Quantum Mechanical Tunneling |

**In General** > s.a. WKB Approximation.

* __Idea__: A particle
whose wave function is initially localized on one side of a potential
barrier, with energy less that the maximum value of the barrier, has a
non-zero probability of being found on the other side later.

@ __References__: Burgess AJP(91)nov [imaginary time];
Cushing FP(95);
Leavens FP(95);
Merzbacher PT(02)aug [history];
Nimtz & Haibel 08;
Razavy 14 [e1 r PT(04)feb];
news guardian(14)oct [how it works].

> __Examples__: see
boundary conditions in quantum cosmology.

**Tunneling Time** > s.a. causality violations.

* __Time scale__: For
thin barriers, the tunneling time is linear in the width; For thicker
ones, it saturates (Hartman effect) and there is a time scale associated
with it; What does it mean? The tunneling speed is usually characterized
by the Wigner velocity, but there are other proposals; The group delay is
proportional to the energy stored, and equal to the dwell time plus a
self-interference delay.

* __Measurement__: It
has been measured in experiments based on the attoclock in ultrafast laser
ionization of Helium atoms.

* __Remark__: The
transit time of a particle between two points is not necessarily well
defined in standard quantum mechanics, whereas it is in Bohm's theory; For
this reason tunneling times may allow us to test the pilot-wave approach
by providing us with a situation in which Bohm's theory can make a
definite prediction when standard quantum mechanics can make none.

@ __General references__: Fertig PRL(90) [distribution];
Chen & Wang PLA(90);
Olkhovsky & Recami PRP(92);
Landauer & Martin RMP(94);
Leavens PLA(95);
Steinberg PRL(95);
Eisenberg & Ashkenazy FP(97)qp/96;
Challinor et al PLA(97);
Abolhasani & Golshani qp/99;
Yamada PRL(99);
Ruseckas PRA(01)qp;
Chuprikov qp/01;
de Carvalho & Nussenzveig PRP(02);
Privitera et al RNC(03)qp/04 [intro];
Olkhovsky et al PRP(04);
Wang et al PRA(04);
Davies AJP(05)jan-qp/04 [clock model];
Chuprikov qp/05 [comparison between definitions];
Winful NJP(06)qp [meaning];
Wu a0804 [imaginary time];
Bernardini AP(09)-a0903 [and scattering delay time];
Nimtz a0903 [rev];
Ordóñez & Hatano PRA(09)-a0905 [non-existence of intrinsic tunneling time];
De Leo & Leonardi JPA(11)-a1103 [phase time formula];
Aichmann & Nimtz FP(14) [misleading interpretations];
Demir & Güner AP(17)-a1512 [entropic formulation].

@ __Phase space approach__: Marinov & Segev PRA(96);
Xavier et al PRL(97) [semiclassical].

@ __Hartman effect__: Winful OE(02),
PRL(03),
PRP(06);
Martínez & Polatdemir PLA(06);
Winful SPIE(07)-a0708;
Bhattacharya & Roy JMP(13)
[precluded for dissipative systems]; > s.a. causality
in quantum theory.

@ __Measurements__: Palao et al PLA(97)qp/99;
Camus et al PRL(17)-a1611
+ news sn(17)jul [evidence];
> s.a. experiments in quantum mechanics.

@ __Related topics__: Zhou et al PLA(01)
[phase transition to crossover]; Winful et al PRA(04) [Dirac particles];
Bernardini EPL(08)-a0804 [relativistic, phase and dwell times];
Xu et al FP(13)
[relativistic extensions]; Bhattacharya PRA(14)
[two-state particle tunneling through a thermal magnetic barrier];
Kelkar et al AP(17)-a1705 [with dissipation].

**Related Topics** > s.a. quantum chaos; quantum
equivalence principle; wigner function.

@ __Dynamics__: Krekora et al PRA(01)
[speed]; Delgado et al PRA(03);
Faria et al FP(06)
[Kramers escape rate theory]; Calzetta & Verdaguer JPA(06)
[decoherence and anomalous diffusion];
Anastopoulos & Savvidou a1611 [quasi-classical paths].

@ __Approaches__: Olavo qp/96 [in classical interpretation of quantum mechanics];
Anastopoulos & Savvidou JMP(06)qp,
JMP(08)-a0706,
Anastopoulos JMP(08)-a0706 [as time-of-arrival problem, and decay];
Levkov et al PRL(07)-a0707 [semiclassical solutions];
Chuprikov a1303 [new approach];
Norsen AJP(13)apr [pilot-wave perspective];
Turok NJP(14)-a1312 [in real time, using complex classical trajectories];
Hipple a1411 ["tunneling" transform];
Espinosa a1805 [calculation of tunneling actions];
> s.a. Polymer Representation.

@ __Apparent superluminality__: Krekora et al PRA(01);
Sokolovski & Liu PRA(01) [semiclassical];
Winful PRL(03);
Sokolovski PRS(04)qp/03;
Nimtz FP(11)-a1004 [violation of special relativity];
Aichmann & Nimtz a1304;
Fayngold a1412.

@ __Other related topics__: Aharonov et al PRA(93) [penetration into classically forbidden regions];
Eddi et al PRL(09) [analog in classical-wave + particle association];
Bender & Hook JPA(11)-a1011 [classical analog for complex-energy particle];
Ivlev a1108 [penetration through classical barriers?];
> s.a. GUP phenomenology; instantons;
quantum systems [tunneling in momentum space, potential step].

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