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Chinese Physics Letters

Wentzel-Kramers-Brillouin Approximation for Dynamic Systems with Kinetic Coupling in Entangled State Representations

Fan Hong‐Yi

Published Jun 28, 2002• 3 citations
Publisher

Abstract

We study the Wentzel-Kramers-Brillouin (WKB) approximation for dynamic systems with kinetic couplings in entangled state representations. The result shows that the kinetic coupling will affect the position of classical turning points where the condition of using the WKB approximation breaks down. The modified WKB approximation formula is derived in the entangled state representation, for example, the common eigenvector of the relative coordinate and the total momentum of two particles. The corresponding Bohr-Sommerfeld quantization rule is also derived.

DOI
10.1088/0256-307x/19/7/304

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We study the Wentzel-Kramers-Brillouin (WKB) approximation for dynamic systems with kinetic couplings in entangled state representations. The result shows that the kinetic coupling will affect the position of classical turning points where the condition of using the WKB approximation breaks down. The modified WKB approximation formula is derived in the entangled state representation, for example, the common eigenvector of the relative coordinate and the total momentum of two particles. The corresponding Bohr-Sommerfeld quantization rule is also derived.
We apply a many-body Wentzel-Kramers-Brillouin (WKB) approach to determine the leading quantum corrections to the semiclassical dynamics of the Josephson model, describing interacting bosons able to tunnel between two localized states. The semiclassical dynamics is known to divide between regular oscillations and self-trapped oscillations where the sign of the imbalance remains fixed. In both cases, the WKB wave functions are matched to Airy functions, yielding a modified Bohr-Sommerfeld quantization condition. At the critical energy dividing normal and self-trapped oscillations, the WKB wave functions should instead be matched to parabolic cylinder functions, leading to a quantization formula that is not just the Bohr-Sommerfeld formula, and recovering the known logarithmic quantum break times at this energy. This work thus provides another illustration of the usefulness of the WKB approach in certain many-body problems.
We extend the Wentzel–Kramers–Brillouin (WKB) approximation to the semi-relativistic spinless Salpeter equation. The Bohr–Sommerfeld quantization rule for a one-dimensional potential is derived by properly connecting the WKB wave functions. Since the latter diverges at the classical turning points (TPs), we develop a uniform WKB approach that yields an approximate wave function valid everywhere including the TPs, along with a modified quantization condition. The method is illustrated for the symmetric linear potential, and comparison with exact numerical results shows excellent agreement, significantly improving upon the standard WKB method.
We propose an extension of the Wentzel-Kramers-Brillouin (WKB) approximation for solving the Schrödinger equation. Based on an ansatz for the wave function, subject to an auxiliary condition on its first derivative, we obtain a set of coupled differential equations that decouple, via an alternating perturbation method, into the well-known Bremmer series. The perturbation improves the wave function amplitudes alternately, and its phase is refined through recursive diagonalization. From this construction, we derive a general quantization formula that encodes the geometric-optics-like physics of the system. When the ratio of the differential reflection coefficient to the classical momentum remains constant, the formula reduces to a closed-form quantization condition consistent with that obtained by re-summing the WKB series to all orders.
The mutual consistency and structural interconnection between the Wentzel-Kramers-Brillouin (WKB) and Wigner-Kirkwood (WK) semiclassical approximations is investigated for nonrelativistic N-particle systems, with mutual scalar interactions and coupling to an external time-varying electromagnetic field. The generalized WK expansion of the propagator 〈x\ensuremath{\Vert}U(t,s)\ensuremath{\Vert}y〉 is obtained from a large-mass expansion of the higher-order WKB approximation. Two techniques are described for computing the WK coefficient functions. One relies on a large-mass expansion of classical paths and the transport representation of the WKB approximation; the other is recursive in nature. For time-independent Hamiltonians H the standard WK expansion of the heat kernel 〈x\ensuremath{\Vert}${e}^{\mathrm{\ensuremath{-}}\ensuremath{\beta}H}$\ensuremath{\Vert}y〉 is recovered.
We demonstrate that the Bargmann representation of quantum mechanics is ideally suited for semiclassical analysis, using as an example the WKB method applied to the bound-state problem in a single well of one degree of freedom. While the WKB expansion formulas are basically the usual ones, in this representation they describe approximations that are uniform and nonsingular in the classically allowed region of phase space because no turning points appear there. The quantization of energy levels relies on a complex contour integral that tests the eigenfunction for analyticity. For the harmonic oscillator, this WKB method trivially gives the exact eigenfunctions in addition to the exact eigenvalues. For an anharmonic well, a self-consistent variational choice of the representation greatly improves the accuracy of the semiclassical ground state. Also, a simple change of scale illuminates the relationship of semiclassical versus linear perturbative expansions, allowing a variety of multidimensional extensions. All in all, the Bargmann representation appears to combine the advantages of a linear description and of a phase-space representation of the quantum state vectors.
Analysis of the Wentzel–Kramers–Brillouin (WKB) exactness in some homogeneous spaces is attempted. CPN as well as its noncompact counterpart DN,1 is studied. U(N+1) or U(N,1) based on the Schwinger bosons leads us to CPN or DN,1 path integral expression for the quantity tr e−iHT, with the aid of coherent states. The WKB approximation terminates in the leading order and yields the exact result provided that the Hamiltonian is given by a bilinear form of the creation and the annihilation operators. An argument on the WKB exactness to more general cases is also made.