Physical Review A

Reconstruction of the quantum state via position distribution and its time derivative

Th. Richter

Published Sep 1, 1996• 3 citations

Abstract

The determination of the harmonic oscillator wave function from position distribution and its time derivative is investigated. Quite simple definitive equations result for the expansion coefficients of the wave function in the Fock basis and yield a unique solution. The effect of a finite spatial resolution of the measurements is taken into account. \textcopyright{} 1996 The American Physical Society.

DOI: 10.1103/physreva.54.2499

Related papers

Reconstruction of the Quantum State via Position Distribution and its Time Derivative

T. Richter

The determination of the harmonic oscillator wave function from position distribution and its time derivative is investigated. Quite simple definitive equations result for the expansion coefficients of the wave function in the Fock basis and yield a unique solution. The effect of a finite spatial resolution of the measurements is taken into account. {copyright} {ital 1996 The American Physical Society.}

European Quantum Electronics Conference (EQEC)Jan 1, 19961 citation

A method to obtain the solution of the time- independent Schrodinger equation for the simple harmonic oscillator

Anirban Bose

Jan 5, 20260 citations

How to determine a quantum state by measurements: The Pauli problem for a particle with arbitrary potential

Stefan Weigert

Physical Review AApr 1, 199649 citations

A method to obtain the solution of the time- independent Schrodinger equation for the simple harmonic oscillator

Anirban Bose

Jul 24, 20260 citations

A method to obtain the solution of the time- independent Schrodinger equation for the simple harmonic oscillator

Anirban Bose

Jan 2, 20260 citations

Theory of Time as the basis of Quantum Mechanics

Romanenko Vladimir Alekseevich

Journal of Physics & Optics SciencesJun 12, 20250 citations

Physical Review A

Reconstruction of the quantum state via position distribution and its time derivative

Th. Richter

Published Sep 1, 1996• 3 citations

Publisher

Abstract

DOI: 10.1103/physreva.54.2499

Related papers

Reconstruction of the Quantum State via Position Distribution and its Time Derivative

T. Richter

The determination of the harmonic oscillator wave function from position distribution and its time derivative is investigated. Quite simple definitive equations result for the expansion coefficients of the wave function in the Fock basis and yield a unique solution. The effect of a finite spatial resolution of the measurements is taken into account. {copyright} {ital 1996 The American Physical Society.}

European Quantum Electronics Conference (EQEC)Jan 1, 19961 citation

A method to obtain the solution of the time- independent Schrodinger equation for the simple harmonic oscillator

Anirban Bose

Jan 5, 20260 citations

How to determine a quantum state by measurements: The Pauli problem for a particle with arbitrary potential

Stefan Weigert

The problem of reconstructing a pure quantum state \ensuremath{\Vert}\ensuremath{\psi}〉 from measurable quantities is considered for a particle moving in a one-dimensional potential V(x). Suppose that the position probability distribution \ensuremath{\Vert}\ensuremath{\psi}(x,t)${\mathrm{\ensuremath{\Vert}}}^{2}$ has been measured at time t, and let it have M nodes. It is shown that after measuring the time evolved distribution at a short-time interval \ensuremath{\Delta}t later, \ensuremath{\Vert}\ensuremath{\psi}(x,t+\ensuremath{\Delta}t)${\mathrm{\ensuremath{\Vert}}}^{2}$, the set of wave functions compatible with these distributions is given by a smooth manifold M in Hilbert space. The manifold M is isomorphic to an M-dimensional torus, ${\mathcal{T}}^{\mathit{M}}$ . Finally, M additional expectation values of appropriately chosen nonlocal operators fix the quantum state uniquely. The method used here is the analog of an approach that has been applied successfully to the corresponding problem for a spin system. \textcopyright{} 1996 The American Physical Society.The problem of reconstructing a pure quantum state \ensuremath{\Vert}\ensuremath{\psi}〉 from measurable quantities is considered for a particle moving in a one-dimensional potential V(x). Suppose that the position probability distribution \ensuremath{\Vert}\ensuremath{\psi}(x,t)${\mathrm{\ensuremath{\Vert}}}^{2}$ has been measured at time t, and let it have M nodes. It is shown that after measuring the time evolved distribution at a short-time interval \ensuremath{\Delta}t later, \ensuremath{\Vert}\ensuremath{\psi}(x,t+\ensuremath{\Delta}t)${\mathrm{\ensuremath{\Vert}}}^{2}$, the set of wave functions compatible with these distributions is given by a smooth manifold M in Hilbert space. The manifold M is isomorphic to an M-dimensional torus, ${\mathcal{T}}^{\mathit{M}}$. Finally, M additional expectation values of appropriately chosen nonlocal operators fix the quantum state uniquely. The method used here is the analog of an approach that has been applied successfully to the corresponding problem for a spin system. \textcopyright{} 1996 The American Physical Society.

Physical Review AApr 1, 199649 citations

A method to obtain the solution of the time- independent Schrodinger equation for the simple harmonic oscillator

Anirban Bose

A method has been introduced to derive the solution of the time-independent Schrodinger equation for the simple harmonic oscillator. A trial solution has been chosen as the product of the divergent part of the approximate asymptotic solution of the Schrodinger equation and an unknown function. Choosing the divergent exponential factor has an advantage. Because if the equation has to have a meaningful bound state solution, the other part of the product is compelled to converge faster than the diverging exponential, so that their product, which is the bound state solution, must converge at large distances. This requirement makes the other part an oscillatory function with (n+1) extrema when it has n nodes. Since its derivative vanishes at each of these extrema, its derivative necessarily possesses at least (n+1) nodes. Such a nodal structure is characteristic of the next higher bound state, motivating an investigation into whether the derivative of the remaining part is directly related to the wave function of the next higher energy eigen state. This investigation demonstrates that the wave functions of the harmonic oscillator can be generated by successively reconstructing the equations of the higher eigen states through repeated differentiation.A method has been introduced to derive the solution of the time-independent Schrodinger equation for the simple harmonic oscillator. A trial solution has been chosen as the product of the divergent part of the approximate asymptotic solution of the Schrodinger equation and an unknown function. Choosing the divergent exponential factor has an advantage. Because if the equation has to have a meaningful bound state solution, the other part of the product is compelled to converge faster than the diverging exponential, so that their product, which is the bound state solution, must converge at large distances. This requirement makes the other part an oscillatory function with (n+1) extrema when it has n nodes. Since its derivative vanishes at each of these extrema, its derivative necessarily possesses at least (n+1) nodes. Such a nodal structure is characteristic of the next higher bound state, motivating an investigation into whether the derivative of the remaining part is directly related to the wave function of the next higher energy eigen state. This investigation demonstrates that the wave functions of the harmonic oscillator can be generated by successively reconstructing the equations of the higher eigen states through repeated differentiation.

Jul 24, 20260 citations

A method to obtain the solution of the time- independent Schrodinger equation for the simple harmonic oscillator

Anirban Bose

Jan 2, 20260 citations

Theory of Time as the basis of Quantum Mechanics

Romanenko Vladimir Alekseevich

The proposed work presents the derivation of the wave function and the Schrödinger equation based on the dual tangential equation of time with an imaginary rate. The derivation uses de Broglie formulas. The wave and complex-conjugate functions are considered, which participate in the probabilistic description of the position of a particle in a potential field. Another interpretation of the product of these functions is presented. Based on it, the classical structure of the electron is derived. The derivation determines the quantum value of the imaginary incident vector. When deriving wave functions, simple solutions immediately arise for energies corresponding to radiation emitted by portions and a microscopic harmonic oscillator. In quantum mechanics, the first solution was used by Planck as a hypothesis when deriving his famous formula for describing equilibrium thermal radiation. The second energy solution was obtained from the Schrödinger equation by cumbersome calculations.The proposed work presents the derivation of the wave function and the Schrödinger equation based on the dual tangential equation of time with an imaginary rate. The derivation uses de Broglie formulas. The wave and complex-conjugate functions are considered, which participate in the probabilistic description of the position of a particle in a potential field. Another interpretation of the product of these functions is presented. Based on it, the classical structure of the electron is derived. The derivation determines the quantum value of the imaginary incident vector. When deriving wave functions, simple solutions immediately arise for energies corresponding to radiation emitted by portions and a microscopic harmonic oscillator. In quantum mechanics, the first solution was used by Planck as a hypothesis when deriving his famous formula for describing equilibrium thermal radiation. The second energy solution was obtained from the Schrödinger equation by cumbersome calculations.

Journal of Physics & Optics SciencesJun 12, 20250 citations

{\mathcal{T}}^{\mathit{M}}

. Finally, M additional expectation values of appropriately chosen nonlocal operators fix the quantum state uniquely. The method used here is the analog of an approach that has been applied successfully to the corresponding problem for a spin system. \textcopyright{} 1996 The American Physical Society.