Two TTCFs, \(\langle \sigma _x(t)\sigma _y(0)\rangle\) and \(\langle \sigma _+(t)\sigma _-(0)\rangle\), are investigated as the example. 1, we select \(\gamma =0.5\) and 2 to simulate the non-Markovian dynamics and compare with the Markovian limit when \(\gamma =10\).
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So it is of great importance to study such problems in the framework of an open quantum system (OQS), \(H_\). The eigenvalues of the system by CFM method are so near to the exact results but the consequence of perturbation predictions does not have this precision.In real experiments, the dynamics of a quantum system can be significantly modified by the coupling with the surrounding environment 1, 2, 3, 4. įinally, the eigenvalues were obtained by the perturbed theory and CFM method. The exact solution leads to Hermite functions and eigenvalues adherence relation of E n = n + 1 2 e ∈ 0 ħ m + m g + e ∈ 0 2 2 e 2 ∈ 0 2. Then, the quantized energies are then given in terms of the zeros of the well-behaved Airy function Ai ( - ζ n ) with E n = ε 0 ζ n.Īlso, we investigated the stark effect in the linear potential that is term of perturbation in the Schrodinger equation. Since the Bi x solution diverges for large positive argument and it does not satisfy the boundary condition ψ ∞ = 0, therefore it is excluded. We show that the Airy functions ( Ai x, Bi x) are eigenfunctions of the unperturbed linear potential. The stark effect in the linear potential was investigated in three methods (Exact, CFM and Perturbed). In this paper, the perturbed linear potentials due to influence of electrical field were studied. 3, the Stark effect for 1D quantum mechanical system described by the linear potentials and the CFM is applied to the Stark effect in the quantum bouncer and the CFM results will be compared with analytical solution of the Stark effect in quantum bouncer (Airy functions). This article is organized as follows: the next section is a description of the CFM and exact solutions of Schrodinger Equation with the Linear Potential in 1D with the appropriate boundary conditions. We show how straightforward use of the most obvious properties of the Airy function solutions gives closed form results for the Stark shifts in this system.
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In this work, we discuss the application of CFM to a system with a linear potential we consider the Stark effect as a perturbation term of desired potential (the second-order shifts in the energy spectrum due to an external constant force) in a one-dimensional model quantum mechanical system described by the linear potentials, the so-called quantum bouncer (defined by V ( z) = Fz for z > 0 and V ( z) = ∞ for z < 0) and the symmetric linear potential (given by V ( z) = F| z|). It stops when the difference between the left and right ratio is below a given desired precision. The integration proceeds simultaneously from this point toward the left and right boundaries evaluating at each step a corresponding ratio. This is done by expressing the solution as a sum of two linearly independent functions (the canonical functions) with specific values at some arbitrary point belonging to the interval defined by the two boundaries.
#Solving schrodinger equation full
The CFM turns the two-point boundary value Schrodinger problem into initial value of Schrodinger problem and allows full and accurate determination of the spectrum. The Canonical Function Method (CFM) can handle a large variety of quantum problems, also the eigenvalues problem making it an extremely versatile, fast and highly accurate. A particle with this potential is studied as “a quantum bouncing ball” in. A system with a symmetrical linear potential V z = F z is a simple model to analyze neutron quantized balances in the gravity field of earth.
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In quantum mechanics, the analysis of systems with various potentials is important.