Semiclassical Time Evolution and Trace Formula for Relativistic Spin-1/2 Particles

Bölte, Jens; Keppeler, Stefan

doi:10.1103/physrevlett.81.1987

Cited by 73 publications

(105 citation statements)

References 14 publications

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“…4.1), calculating the spin part of the path integral exactly. The trace formula we obtain is a generalization of the results by Bolte and Keppeler [5] to arbitrary spin. A similar approach leads, under certain restrictions, to the trace formula in the adiabatic (strong-coupling) limit (Sec.…”

Section: Semiclassics With Spin-orbit Couplingmentioning

confidence: 74%

“…For s = 1/2 the modulation factor M po (E) was derived in [5] by a different method, which, as ours, treats the spin degrees of freedom on the quantum-mechanical level.…”

Section: Weak-coupling Limit: Quantum-mechanical Treatment Of Spinmentioning

confidence: 99%

“…We intend to generalize the path-integral representation for the partition function (5) to include the spin degree of freedom. The main issue then is to be able to describe spin on the quantum-mechanical level by a continuous variable.…”

Section: Spin Coherent States and Quantum Partition Functionmentioning

confidence: 99%

“…Imposing periodic boundary conditions on the propagation and thus integrating over closed paths, we arrive at the expression for the partition function [cf. (5)]:…”

Section: Spin Coherent States and Quantum Partition Functionmentioning

confidence: 99%

“…Frisk and Guhr [13] found that in certain cases this problem can be corrected by a heuristic procedure. Bolte and Keppeler [5] studied the opposite situation when the interaction is weak. They derived the trace formula for the Dirac equation and Pauli equation with spin 1/2.…”

Section: Introductionmentioning

confidence: 99%

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Semiclassical theory of spin orbit interaction in the extended phase space

Pletyukhov

Zaitsev

2003

J. Phys. A: Math. Gen.

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Section: Semiclassics With Spin-orbit Couplingmentioning

confidence: 74%

“…For s = 1/2 the modulation factor M po (E) was derived in [5] by a different method, which, as ours, treats the spin degrees of freedom on the quantum-mechanical level.…”

Section: Weak-coupling Limit: Quantum-mechanical Treatment Of Spinmentioning

confidence: 99%

Section: Spin Coherent States and Quantum Partition Functionmentioning

confidence: 99%

“…Imposing periodic boundary conditions on the propagation and thus integrating over closed paths, we arrive at the expression for the partition function [cf. (5)]:…”

Section: Spin Coherent States and Quantum Partition Functionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Semiclassical theory of spin orbit interaction in the extended phase space

Pletyukhov

Zaitsev

2003

J. Phys. A: Math. Gen.

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Space–time approach to quantum chaos

Steiner¹

2003

Physica Status Solidi (b)

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Dedicated to Professor Jozef T. Devreese on the occasion of his 65th birthday PACS 03.65. Sq, 05.45.Mt In the first part of this paper, it is shown that the energy levels of a quantum system, whose classical limit is chaotic, encode certain space-time properties of the corresponding classical system. To see this, one considers the semiclassical limit as Planck's constant tends to zero. As a generalization of Mark Kac's famous question, it is demonstrated that "one can hear the periodic orbits of a quantum billiard". In the second part, some mathematical aspects of the semiclassical limit are reviewed. In order to deal with expressions that are mathematically easier to control, one does not work with the path integral directly, but instead with a smoothed kernel corresponding to a well-defined Fourier integral operator. Applying then the techniques from microlocal analysis and pseudodifferential operators, one arrives at a semiclassical trace formula which is a generalization of the Gutzwiller trace formula originally derived from the path integral. In the third part of this paper, the Hadamard-Gutzwiller model is discussed whose classical limit is a strongly chaotic (Anosov) system. In order to derive exact orbit sum rules for this model, one requires the path integral on hyperbolic D-space ðD ! 2Þ which can be exactly solved by using the general lattice definition of path integrals in curvilinear coordinates.1 What is semiclassical about quantum mechanics? In the field of quantum chaos, one considers quantum systems whose classical limit is chaotic (see, e.g., [1] for a recent review). A central question then is whether one can find some (possibly unique and universal) fingerprints of classical chaos left on the corresponding quantum systems. In view of the correspondence principle, one expects such signatures of quantum chaos to be seen in the semiclassical limit as Planck's constant h tends to zero. It turns out that the semiclassical theory based on the semiclassical approximation to the Feynman path integral (see, e.g., Chapter 5 in [2]) leads to a deep understanding of quantum spectra and wave functions of complex quantum systems in terms of space-time properties of the corresponding classical dynamics.The classical dynamics as well as the quantum mechanics of classically chaotic systems is quite complicated. Therefore, one often studies simple model systems which nevertheless show the main features of typical chaotic systems. Prototype examples of strongly chaotic systems are Euclidean and non-Euclidean billiards. In the Euclidean case, they are defined by the free motion of a point particle with mass m inside a compact Euclidean domain W & R 2 with elastic reflections at the boundary @W. The corresponding quantum systems are governed by the Schraedinger equation inside Ŵ H Hw n ðx Þ ¼ E n w n ðx Þ ; x 2 W ;

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Carbon nanotube in a threading magnetic field: Conductance oscillations persistent current and magnetization

Onorato

2013

Physica Status Solidi (b)

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987752The magnetic response of electrons in a zigzag carbon nanotube (CN) threaded by an Aharonov Bohm flux f is carefully analyzed by using a Hartree Fock approximation. The ground state energy as a function of the magnetic field, B, and the electron-electron repulsion strength, U, are obtained for N e interacting electrons confined in short ($100 nm) and long ($1 mm) CNs. The spin and orbital magnetization responses to the magnetic field variations are discussed. The CN transport properties are investigated and periodic field dependent oscillations are predicted for both longitudinal ballistic and persistent currents. The behavior of persistent current is investigated as a function of U and a significant suppression in current amplitude is observed when the strength of the electron-electron repulsion increases.

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Semiclassical Time Evolution and Trace Formula for Relativistic Spin-1/2 Particles

Cited by 73 publications

References 14 publications

Semiclassical theory of spin orbit interaction in the extended phase space

Semiclassical theory of spin orbit interaction in the extended phase space

Space–time approach to quantum chaos

Carbon nanotube in a threading magnetic field: Conductance oscillations persistent current and magnetization

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