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Physica Scripta

Bogoliubov's canonical transformation approach on a harmonic oscillator: exponential decaying mass

Biswanath Rath

Published Dec 1, 2008• 2 citations
Publisher

Abstract

We solve decaying mass oscillator Hamiltonian using Bogoliubov's canonical transformation approach. It has been argued that the decay constant γ of the exponential decaying mass of the oscillator cannot be varied arbitrarily. However, γ is restricted to vary within the range 0<γ<w0 because of the uncertainty product and the discrete nature of energy states where w2=w02−γ2.

DOI
10.1088/0031-8949/78/06/065012

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We solve decaying mass oscillator Hamiltonian using Bogoliubov's canonical transformation approach. It has been argued that the decay constant γ of the exponential decaying mass of the oscillator cannot be varied arbitrarily. However, γ is restricted to vary within the range 0<γ<w0 because of the uncertainty product and the discrete nature of energy states where w2=w02−γ2.
The problem of a harmonic oscillator with varying mass parameter is reduced by canonical transformation to the corresponding constant mass problem and is solved in the case of an exponentially decaying mass. The constructed canonical Hamiltonian has time-independent eigenvalues and eigenvectors. The cases of undercritical and overcritical damping are considered in detail. The Green function is calculated and the behaviour of coherent states is discussed. The theory is related to the case of a cavity oscillator with a decaying field as in threshold laser operation. In particular, the energy of the field is considered.
Exact solutions to the harmonic oscillator with an exponentially decaying mass are explicitly represented in terms of Bessel functions. The dynamical invariant quantity of the system has the form of a rosette-shaped orbit in phase space. From this, it is confirmed that this system is bounded. By using the invariant operator represented in terms of lowering and raising operators, we have obtained wavefunctions and the propagator. Finally, minimum uncertainty states and conditions are evaluated by using other operators which are obtained from the above ones.
A time-dependent unitary (canonical) transformation is found that maps the Hamiltonian for a harmonic oscillator with time-dependent real mass and real frequency to that of a generalized harmonic oscillator with time-dependent real mass and imaginary frequency. The latter may be reduced to an ordinary harmonic oscillator by means of another unitary (canonical) transformation. A simple analysis of the resulting system leads to the identification of a previously unknown class of exactly solvable time-dependent oscillators. Furthermore, it is shown how one can apply these results to establish a canonical equivalence between some real and imaginary frequency oscillators. In particular it is shown that a harmonic oscillator whose frequency is constant and whose mass grows linearly in time is canonically equivalent with an oscillator whose frequency changes from being real to imaginary and vice versa repeatedly.
A canonical description of a harmonic oscillator with energy dissipation is sought which combines the advantages of the Kanai-Caldirola Hamiltonian and a simple model of strangulation previously considered by the authors. The treatment is in the Heisenberg picture of quantum mechanics or alternatively in classical mechanics. A strangulation \ensuremath{\beta} is superimposed on the damping (or growth) \ensuremath{\gamma}, represented by an oscillator mass m(t)=m0{m}_{0}m0​exp(2\ensuremath{\gamma}t). In general, the expected dissipation of energy occurs only if \ensuremath{\beta}>\ensuremath{\Vert}\ensuremath{\gamma}\ensuremath{\Vert}. If \ensuremath{\beta}<\ensuremath{\Vert}\ensuremath{\gamma}\ensuremath{\Vert} the attempted strangulation induces a long-term growth in energy unless a special initial motion is chosen. A slight deviation from this initial motion results in a temporary decay in energy followed by growth. \ensuremath{\beta}<0 always induces a growth in energy.
The non-Hermitian quadratic Hamiltonian H=ωa†a+αa2+βa†2 is analyzed, where a† and a are harmonic oscillator creation and annihilation operators and ω, α, and β are real constants. For the case that ω2−4αβ⩾0, it is shown using operator techniques that the Hamiltonian possesses real and positive eigenvalues. A generalized Bogoliubov transformation allows the energy eigenstates to be constructed from the algebra and states of the harmonic oscillator. The eigenstates are shown to possess an imaginary norm for a large range of the parameter space. Finding the orthonormal dual space allows the inner product to be redefined using the complexification procedure of Bender et al. for non-Hermitian Hamiltonians. Transition probabilities governed by H are shown to be manifestly unitary when the complexification procedure is followed. A specific transition element between harmonic oscillator states is evaluated for both the Hermitian and non-Hermitian cases to identify the differences in time evolution.