Quasiaccurately solvable quantum mechanics problems and the anharmonic oscillator problem

1993 ◽  
Vol 36 (2) ◽  
pp. 161-172
Author(s):  
A. S. Vshivtsev ◽  
V. Ch. Zhukovskii ◽  
R. A. Potapov ◽  
A. O. Starinets
Keyword(s):  
Quantum Mechanics ◽  
Anharmonic Oscillator ◽  
Solvable Quantum

Quantum mechanics of the anharmonic oscillator

1973 ◽  
Vol 14 (2) ◽  
pp. 219-227 ◽  
Author(s):  
Francis R. Halpern
Keyword(s):  
Quantum Mechanics ◽  
Anharmonic Oscillator

scholarly journals NEW CONCEPT OF SOLVABILITY IN QUANTUM MECHANICS

2013 ◽  
Vol 53 (5) ◽  
pp. 473-482 ◽  
Author(s):  
Miloslav Znojil
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Ad Hoc ◽  
Current Concept ◽  
Quantum Models ◽  
Solvable Quantum ◽  
New Type

In quite a few recent quantum models one is allowed to make a given Hamiltonian <em>H</em> self-adjoint only after an ad hoc generalization of Hermitian conjugation, <em>H</em><sup>†</sup>→<em>H</em><sup>‡</sup>:= Θ <sup>−1</sup><em>H</em><sup>†</sup>Θ wherethe suitable operator Θ is called Hilbert-space metric. In the generalized, hidden-Hermiticity scenario with nontrivial metric Θ≠<em> I</em> the current concept of solvability (meaning, most often, the feasibility of a non-numerical diagonalization of <em>H</em>) requires a generalization (allowing for a non-numerical tractabilityof Θ). A few very elementary samples of "solvable" quantum models of this new type are presented.

scholarly journals A family of analytically solvable Schrödinger equations related by Levi-Civita transformation

2019 ◽  
Vol 16 (3) ◽  
pp. 103
Author(s):  
Le Dai Nam ◽  
Phan Anh Luan ◽  
Luu Phong Su ◽  
Phan Ngoc Hung
Keyword(s):  
Exact Solution ◽  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Anharmonic Oscillator ◽  
Relativistic Quantum ◽  
Relativistic Quantum Mechanics ◽  
Schrödinger Equations ◽  
Two Dimensional ◽  
Spatial Transformations ◽  
Solvable Problems

Some two-dimensional problems in non-relativistic quantum mechanics can connect to each other by certain spatial transformations such as Levi-Civita transformation. This property allows forming a series of two-dimensional problems into an interrelated family. Starting from two related problems namely Coulomb plus harmonic oscillator and sextic double-well anharmonic oscillator potentials, such family is constructed via repeatedly applying Levi-Civita transformations. Obviously, this family contains various of exactly analytically solvable problems. The quasi-exact solution for each unknown member of this family is also obtained and systematically investigated.

On the Field Theoretic Functional Calculus for the Anharmonic Oscillator. II

1967 ◽  
Vol 22 (12) ◽  
pp. 1842-1865 ◽  
Author(s):  
W. Schuler ◽  
H. Stumpf
Keyword(s):  
Functional Equation ◽  
Quantum Mechanics ◽  
Finite Number ◽  
Functional Calculus ◽  
Functional Equations ◽  
Anharmonic Oscillator ◽  
Quantum Field ◽  
Operator Representation ◽  
Physical Conditions ◽  
Technical Details

The theory of solution for quantum field functional equations is developed for a suitable testproblem of quantum mechanics. In Sect. 1 the anharmonic oscillator is described in a field theoretic fashion. In Sect. 2 its functional equations are derived and in Sect. 3 these equations are symmetrized due to physical conditions. In Sect. 4, 5 the expansion of the physical functionals into series of base functionals is discussed and a convenient notation for the operator representation is introduced. In Sect. 6 the representation of the functional equation for an expansion into Dyson base functionals is given. In Sect. 7 and 8 functionals are approximated by expansions with only a finite number of terms and the resulting equations are prepared for integration. In Sect. 9, 10 the integration of the resulting equations for N = 2 and N = 4 is discussed in detail so that one finally obtains eigenvalue equations which contain only integrals to be solved. In the appendices technical details are derived.

QUANTUM AND SEMICLASSICAL SCALES FOR THE xn ANHARMONIC OSCILLATOR

1992 ◽  
Vol 07 (09) ◽  
pp. 763-766
Author(s):  
J.T. ANDERSON
Keyword(s):  
Quantum Mechanics ◽  
Fine Structure ◽  
Singular Point ◽  
Anharmonic Oscillator ◽  
Semiclassical Approximation ◽  
Canonical Quantization ◽  
Length Scale ◽  
Quantum Invariants

The classical fine structure around a singular point has been shown to be smoothed out by h in quantum mechanics. For the xn anharmonic oscillator in the semiclassical approximation this result is shown to set a quantum length scale [Formula: see text] below which the fine structure is not observable. The length scale allows canonical quantization in terms of the ratio of the quantum to semiclassical scales [Formula: see text]. This ratio is equivalent to ratios of the quantum to semiclassical scales in action and energy and provides a measure of the departure of semiclassical from quantum invariants.

Quantum mechanics of the isotropic three-dimensional anharmonic oscillator

1964 ◽  
Vol 60 (2) ◽  
pp. 273-278 ◽  
Author(s):  
I. J. Zucker
Keyword(s):  
Quantum Mechanics ◽  
Power Series ◽  
Potential Function ◽  
Anharmonic Oscillator ◽  
Three Dimensional ◽  
Spherically Symmetric ◽  
Symmetric Potential ◽  
Eigen Values

AbstractA method of determining numerically to any degree of accuracy the eigen-values of Hamiltonians in the form of power series is presented. The case of a spherically symmetric potential function of the form V = ar2 + br4 + cr6 is treated in detail.

scholarly journals On the Field Theoretic Functional Calculus for the Anharmonic Oscillator III

1968 ◽  
Vol 23 (6) ◽  
pp. 902-917 ◽  
Author(s):  
W. Schuler ◽  
H. Stumpf
Keyword(s):  
Quantum Mechanics ◽  
Functional Equations ◽  
Physical State ◽  
Anharmonic Oscillator ◽  
Explicit Representation ◽  
Necessary Condition ◽  
Quantum Field ◽  
Functional Formulation ◽  
Technical Details ◽  
Lowest Eigenvalue

The theory of solution for quantum field functional equations is developped for a suitable testproblem of quantum mechanics. In Sect. 1 the functional formulation of the anharmonic oscillator in its spinorial representation is given, and in Sect. 2 translational equivalent functional equations are discussed. The expansion of the physical state functionals into series of basefunctionals and the symmetrical representation of the functional equations for such an expansion is discussed in Sect. 3. In the following Sect. 4 the special symmetric orthogonal Hermitean functionals are used and the explicit representation is derived. In Sect. 5 the functionals are approximated by expansions with only a finite number of terms and the resulting equations are prepared for integration and in Sect. 6 a necessary condition of stationarity is considered. In Sect. 7 the simplest equation for N=1 is discussed in detail and the lowest eigenvalue is obtained. In the appendices technical details are derived.

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