvectors are the W-harmonic oscillator eigenstates, with the spectrum (p/m i, p/m i). The W, p_W satisfy the uncertainty product relation: Delta-W * Delta-p_W gte 1/2.
Using ideas based on supersymmetric quantum mechanics, we design canonical transformations of the usual position and momentum to create generalized “Cartesian-like” positions, W, and momenta, Pw , with unit Poisson brackets. These are quantized by the usual replacement of the classical , x Px by quantum operators, leading to an infinite family of potential “operator observables”. However, all but one of the resulting operators are not Hermitian (formally self-adjoint) in the original position representation. Using either the chain rule or Dirac quantization, we show that the resulting operators are “quasi-Hermitian” relative to the x-representation and that all are Hermitian in the W-representation. Depending on how one treats the Jacobian of the canonical transformation in the expression for the classical momentum, Pw , quantization yields a) continuous mutually unbiased bases (MUB), b) orthogonal bases (with Dirac delta normalization), c) biorthogonal bases (with Dirac delta normalization), d) new W-harmonic oscillators yielding standard orthonormal bases (as functions of W) and associated coherent states and Wigner distributions. The MUB lead to W-generalized Fourier transform kernels whose eigenvectors are the W-harmonic oscillator eigenstates, with the spectrum (±1,±i) , as well as “W-linear chirps”. As expected, W, Pw satisfy the uncertainty product relation: ΔWΔPw ≥1/2 , h=1.
In quantum mechanics the linear harmonic oscillator representatives of the canonical observables q and p are certain well-known infinite square matrices, qmn and pmn, with denumerable sets of rows and columns. It is pedagogically useful to investigate various problems which arise when one considers the finite matrices Qmn and Pmn which result when qmn and pmn are truncated after the Nth row and column, where N( 2≥2) is some preassigned positive integer. So that the enquiry may be more instructive, it begins with an abstract formulation of the problem of the “truncated harmonic oscillator.” Formally, the latter has the usual Hamiltonian H = 12( Q2+P2 ) , but Q and P are not canonical—their commutator [Q,P] now differing from i by a certain projection operator. The spectrum of H and the (finite) basis { |n〉} generated by H are found by standard methods, though there are now some complications which are absent from the theory of the linear harmonic oscillator. The diagonalization of Q and P presents no difficulties if full use is made of the classical theory of Hermite polynomials. The spectrum of Q is the set of N zeros hs of the Nth Hermite polynomial. Although the wavefunctions 〈 Q′|n〉 are therefore functions of the discrete variable hs, one can formally contemplate them as functions of a continuous real variable h; and some curious intimations of Schrödinger representation appear. The analog of a component of orbital angular momentum is considered, and its spectrum obtained. Finally, the asymptotic limit N → ∞ is briefly discussed, in order thus explicitly to dispel the impression that this limit might somehow represent the “usual” linear harmonic oscillator.
A previously developed novel theory for the formal canonical quantization of classically dissipating systems will be the starting point for a detailed discussion of the quantum statistical aspects of the simple linearly damped harmonic oscillator. The formalism essentially involves complex classical canonical coordinates and momenta, and thus quite naturally leads to the possibility of creation and annihilation operators. Furthermore, the occurrence of quantal noise operators appears to be of principal importance for the conservation of the fundamental commutator in the course of time, as will be expressed in a simple fluctuation-dissipation relation. Making a canonical transformation back to the real, Cartesian Hermitian position and momentum an "effective" Hermitian Hamiltonian will be derived, with which a transformation is made from the Heisenberg frame to the Schr\"odinger frame where the density operator equation will be computed. This will make it obvious that no proper Schr\"odinger equation exists for the dissipative subsystem on its own, thus reflecting an incomplete knowledge. The master equation will then be translated into its Wigner representation. The intimate connection between the diffusion coefficients in the resulting Fokker-Planck equation and the uncertainty relation will be demonstrated in a clear fashion.
Angle and action operators (w, j) for the simple harmonic oscillator are treated as resulting from a canonical transformation of coordinate and momentum operators (q, k) generated by a one-sided unitary operator U such that U†U = 1 and UU† commutes with k but not with q. From the discrete spectrum of the number operator n, eigenvectors |η〉 are constructed for every real value of η; the set {|η〉} is complete and orthogonal. Another complete set {|W〉} is obtained, consisting of the Fourier transforms of the kets in the set {|η〉}. The angle operator is w=U†qU=∫dW |W〉 W 〈W|. The set {|W〉} is not orthogonal; |W〉 is not an eigenvector of w. If v is defined as ∫dW|W〉exp (−2πiW) 〈W|, then the creation and destruction operators are given by a = vn½, a† = n½v†. v is a one-sided unitary operator such that vv† = 1, but v†v = 1 − |0〉 〈0|, where |0〉 is the ground state of the oscillator; v and v† are similar to the operators E- and E+ of Carruthers and Nieto. The Weyl transforms of w and j = 2πℏ(n + ½1) are the classical angle and action variables of the oscillator. The Weyl transform is formulated in terms of the coherent states of the oscillator. A time operator canonical to the Hamiltonian is defined as t = 2πw/ω (ω/2π = frequency). The observables for the oscillator are also given in the Heisenberg picture and their classical limits are considered.
The development of supersymmetric (SUSY) quantum mechanics has shown that some of the insights based on the algebraic properties of ladder operators related to the quantum mechanical harmonic oscillator (QMHO) carry over to the study of more general systems. At this level of generality, pairs of eigenfunctions of so-called partner Hamiltonians are transformed into each other, but the entire spectrum of any one of them cannot be deduced from this intertwining relationship in general – except in special cases. In this paper, we present a more general structure that provides all eigenvalues for a class of Hamiltonians that do not factor into a pair of operators satisfying canonical commutation relations. Instead of a pair of partner Hamiltonians, we consider two pairs that differ by an overall shift in their spectrum. This is called coupled supersymmetry. In that case, we also develop coherent states and present some uncertainty principles which generalise the Heisenberg uncertainty principle. Coupled SUSY is explicitly realised by an infinite family of differential operators which admit orthonormal bases of eigenfunctions of generalised harmonic oscillators.
For a non-Hermitian Hamiltonian H possessing a real spectrum, we introduce a canonical orthonormal basis in which a previously introduced unitary mapping of H to a Hermitian Hamiltonian h takes a simple form. We use this basis to construct the observables O α of the quantum mechanics based on H . In particular, we introduce pseudo-Hermitian position and momentum operators and a pseudo-Hermitian quantization scheme that relates the latter to the ordinary classical position and momentum observables. These allow us to address the problem of determining the conserved probability density and the underlying classical system for pseudo-Hermitian and in particular PT -symmetric quantum systems. As a concrete example we construct the Hermitian Hamiltonian h , the physical observables O α , the localized states and the conserved probability density for the non-Hermitian PT -symmetric square well. We achieve this by employing an appropriate perturbation scheme. For this system, we conduct a comprehensive study of both the kinematical and dynamical effects of the non-Hermiticity of the Hamiltonian on various physical quantities. In particular, we show that these effects are quantum mechanical in nature and diminish in the classical limit. Our results provide an objective assessment of the physical aspects of PT -symmetric quantum mechanics and clarify its relationship with both conventional quantum mechanics and classical mechanics.