TY - CHAP

T1 - Quantum Chaotic Scattering and Conductance Fluctuations in Nanostructures

AU - Lai, Ying Cheng

AU - Tél, Tamás

N1 - Publisher Copyright:
© 2011, Springer Science+Business Media, LLC.

PY - 2011

Y1 - 2011

N2 - This chapter is devoted to the manifestation of classical chaotic scattering in the quantum world. The major characteristic that distinguishes a quantum system from its classical counterpart is that in quantum mechanics, the system is characterized by a nonzero value of the Planck constant. Let ℏ denote the Planck constant nondimensionalized by normalizing to characteristic length and momentum values, so that ℏ → 0 corresponds to the classical limit, ℏ ≪ 1 to the semiclassical regime, and ℏ ∼ 1 to the fully quantum-mechanical regime. To study the quantum manifestation of classical Hamiltonian chaos, the semiclassical regime is of particular importance because this is the regime in which both quantum and classical effects are relevant. In particular, we shall be interested in signatures of chaotic scattering when the same system is treated quantum-mechanically in the semiclassical regime. The mathematical methods needed to study the semiclassical regime differ from those used so far. This chapter is therefore of different character than the others. Our aim is to flesh out the most important phenomena only, where fingerprints of the classical transient chaos appear at the semiclassical level, motivating the reader to pursue more detailed studies.

AB - This chapter is devoted to the manifestation of classical chaotic scattering in the quantum world. The major characteristic that distinguishes a quantum system from its classical counterpart is that in quantum mechanics, the system is characterized by a nonzero value of the Planck constant. Let ℏ denote the Planck constant nondimensionalized by normalizing to characteristic length and momentum values, so that ℏ → 0 corresponds to the classical limit, ℏ ≪ 1 to the semiclassical regime, and ℏ ∼ 1 to the fully quantum-mechanical regime. To study the quantum manifestation of classical Hamiltonian chaos, the semiclassical regime is of particular importance because this is the regime in which both quantum and classical effects are relevant. In particular, we shall be interested in signatures of chaotic scattering when the same system is treated quantum-mechanically in the semiclassical regime. The mathematical methods needed to study the semiclassical regime differ from those used so far. This chapter is therefore of different character than the others. Our aim is to flesh out the most important phenomena only, where fingerprints of the classical transient chaos appear at the semiclassical level, motivating the reader to pursue more detailed studies.

KW - Chaotic Saddle

KW - Conductance Fluctuation

KW - Semiclassical Regime

KW - Semiclassical Theory

KW - Unstable Periodic Orbit

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U2 - 10.1007/978-1-4419-6987-3_7

DO - 10.1007/978-1-4419-6987-3_7

M3 - Chapter

AN - SCOPUS:85067991807

T3 - Applied Mathematical Sciences (Switzerland)

SP - 239

EP - 262

BT - Applied Mathematical Sciences (Switzerland)

PB - Springer

ER -