Accurate analytic presentation of solution of the Schrödinger equation with arbitrary physical potential

Liverts, Evgeny Z.; Drukarev, E. G.; Mandelzweig, V. B.

doi:10.1016/j.aop.2007.02.003

Cited by 20 publications

(27 citation statements)

References 27 publications

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“…The latter is the generalization of the Bohr-Sommerfeld quantization rule [9] and the WKB [10][11][12]. Except for these approaches, the quasilinearization method (QLM) has played an important role in dealing with arbitrary physical potentials numerically [13][14][15][16][17][18][19][20]. Recently, Yin et al have shown why the SWKB is exact for all shape invariant potentials [21].…”

Section: Introductionmentioning

confidence: 99%

Energy spectrum for a modified Rosen-Morse potential solved by proper quantization rule and its thermodynamic properties

Dong

Cruz‐Irisson

2011

J Math Chem

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We apply our recently proposed proper quantization rule, (x)]/h to obtain the energy spectrum of the modified Rosen-Morse potential. The beauty and symmetry of this proper rule come from its meaning-whenever the number of the nodes of φ(x) or the number of the nodes of the wave function ψ(x) increases by one, the momentum integralx Bx A k(x)dx will increase by π . Based on this new approach, we present a vibrational high temperature partition function in order to study thermodynamic functions such as the vibrational mean energy U , specific heat C, free energy F and entropy S. It is surprising to note that the specific heat C(k = 1) first increases with β and arrives to the maximum value and then decreases with it. However, it is shown that the entropy S(k = 1) first increases with the deepness of potential well λ and then decreases with it.

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Section: Introductionmentioning

confidence: 99%

Energy spectrum for a modified Rosen-Morse potential solved by proper quantization rule and its thermodynamic properties

Dong

Cruz‐Irisson

2011

J Math Chem

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“…Since the QLM iterations have very fast quadratic convergence [1,2], one can expect that even the first iteration which is given by an analytic expression [3] …”

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confidence: 99%

“…(1). Its explicit form is given by y 1 (r) = y 0 (r) + Φ(r) χ 2 0 (r) where [3] Φ(r) = 2µ Here Ei(z) = − ∞ −z e −t t dt is exponential integral function. Inserting y 1 (r) into the expression for E 1 , one can calculate the first iteration energy.…”

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confidence: 99%

Analytic presentation of a solution of the Schrödinger equation

Liverts

Drukarev²,

Krivec

et al. 2008

Few-Body Syst

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High-precision approximate analytic expressions for energies and wave functions are found for arbitrary physical potentials. The Schrödinger equation is cast into nonlinear Riccati equation, which is solved analytically in first iteration of the quasi-linearization method (QLM). The zeroth iteration is based on general features of the exact solution near the boundaries. The approach is illustrated on the Yukawa potential. The results enable accurate analytical estimates of effects of parameter variations on physical systems.We find accurate analytic presentation of wave functions and energies for an arbitrary physical potential U (r). We use the quasilinearization method (QLM) suggested recently for solving the Schrödinger equation after conversion to Riccati form [1,2]. In QLM the nonlinear terms of the differential equation are approximated by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter.Substitution of expression y(r)The corresponding QLM equation [1,2] is y ′ n+1 (r)+ 2y n+1 y n (r) = y 2 n (r) + k 2 n (r), where k 2 n is obtained from k 2 (r) by replacing there E by energy of n- will be accurate if the zeroth iteration based on general features of solutions near the boundaries is chosen. For illustration we find the wave functions and binding energies of Yukawa potential analytically in the first QLM iteration. To estimate the precision of our analytic solution we solve the Schrödinger equation numerically as well.The Yukawa potential U (r) = −g e −λr r , g > 0 was suggested in the early days of quantum mechanics for description of nucleon interactions. During last decades the Yukawa potential have been used in atomic physics applications, such as the screening of nucleon electromagnetic field by electron cloud, or atoms under external pressure and in connection to quark interactions with parameters λ and g depending on the temperature of the quark-gluon plasma.Let us try to guess the simplest form of the wave function. The large distance behavior is ψ(r) ∼ e −ηr with η = √ −2mE, while the small distance behavior is determined by the Kato condition [4] ψ(0) = − ψ ′ (0) µ with µ = mg. Noting also that the radial wave function should have a nonzero value at the origin, we come to the following initial guess function ψ 0 (r) ∼ e −ηr −e −ar r with a chosen to satisfy the Kato condition which leads to a = 2µ − η. Thus we find for the initial guess χ 0 (r) = rψ 0 (r) = N e −ηr − e −(2µ−η)r , and therefore y 0 (r) = −µ + (µ − η) coth [(µ − η) r] whereis the normalization factor.Inserting this into the equation for E 0 and using a straightforward integration, one obtains for the zeroth order ground state energy Insert PSN Here

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“…The method provides final and reasonable results for both small and large values of the coupling constant and is able to handle even super singular potentials for which each term of the perturbation theory is infinite and the perturbation expansion does not exist [46][47][48][49].…”

Section: Discussionmentioning

confidence: 94%

“…Earlier, the accurate analytical presentation of solutions, deduced by using the first QLM iteration were presented for quartic oscillator [47], as well as for the generalized anharmonic oscillator [48] and for the arbitrary physical potential vanishing at large distances [49].…”

Section: Introductionmentioning

confidence: 99%

Accurate analytic presentation of solution for the spiked harmonic oscillator problem

Liverts

Mandelzweig

2007

Annals of Physics

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Accurate analytic presentation of solution of the Schrödinger equation with arbitrary physical potential

Cited by 20 publications

References 27 publications

Energy spectrum for a modified Rosen-Morse potential solved by proper quantization rule and its thermodynamic properties

Energy spectrum for a modified Rosen-Morse potential solved by proper quantization rule and its thermodynamic properties

Analytic presentation of a solution of the Schrödinger equation

Accurate analytic presentation of solution for the spiked harmonic oscillator problem

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