Probabilistic calculations on the ground state of the harmonic oscillator

Sergio A Albeverio, Raphael J. Høegh-Krohn

Ashot S. Gevorkyan

The dissipation and decoherence (for example, the effects of noise in quantum computations), interaction with thermostat or in general with physical vacuum, measurement and many other complicated problems of open quantum systems are a consequence of interaction of quantum system with the environment. These problems are described mathematically in terms of complex probabilistic process (CPP). Particularly, treating the environment as a Markovian process we derive an Langevin-Schroedinger type stochastic differential equation (SDE) for describing the quantum system interacting with environment. For the 1D randomly quantum harmonic oscillator (QHO) model L-Sh SDE is a solution in the form of orthogonal CPP. On the basis of orthogonal CPP the stochastic density matrix (SDM) method is developed and in its framework relaxation processes in the uncountable dimension closed system of "QHO+environment" is investigated. With the help of SDM method the thermodynamical potentials, like nonequilibrium entropy and the energy of ground state are exactly constructed. The dispersion for different operators are calculated. In particular, the expression for uncertain relations depending on parameter of interaction with environment is obtained. The Weyl transformation for stochastic operators is specified. Ground state Winger function is developed in detail.

Charles E. Treanor

Valdir C. Aguilera-Navarro, Rosa Ma. Méndez V.

Variational calculations play an important role in quantum mechanics, particularly in determining the ground-state energy of physical systems. Didactic examples in which one can see how the exact energy and wave function are approached by using a family of trial functions are not very numerous. An example was given in a recent book [M. Moshinsky, The Harmonic Oscillator in Modern Physics: From Atoms to Quarks (Gordon and Breach, New York, 1969)] which discusses the ground state of the hydrogen atom using as trial wave function a linear combination of harmonic-oscillator states. In the present paper we carry out a similar analysis for the ground state of a particle of mass m in a three-dimensional square-well potential. We discuss not only the approach to the exact energy when we vary the frequency and the number of oscillator states but also analyze the overlap of the exact and variational wave functions and compare the exact and approximate form factors.

Daegene Song

Unlike its classical counterpart, the ground state of a quantum harmonic oscillator yields a non-zero value which has led to a significant problem regarding the cosmological constant. We will show that since observables, including that of energy, serve as a reference frame for an observer, not only is the harmonic oscillator fluctuating at the ground level, so is the reference frame of the observer when the measurement is realized. This may provide an explanation of why vacuum energy calculated from quantum theory is non-zero, yet is not observed, because the observer is also at the same ground level of energy. 1