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Advanced Theory and Simulations

Quantum Dynamics in Confined Pseudo‐Harmonic Oscillator in a Time‐Dependent Moving Boundary

Akash Halder, Amlan K. Roy, Debraj Nath

Published Dec 29, 2025• 0 citations
Publisher

Abstract

ABSTRACT In this work, we present an analytical solution of Schrödinger equation of confined pseudoharmonic potential in the presence of a moving boundary condition for an arbitrary angular momentum state. It turns out that an important quantity to probe the problem is the internuclear distance ratio, which depends on the solution of Ermakov equation. The minimum value of the time‐dependent (TD) Heisenberg uncertainty product always remains greater than that of the minimum uncertainty product . The TD average energy is derived analytically in a closed form, and the corresponding average force and average pressure are defined. Moreover, the time correlation function of two states for the case of six selected diatomic molecules (CO, NO, ScH, CH, , ) is obtained. It is found to depend on the internuclear distance ratio at two different time domains. The TD survival probability and average life‐time of a molecule in a confined quantum system are defined. Expressions are offered for quantum similarity measure, dissimilarity, and quantum similarity index. The latter is given for a pair of molecules. The obtained results are compared with available literature, wherever possible. To our knowledge this is the first detailed report of a non‐harmonic central potential in a TD moving boundary condition.

DOI
10.1002/adts.202501089

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ABSTRACT In this work, we present an analytical solution of Schrödinger equation of confined pseudoharmonic potential in the presence of a moving boundary condition for an arbitrary angular momentum state. It turns out that an important quantity to probe the problem is the internuclear distance ratio, which depends on the solution of Ermakov equation. The minimum value of the time‐dependent (TD) Heisenberg uncertainty product always remains greater than that of the minimum uncertainty product . The TD average energy is derived analytically in a closed form, and the corresponding average force and average pressure are defined. Moreover, the time correlation function of two states for the case of six selected diatomic molecules (CO, NO, ScH, CH, , ) is obtained. It is found to depend on the internuclear distance ratio at two different time domains. The TD survival probability and average life‐time of a molecule in a confined quantum system are defined. Expressions are offered for quantum similarity measure, dissimilarity, and quantum similarity index. The latter is given for a pair of molecules. The obtained results are compared with available literature, wherever possible. To our knowledge this is the first detailed report of a non‐harmonic central potential in a TD moving boundary condition.
In this work, we present analytical solution of Schrödinger equation of confined pseudoharmonic potential in presence of a moving boundary condition, for an arbitrary angular momentum state. It turns out that an important quantity to probe the problem is internuclear distance ratio, which depends on the solution of Ermakov equation. The minimum value of time-dependent (TD) Heisenberg uncertainty product always remains greater than that of the minimum uncertainty product h/2. The TD average energy is derived analytically in a closed form and the corresponding average force and average pressure are defined. Moreover, time correlation function of two states for the case of six selected diatomic molecules (CO, NO, ScH, CH, H2, N2) is obtained. It is found to depend on internuclear distance ratio at two different time domains. The TD survival probability and average life-time of molecule in a confined quantum system are defined. Expressions are offered for quantum similarity measure, dissimilarity and quantum similarity index. The latter is given for a pair of molecules. The obtained results are compared with available literature, wherever possible. To our knowledge this is the first detailed report of a non-harmonic central potential in a TD moving boundary condition.
Abstract For the Deng-Fan potential within a moving boundary condition, the time-dependent Schrödinger equation is considered analytically. The eigenvalue equation is solved by using a combination of Pekeris and Greene-Aldrich approximations. Various time-dependent quantities including density distribution function, auto-correlation function, disequilibrium, average energy, quantum similarity, quantum similarity index are obtained for selected eight diatomic molecules. The motion of the peak of soliton with moving boundary condition is investigated for ground states of some diatomic molecules along with the corresponding peak values.
By applying the standard analytical techniques of solving partial differential equations, we have obtained the exact solution in terms of the Fourier sine series to the time-dependent Schrödinger equation describing a quantum one-dimensional harmonic oscillator of time-dependent frequency confined in an infinite square well with the two walls moving along some parametric trajectories. Based upon the orthonormal basis of quasi-stationary wave functions, the exact propagator of the system has also been analytically derived. Special cases like (i) a confined free particle, (ii) a confined time-independent harmonic oscillator, and (iii) an aging oscillator are examined, and the corresponding time-dependent wave functions are explicitly determined. Besides, the approach has been extended to solve the case of a confined generalized time-dependent harmonic oscillator for some parametric moving boundaries as well.
Abstract We present an exact solution of a confined model of the non-relativistic quantum harmonic oscillator, where the effective mass and the angular frequency are dependent on the position. The free Hamiltonian of the proposed model has the form of the BenDaniel–Duke kinetic energy operator. The position-dependency of the mass and the angular frequency is such that the homogeneous nature of the harmonic oscillator force constant k and hence the regular harmonic oscillator potential is preserved. As a consequence thereof, a quantization of the confinement parameter is observed. It is shown that the discrete energy spectrum of the confined harmonic oscillator with position-dependent mass and angular frequency is finite, has a non-equidistant form and depends on the confinement parameter. The wave functions of the stationary states of the confined oscillator with position-dependent mass and angular frequency are expressed in terms of the associated Legendre or Gegenbauer polynomials. In the limit where the confinement parameter tends to ∞, both the energy spectrum and the wave functions converge to the well-known equidistant energy spectrum and the wave functions of the stationary non-relativistic harmonic oscillator expressed in terms of Hermite polynomials. The position-dependent effective mass and angular frequency also become constant under this limit.
Abstract The stationary states of a particle under the influence of a delta potential confined by impenetrable walls are investigated using the method of expansion in orthogonal functions. The eigenfunctions of the time-independent Schrödinger equation are expressed in closed form by using a pair of closed-form expressions for series available in the literature. The analysis encompasses both attractive and repulsive potentials with arbitrary couplings. Confinement significantly impacts the quantum states and introduces a scenario of double degeneracy including the ground state. Analysis extends to discuss the transition to unconfinement. This research holds particular significance for educators and students engaged in mathematical methods applied to physics and quantum mechanics within undergraduate courses, offering valuable insights into the complex relationships among profiles of potentials, boundary conditions, and the resulting quantum phenomena.
Quantum confinement is studied by numerically solving time-dependent (TD) Schrödinger equation (SE). An imaginary-time evolution technique is employed in conjunction with the minimization of an expectation value, to reach the global minimum. Excited states are obtained by imposing the orthogonality constraint with all lower states. Applications are made on three important model quantum systems, namely, harmonic, repulsive and quartic oscillators; enclosed inside an impenetrable box. The resulting diffusion equation is solved using finite-difference method. Both symmetric and asymmetric confinement are considered for attractive potential; for others only symmetrical confinement. Accurate eigenvalue, eigenfunction and position expectation values are obtained, which show excellent agreement with existing literature results. Variation of energies with respect to box length is followed for small, intermediate and large sizes. In essence, a simple accurate and reliable method is proposed for confinement in quantum systems.