The Morse oscillator in position space, momentum space, and phase space

Dahl, Jens Peder; Springborg, Michael

doi:10.1063/1.453761

Cited by 251 publications

(188 citation statements)

References 33 publications

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“…The explicit expressions of the Morse stationary bound states have been found in various forms [17,18]. Again, we obtain the position ground state by means of the factorization method.…”

Section: à2mentioning

confidence: 99%

Information-theoretic measures for Morse and Pöschl–Teller potentials

Dehesa

Martínez–Finkelshtein

Сорокин

2006

Molecular Physics

112

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The spreading of the quantum-mechanical probability cloud for the ground state of the Morse and modified Po¨schl-Teller potentials, which controls the chemical and physical properties of some molecular systems, is studied in position and momentum space by means of global (Shannon's information entropy, variance) and local (Fisher's information) informationtheoretic measures. We establish a general relation between variance and Fisher's information, proving that, in the case of a real-valued and symmetric wavefunction, the well-known Cramer-Rao and Heisenberg uncertainty inequalities are equivalent. Finally, we discuss the asymptotics of all three information measures, showing that the ground state of these potentials saturates all the uncertainty relations in an appropriate limit of the parameter.

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“…The explicit expressions of the Morse stationary bound states have been found in various forms [17,18]. Again, we obtain the position ground state by means of the factorization method.…”

Section: à2mentioning

confidence: 99%

Information-theoretic measures for Morse and Pöschl–Teller potentials

Dehesa

Martínez–Finkelshtein

Сорокин

2006

Molecular Physics

112

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“…j in Equation (47) has the same meaning as in Equation (42). ρ eq determines W eq (x, q), through Equation (39).…”

Section: Equilibrium Wigner Functionmentioning

confidence: 99%

“…See also [40,41]. As a nontrivial illustration, the Wigner functions, W , associated with several eigenstates of the Morse potential, have been studied by combining analytical and numerical methods: negative values of the W associated with the ground state are reported in [42,43]. The latter two references and [44] present negative values of the W 's associated with some excited states.…”

Section: W and W St As Quasi-definite Functionals In Momentummentioning

confidence: 99%

Non-Equilibrium Liouville and Wigner Equations: Moment Methods and Long-Time Approximations

Alvarez‐Estrada

2014

Entropy

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Abstract:We treat the non-equilibrium evolution of an open one-particle statistical system, subject to a potential and to an external "heat bath" (hb) with negligible dissipation. For the classical equilibrium Boltzmann distribution, W c,eq , a non-equilibrium three-term hierarchy for moments fulfills Hermiticity, which allows one to justify an approximate long-time thermalization. That gives partial dynamical support to Boltzmann's W c,eq , out of the set of classical stationary distributions, W c,st , also investigated here, for which neither Hermiticity nor that thermalization hold, in general. For closed classical many-particle systems without hb (by using W c,eq ), the long-time approximate thermalization for three-term hierarchies is justified and yields an approximate Lyapunov function and an arrow of time. The largest part of the work treats an open quantum one-particle system through the non-equilibrium Wigner function, W . W eq for a repulsive finite square well is reported. W 's (< 0 in various cases) are assumed to be quasi-definite functionals regarding their dependences on momentum (q). That yields orthogonal polynomials, H Q,n (q), for W eq (and for stationary W st ), non-equilibrium moments, W n , of W and hierarchies. For the first excited state of the harmonic oscillator, its stationary W st is a quasi-definite functional, and the orthogonal polynomials and three-term hierarchy are studied. In general, the non-equilibrium quantum hierarchies (associated with W eq ) for the W n 's are not three-term ones. As an illustration, we outline a non-equilibrium four-term hierarchy and its solution in terms of generalized operator continued fractions. Such structures also allow one to formulate long-time approximations, but make it more difficult to justify thermalization. For large thermal and de Broglie wavelengths, the dominant W eq and a non-equilibrium equation for W are reported: the non-equilibrium hierarchy could plausibly be a three-term one and possibly not far from Gaussian, and thermalization could possibly be justified. Liouville and Wigner distributions and non-equilibrium equations; equilibrium solutions and orthogonal polynomials; non-equilibrium moments; long-time approximations; low temperature regime

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“…, Equilibrium Wigner functions have also been calculated for various simple quantum systems such as a particle in an infinite square well [104], the Morse oscillator [106,107], the anharmonic quartic oscillator [108,109], the inverted harmonic oscillator [110], etc.…”

Section: The Wigner Distribution Function For Particlesmentioning

confidence: 99%