Values
The semiclassical propagation scheme allows for the analytical calculation of expectation values from the coherent states that comprise the final density matrix. For the unified sampling according to Chap. 3, each pair of trajectories provides a contribution to the final result which is incoherently (i. e., on the level of the density matrix) added to the other
D.4 Computation of Matrix Elements and Expectation Values 93 ones. Inserting the propagator from Eq. (3.5) into the density propagation scheme Eq. (3.4) results in D b AE = tr bAρf = Z dP Z dxidx′i dxf dx′f D x′f bA xf E b Kξ,νa (xf, xi) bKξ,νb∗ x′f, x′i ρ xi, x′i = Z dP 1 (2π~)2 D gσ p′f, q′f bA gσ(pf, qf) E R (pi, qi, t) R∗ p′i, qi′, t e~i(S(pi,qi,t)−S(p ′ i,qi′,t)) hg σ(pi, qi) | Ψ0iΨ0 gσ p′i, qi′ (D.51) where the integrations over the forward and backward, initial and final coordinate spaces have been eliminated and the remaining one is over the noise force distribution and ini- tial forward and backward phase spaces P = (ξ, ν, pi, qi, p′i, qi′). Since the semiclassically
propagated wave function is represented by a number of (phase-corrected) Gaussians, any expectation value is composed of a set of such Gaussians sandwiching the respective observ- able. Calculation of the overlaps, prefactors (R (pf, qf)), and action expressions has already
been dealt with in Sec. 3.1. The Gaussian expectation valuesDg bA gEfor a few observables will be given here.
The first moment of position evaluates to gσ p′, q′ bx gσ(p, q) = = Z dx r σ πexp n −σ 2 x − q ′∗2 − ip′∗ x − q′∗−σ 2 (x − q) 2+ ip′′ (x − q)ox = Z dx r σ πexp n −σx2+ σq′∗+ σq − ip′∗+ ipx + ip′∗q′∗− ipq −σ 2q ′∗2−σ 2q 2ox = Z dx r σ πexp n −σ (x − ¯r)2+ σ¯r2+ ip′∗q′∗− ipq − σ 2q ′∗2− σ 2q 2ox = ¯r expnσ¯r2+ ip′∗q′∗− ipq − σ 2q ′∗2−σ 2q 2o (D.52)
with ¯r = q+q2′∗ + ip−p2σ′∗. Similarly, the second moment of position results in gσ p′, q′ bx2 gσ(p, q) = 1 2σ + ¯r 2 expnσ¯r2+ ip′∗q′∗− ipq − σ 2q ′∗2−σ 2q 2o. (D.53) The energy expectation value is calculated from the sum of the kinetic and the potential part. With the second derivative of a Gaussian
hx| △ |gσ(p, q) i = = △ expn−σ2 (x − q)2+ ip (x − q)o = ▽ (−σ (x − q) + ip) hx| gσ(p, q) i = −σ hx| gσ(p, q) i + (−σ (x − q) + ip)2hx| gσ(p, q) i = σ2x2− 2σxq + σ2q − 2σ (x − q) ip − p2− σhx| gσ(p, q) i = σ2x2− 2 σ2q + σipx + σ2q2+ 2iσqp − p2− σhx| gσ(p, q) i , (D.54)
the kinetic energy results in gσ p′, q′ − 1 2m△ gσ(p, q) = −2m1 σ2 1 2σ + ¯r 2 − 2 σ2q + σipr + σ¯ 2q2+ 2iσqp − p2− σ expnσ¯r2+ ip′∗q′∗− ipq − σ 2q′∗2− σ 2q 2o. (D.55)
Inserting the Morse oscillator discussed in Chap. 5, i.e.,
V (x) = D [1 − exp {−κx}]2 = D − 2D exp {−κx} + D exp {−2κx} (D.56) into a Gaussian bracket results in
D gσ p′, q′ bV gσ(p, q) E = − 2D exp σr −¯ κ 2σ 2 + ip′∗q′∗− ipq − σ 2q′∗2− σ 2q 2 + D expnσ¯r2+ ip′∗q′∗− ipq − σ 2q ′∗2−σ 2q 2o + D exp σ¯r − κ σ 2 + ip′∗q′∗− ipq − σ 2q ′∗2− σ 2q 2 . (D.57) With the potential and kinetic contribution the total energy expectation value yields
D gσ p′, q′ bE gσ(p, q) E = −2D exp −¯rκ 2 + κ2 4σ + D + D exp −¯rκ + κ 2 σ −2m1 σ 2 q − q ′∗+ i1 2 p + p ′∗2−σ 2 !! expnσ¯r2+ ip′∗q′∗− ipq − σ 2q′∗2− σ 2q 2o. (D.58)
For the harmonic oscillator potential V (x) = mω22x2 studied in Chap. 4, the Gaussian
bracket is much simpler D gσ p′, q′ bV gσ(p, q) E = mω2 2 1 2σ + ¯r 2 expnσ¯r2+ ip′∗q′∗− ipq − σ 2q′∗2− σ 2q 2o. (D.59)
With this expression, the total energy expectation value in the harmonic case results in D gσ p′, q′ bE gσ(p, q) E = −2m1 σ 2 q − q′∗ + i1 2 p + p′∗ 2 − σ2 ! +mω 2 2 1 2σ + ¯r 2 . (D.60)
D.4 Computation of Matrix Elements and Expectation Values 95 Since the Gaussians are supposed to be coherent states of the harmonic oscillator, i.e., σ = ωm, this can be simplified to
D gσ p′, q′ bE gσ(p, q) E = 1 2m σ2 q + i σp q′+ i σp ′ ∗ + σ . (D.61)
All of the expressions given in this section can now be computed during the trajectory evaluation. Compared with the propagation of the EOMs of the trajectories, their com- putation is relatively costly as it involves the evaluation of complex exponential functions. Since the propagation does not depend on them, they do not need to be evaluated at every time step but only when needed.
Bibliography
[1] Howard, R.A.
Dynamic Probabilistic Systems, Volume I: Markov Models. Dover Publications, 2007, ISBN 978-0486458700.
[2] Gardiner, C. and Zoller, P. Quantum Noise.
Springer, Berlin, Heidelberg, New York, 3rd edition, 2004, ISBN 978-3-540-22301-6. [3] Schlosshauer, M.A.
Decoherence and the Quantum-To-Classical Transition. Springer, Berlin, Heidelberg, 2007, ISBN 978-3-540-35773-5. [4] Caldeira, A.O. and Leggett, A.J.
Influence of Dissipation on Quantum Tunneling in Macroscopic Systems. Phys. Rev. Lett., 46(4):211–214, Jan 1981.
[5] Widom, A. and Clark, T.D.
Probabilities for Quantum Tunneling through a Barrier with Linear Passive Dissipation. Phys. Rev. Lett., 48(2):63–65, Jan 1982.
[6] Hakim, V. and Ambegaokar, V.
Quantum theory of a free particle interacting with a linearly dissipative environment. Phys. Rev. A, 32(1):423–434, Jul 1985.
[7] Caldeira, A.O. and Leggett, A.J.
Path integral approach to quantum Brownian motion. Phys. A, 121:587–616, 1983.
[8] Grabert, H., Schramm, P., and Ingold, G.L.
Quantum Brownian motion: The functional integral approach. Phys. Rep., 168:115–207, 1988.
[9] Devoret, M.H., Martinis, J.M., and Clarke, J.
Measurements of Macroscopic Quantum Tunneling out of the Zero-Voltage State of a Current-Biased Josephson Junction.
Phys. Rev. Lett., 55(18):1908–1911, Oct 1985.
[10] Roos, C., Zeiger, T., Rohde, H., N¨agerl, H.C., Eschner, J., Leibfried, D., Schmidt- Kaler, F., and Blatt, R.
Quantum State Engineering on an Optical Transition and Decoherence in a Paul Trap. Phys. Rev. Lett., 83(23):4713–4716, Dec 1999.
[11] Ithier, G., Collin, E., Joyez, P., Meeson, P.J., Vion, D., Esteve, D., Chiarello, F., Shnirman, A., Makhlin, Y., Schriefl, J., and Sch¨on, G.
Decoherence in a superconducting quantum bit circuit. Phys. Rev. B, 72(13):134519, Oct 2005.
[12] Stockburger, J.T. and Grabert, H. Non-Markovian quantum state diffusion. Chem. Phys., 268:249–256, 2001.
[13] Stockburger, J.T. and Grabert, H.
Exact c-Number Representation of Non-Markovian Quantum Dissipation. Phys. Rev. Lett., 88(17):170407, Apr 2002.
[14] Stockburger, J.T.
Simulating spin-boson dynamics with stochastic Liouville-von Neumann equations. Chem. Phys., 296:159–169, 2004.
[15] Strunz, W.T., Di´osi, L., and Gisin, N.
Open System Dynamics with Non-Markovian Quantum Trajectories. Phys. Rev. Lett., 82(9):1801–1805, Mar 1999.
[16] Breuer, H.P., Kappler, B., and Petruccione, F.
Stochastic wave-function method for non-Markovian quantum master equations. Phys. Rev. A, 59(2):1633–1643, Feb 1999.
[17] Zhou, Y., Yan, Y., and Shao, J.
Stochastic simulation of quantum dissipative dynamics. Europhys. Lett., 72:334–340, 2005.
[18] Breuer, H.P., Laine, E.M., and Piilo, J.
Measure for the Degree of Non-Markovian Behavior of Quantum Processes in Open Systems.
Phys. Rev. Lett., 103(21):210401, 2009. [19] Weiss, U.
Quantum dissipative systems.
Number 13 in Series in modern condensed matter physics. World Scientific, Singapore, 3rd edition, 2008.
[20] Goletz, C.M. and Grossmann, F.
Decoherence and dissipation in a molecular system coupled to an environment: An application of semiclassical hybrid dynamics.
J. Chem. Phys., 130(24):244107, 2009.
[21] Hughes, K.H., Christ, C.D., and Burghardt, I.
Effective-mode representation of non-Markovian dynamics: A hierarchical approxima- tion of the spectral density. I. Application to single surface dynamics.
J. Chem. Phys., 131(2):024109, 2009. [22] Mintert, F. and Heller, E.J.
Simulation of open quantum systems. Europhys. Lett., 86(5):50006, 2009.
99 [23] Yu, T., Di´osi, L., Gisin, N., and Strunz, W.T.
Post-Markov master equation for the dynamics of open quantum systems. Phys. Lett. A, 265(5-6):331 – 336, 2000, ISSN 0375-9601.
[24] Gnutzmann, S. and Haake, F.
Positivity violation and initial slips in open systems. Z. Phy. B Con. Mat., 101:263–273, 1996.
[25] Feynman, R.P. and Vernon, F.L.
The theory of a general quantum system interacting with a linear dissipative system. Ann. Phys., 24:118–173, 1963.
[26] Breuer, H.P. and Petruccione, F. The Theory of Open Quantum Systems. Oxford University Press, Oxford, 2002. [27] Gisin, N. and Percival, I.C.
The quantum-state diffusion model applied to open systems. J. Phys. A, 25(21):5677–5691, 1992.
[28] Dalibard, J., Castin, Y., and Mølmer, K.
Wave-function approach to dissipative processes in quantum optics. Phys. Rev. Lett., 68(5):580–583, Feb 1992.
[29] Ankerhold, J., Pechukas, P., and Grabert, H.
Strong Friction Limit in Quantum Mechanics: The Quantum Smoluchowski Equation. Phys. Rev. Lett., 87(8):086802, Aug 2001.
[30] Lindblad, G.
On the generators of quantum dynamical semigroups. Commun. Math. Phys., 48:119–130, 2005.
[31] Di´osi, L.
On High-Temperature Markovian Equation for Quantum Brownian Motion. Europhys Lett., 22(1):1–3, 1993.
[32] Vacchini, B.
Completely Positive Quantum Dissipation. Phys. Rev. Lett., 84(7):1374–1377, Feb 2000.
[33] Zhang, S. and Pollak, E.
Quantum dynamics for dissipative systems: A numerical study of the Wigner–Fokker– Planck equation.
J. Chem. Phys., 118(10):4357–4364, 2003.
[34] Burghardt, I. and Møller, K.B.
Quantum dynamics for dissipative systems: A hydrodynamic perspective. J. Chem. Phys., 117(16):7409–7425, 2002.
[35] Cartling, B.
Kinetics of activated processes from nonstationary solutions of the Fokker–Planck equa- tion for a bistable potential.
J. Chem. Phys., 87(5):2638–2648, 1987. [36] Grossmann, F. and Koch, W.
A finite-difference implementation of the Caldeira–Leggett master equation. J. Chem. Phys., 130(3):034105, 2009.
[37] Fleck, J.A., Morris, J.R., and Feit, M.D.
Time-dependent propagation of high energy laser beams through the atmosphere. Appl. Phys. A-Mater., 10, Issue 2:129 – 160, Jun 1976.
[38] Press, W., Teukolsky, S., Vetterling, W., and Flannery, B. Numerical Recipes in C.
Cambridge University Press, Cambridge, UK, 2nd edition, 1992. [39] Hubbard, J.
Calculation of Partition Functions. Phys. Rev. Lett., 3(2):77–78, Jul 1959. [40] Feynman, R.P. and Hibbs, A.R.
Quantum Mechanics and Path Integrals.
McGraw-Hill Companies, 1965, ISBN 978-0070206502. [41] Stockburger, J.T.
Stochastic methods for the dynamics of open quantum systems.
Habilitation, Institut f¨ur Theoretische Physik, Universit¨at Stuttgart, 2006. [42] Frigo, M. and Johnson, S.G.
The Design and Implementation of FFTW3. Proceedings of the IEEE, 93(2):216–231, 2005.
Special issue on ’Program Generation, Optimization, and Platform Adaptation’. [43] Heller, E.J.
Frozen Gaussians: A very simple semiclassical approximation. J. Chem. Phys., 75(6):2923–2931, 1981.
[44] Herman, M.F. and Kluk, E.
A semiclasical justification for the use of non-spreading wavepackets in dynamics calcu- lations.
Chem. Phys., 91:27–34, 1984. [45] Thoss, M. and Wang, H.
Semiclassical description of molecular dynamics based on initial-value representation methods.
Annu. Rev. Phys. Chem., 55(1):299–332, 2004. [46] van de Sand, G. and Rost, J.M.
Irregular Orbits Generate Higher Harmonics. Phys. Rev. Lett., 83(3):524–527, Jul 1999.
101 [47] Hochman, G. and Kay, K.G.
Tunneling by a semiclassical initial value method with higher order corrections. J. Phys. A-Math. Theor., 41(38):385303, 2008.
[48] Zhang, S. and Pollak, E.
Monte Carlo Method for Evaluating the Quantum Real Time Propagator. Phys. Rev. Lett., 91:190201, 2003.
[49] Kluk, E., Herman, M.F., and Davis, H.L.
Comparison of the propagation of semiclassical frozen Gaussian wave functions with quantum propagation for a highly excited anharmonic oscillator.
J. Chem. Phys., 84(1):326–334, 1986.
[50] Großmann, F.
Theoretical Femtosecond Physics. Springer, 2008, ISBN 978-3-540-77896-7. [51] Metropolis, N. and Ulam, S.
The Monte Carlo Method.
J. Am. Stat. Assoc., 44:335, 1949.
[52] Stockburger, J.T.
Guide Trajectory for SLN semiclassics, 2003. Private communication.
[53] Gray, S.K., Noid, D.W., and Sumpter, B.G.
Symplectic integrators for large scale molecular dynamics simulations: A comparison of several explicit methods.
J. Chem. Phys., 101(5):4062–4072, 1994. [54] de Broglie, L.
La m´ecanique ondulatoire et la structure atomique de la mati`ere et du rayonnement. J. Phys. Radium, 8(5):225–241, 1927.
[55] Bohm, D.
A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. I. Phys. Rev., 85(2):166–179, Jan 1952.
[56] Wyatt, R.E.
Quantum Dynamics with Trajectories. Springer, 2005, ISBN 978-0-387-22964-5. [57] Lopreore, C.L. and Wyatt, R.E.
Quantum Wave Packet Dynamics with Trajectories. Phys. Rev. Lett., 82(26):5190–5193, Jun 1999.
[58] Liu, J. and Makri, N.
Monte Carlo Bohmian Dynamics from Trajectory Stability Properties. J. Phys. Chem. A, 108(25):5408–5416, 2004.
[59] Goldfarb, Y., Degani, I., and Tannor, D.J.
Bohmian mechanics with complex action: A new trajectory-based formulation of quan- tum mechanics.
J. Chem. Phys, 125(23):231103, 2006. [60] Trahan, C.J., Hughes, K., and Wyatt, R.E.
A new method for wave packet dynamics: Derivative propagation along quantum tra- jectories.
J. Chem. Phys., 118(22):9911–9914, 2003. [61] Kloeden, P.E. and Platen, E.
Numerical Solution of Stochastic Differential Equations. Springer, 3 edition, 1999, ISBN 3-540-54062-8.
[62] Fisher, B., Perkins, S., Walker, A., and Wolfart, E. Hypermedia Image Processing Reference.
http://www.cee.hw.ac.uk/hipr/html/hipr\_top.html, 1994. [63] Suzuki, M., editor.
Quantum Monte Carlo Methods in Condensed Matter Physics. World Scientific Publishing Company, 1993, ISBN 978-9810236830. [64] Mak, C.H. and Egger, R.
Monte Carlo Methods for Real-Time Path Integration. Adv. Chem. Phys., pages 39–76, 2007.
[65] Karrlein, R. and Grabert, H.
Exact time evolution and master equations for the damped harmonic oscillator. Phys. Rev. E, 55(1):153–164, Jan 1997.
[66] Morse, P.M.
Diatomic Molecules According to the Wave Mechanics. II. Vibrational Levels. Phys. Rev., 34(1):57–64, Jul 1929.
[67] Walker, R.B. and Preston, R.K.
Quantum versus classical dynamics in the treatment of multiple photon excitation of the anharmonic oscillator.
J. Chem. Phys., 67(5):2017–2028, 1977. [68] Davis, M.J. and Wyatt, R.E.
Surface-of-section analysis in the classical theory of multiphoton absorption. C. Phys. Lett., 86(3):235 – 241, 1982, ISSN 0009-2614.
[69] Miller, W.H.
The Semiclassical Initial Value Representation: A Potentially Practical Way for Adding Quantum Effects to Classical Molecular Dynamics Simulations.
J. Phys. Chem. A, 105(13):2942–2955, 2001. [70] Grossmann, F.
A semiclassical approach to dissipation in quantum mechanics. J. Chem. Phys., 103:3696–3704, 1995.
103 [71] Tanimura, Y. and Maruyama, Y.
Gaussian–Markovian quantum Fokker–Planck approach to nonlinear spectroscopy of a displaced Morse potentials system: Dissociation, predissociation, and optical Stark effects.
J. Chem. Phys., 107(6):1779–1793, 1997. [72] Makri, N.
Quantum Dissipative Dynamics: A Numerically Exact Methodology. J. Phys. Chem. A, 102(24):4414–4427, 1998.
[73] Doye, J.P.K. and Wales, D.J.
The effect of the range of the potential on the structure and stability of simple liquids: from clusters to bulk, from sodium to C60.
J. Phys. B: At. Mol. Opt., 29(21):4859, 1996. [74] Baletto, F. and Ferrando, R.
Structural properties of nanoclusters: Energetic, thermodynamic, and kinetic effects. Rev. Mod. Phys., 77(1):371–423, May 2005.
[75] Nest, M. and Saalfrank, P.
Open-system quantum dynamics for gas-surface scattering: Nonlinear dissipation and mapped Fourier grid methods.
J. Chem. Phys., 113(19):8753–8761, 2000. [76] Kao, C.L. and Madix, R.J.
The Adsorption Dynamics of Molecular Methane, Propane, and Neopentane on Pd(111): Theory and Experiment†.
J. Phys. Chem. B, 106(33):8248–8257, 2002. [77] Andrianov, I. and Saalfrank, P.
Free vibrational relaxation of H adsorbed on a Si(1 0 0) surface investigated with the multi-configurational time-dependent Hartree method.
Chem. Phys. Lett., 433(1-3):91 – 96, 2006, ISSN 0009-2614. [78] Stacey, F.D. and Davis, P.M.
High pressure equations of state with applications to the lower mantle and core. Phys. Earth Plan. In., 142(3-4):137 – 184, 2004, ISSN 0031-9201.
[79] ter Haar, D.
The Vibrational Levels of an Anharmonic Oscillator. Phys. Rev., 70(3-4):222–223, Aug 1946.
[80] Gallas, J.A.C.
Some matrix elements for Morse oscillators. Phys. Rev. A, 21(6):1829–1834, Jun 1980. [81] Gradshteyn, I.S. and Ryzhik, I.M.
Table of Integrals, Series, and Products. Academic Press, 1980.
[82] H¨anggi, P., Talkner, P., and Borkovec, M. Reaction-rate theory: fifty years after Kramers. Rev. Mod. Phys., 62(2):251–341, Apr 1990. [83] Yu, L.H.
Quantum tunneling in a dissipative system. Phys. Rev. A, 54(5):3779–3782, Nov 1996. [84] Isar, A., Sandulescu, A., and Scheid, W.
Dissipative tunneling through a parabolic potential in the Lindblad theory of open quantum systems.
The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics, 12:3– 10, 2000.
[85] Tanimura, Y. and Wolynes, P.G.
The interplay of tunneling, resonance, and dissipation in quantum barrier crossing: A numerical study.
The Journal of Chemical Physics, 96(11):8485–8496, 1992. [86] Ueda, M.
Transmission spectrum of a tunneling particle interacting with dynamical fields: Real- time functional-integral approach.
Phys. Rev. B, 54(12):8676–8687, Sep 1996.
[87] Konno, K., Nishida, M., Tanda, S., and Hatakenaka, N. The effect of dissipation on quantum transmission resonance. Physics Letters A, 368:442–449, 2007.
[88] Ankerhold, J., Grabert, H., and Ingold, G.L.
Dissipative quantum systems with a potential barrier: General theory and the parabolic barrier.
Phys. Rev. E, 51(5):4267–4281, May 1995. [89] Ankerhold, J. and Grabert, H.
Dissipative quantum systems with a potential barrier. II. Dynamics near the barrier top. Phys. Rev. E, 52(5):4704–4723, Nov 1995.
[90] Ankerhold, J. and Grabert, H.
Dissipative quantum systems with a potential barrier. III. Steady state nonequilibrium flux and reaction rate.
Phys. Rev. E, 55(2):1355–1374, Feb 1997. [91] Fr¨obrich, P., Lipperheide, R., and M¨ohring, K.
On tunnelling with dissipation.
Zeitschrift f¨ur Physik B Condensed Matter, 78:325– 332, 1990. [92] Takigawa, N., Hagino, K., and Abe, M.
Dynamical norm method for nonadiabatic macroscopic quantum tunneling. Phys. Rev. C, 51(1):187–197, Jan 1995.
105 [93] Reisdorf, W., Hessberger, F.P., Hildenbrand, K.D., Hofmann, S., M¨unzenberg, G., Schmidt, K.H., Schneider, J.H.R., Schneider, W.F.W., S¨ummerer, K., Wirth, G., Kratz, J.V., and Schlitt, K.
Influence of Collective Surface Motion on the Threshold Behavior of Nuclear Fusion. Phys. Rev. Lett., 49(25):1811–1815, Dec 1982.
[94] Dasso, C.H., Landowne, S., and Winther, A.
Channel-coupling effects in heavy-ion fusion reactions. Nucl. Phys. A, 405(2):381 – 396, 1983, ISSN 0375-9474. [95] Grabert, H. and Wipf, H.
Tunneling of hydrogen in metals.
In U. R¨ossler, editor, Festk¨orperprobleme-Advances in Solid State Physics, volume 30, pages 1–23. Vieweg, Braunschweig, 1990.
[96] Caldeira, A.O. and Leggett, A.J.
Quantum tunnelling in a dissipative system. Ann. Phys., 149:374–456, 1983.
[97] Grabert, H., Olschowski, P., and Weiss, U.
Temperature dependence of quantum decay rates in dissipative systems. Phys. Rev. B, 32(5):3348–3350, Sep 1985.
[98] Garg, A., Onuchic, J.N., and Ambegaokar, V. Effect of friction on electron transfer in biomolecules. The Journal of Chemical Physics, 83(9):4491–4503, 1985.
[99] Leggett, A.J., Chakravarty, S., Dorsey, A.T., Fisher, M.P.A., Garg, A., and Zwerger, W.
Dynamics of the dissipative two-state system. Rev. Mod. Phys., 59(1):1–85, Jan 1987. [100] Pollak, E.
Transition-state theory for tunneling in dissipative media. Phys. Rev. A, 33(6):4244–4252, Jun 1986.
[101] Eckart, C.
The Penetration of a Potential Barrier by Electrons. Phys. Rev., 35(11):1303–1309, Jun 1930.
[102] Johnston, H.S. and Heicklen, J.
Tunneling corrections for unsymmetrical Eckart potential energy barriers. J. Phys. Chem., 66(3):532–533, 1962.
[103] Voth, G.A., Chandler, D., and Miller, W.H.
Time correlation function and path integral analysis of quantum rate constants. J. Phys. Chem., 93(19):7009–7015, 1989.
[104] Liao, J.L. and Pollak, E.
A test of quantum transition state theory for a system with two degrees of freedom. J. Chem. Phys., 110(1):80–87, 1999.
[105] Di´osi, L. and Strunz, W.T.
The non-Markovian stochastic Schr¨odinger equation for open systems. Phys. Lett. A, 235(6):569 – 573, 1997, ISSN 0375-9601.
[106] Koch, W., Grossmann, F., Stockburger, J.T., and Ankerhold, J. Non-Markovian Dissipative Semiclassical Dynamics.
Phys. Rev. Lett., 100(23):230402, 2008. [107] van K¨ampen, N.G.
Itˆo versus Stratonovich.
J. Stat. Phys., 24(1):175–187, 1981. [108] Gardiner, C.W.
Handbook of Stochastic Methods: For the Natural and Social Sciences. Springer, Berlin, 4th edition, 2009, ISBN 978-3540707127.
[109] Milshtein, G. and Tret’yakov, M.
Numerical solution of differential equations with colored noise. J. Stat. Phys., 77(3):691–715, 1994.
[110] Stratonovich, R.L.
ОБ ОДНОМ МЕТОДЕ ВЫЧИСЛЕНИЯ КВАНТОВЫХ ФУНКЦИИ
РАСПРЕДЕЛЕНИЯ.
Dokl. Akad. Nauk S.S.S.R., 115:1097, 1957. [111] Wolfram Mathematica Online Integrator. http://integrals.wolfram.com, Jan 2010.
List of Publications
Scientific Journals
[1] Brisker, D., Cherkes, I., Gnodtke, C., Jarukanont, D., Klaiman, S., Koch, W., Weissman, S., Volkovich, R., Toroker, M.C., and Peskin, U.
Controlled electronic transport through branched molecular conductors. Molecular Physics, 106:281–287, 2008.
[2] Koch, W., Grossmann, F., Stockburger, J.T., and Ankerhold, J. Non-Markovian Dissipative Semiclassical Dynamics.
Phys. Rev. Lett., 100(23):230402, 2008. [3] Grossmann, F. and Koch, W.
A finite-difference implementation of the Caldeira–Leggett master equation. J. Chem. Phys., 130(3):034105, 2009.
[4] Koch, W., Großmann, F., Stockburger, J.T., and Ankerhold, J.
Semiclassical non-Markovian Brownian motion in anharmonic potentials. Chemical Physics, 370(1-3):34 – 41, 2010, ISSN 0301-0104.
Dynamics of molecular systems: From quantum to classical. [5] Koch, W., Großmann, F., Ankerhold, J., and Stockburger, J.T.
Semiclassical formulation of non-Markovian quantum Brownian motion. Physica E, 42(3):388 – 393, 2010, ISSN 1386-9477.
Proceedings of the international conference Frontiers of Quantum and Mesoscopic Ther- modynamics FQMT ’08.
[6] Goletz, C.M., Koch, W., and Grossmann, F.
Semiclassical dynamics of open quantum systems: Comparing the finite with the infinite perspective.
Chemical Physics, In Press, Corrected Proofs:–, 2010, ISSN 0301-0104. [7] Koch, W., Lubk, A., Grossmann, F., Lichte, H., and Schmidt, R.
Coherent and incoherent effects on the imaging and scattering process in transmission electron microscopy and off-axis electron holography.
Ultramicroscopy, 110(11):1397 – 1403, 2010, ISSN 0304-3991.
Proceedings
[1] Koch, W. and Grossmann, F.
Non-Markovian thermalization in the Morse oscillator.
In D.V. Shalashilin and M.P. de Miranda, editors, Multidimensional Quantum Mechanics with Trajectories. CCP6, Daresbury Laboratory, 2009, ISBN 978-0-9545289-8-0.
Danke
Wenn ich die vergangenen drei Jahre Revue passieren lasse, fallen mir eine ganze Menge Menschen ein, die mir in dieser Zeit wichtig waren. Einigen davon m¨ochte ich an dieser Stelle im Speziellen meine Dankbarkeit aussprechen, da ohne sie ein erfolgreicher Abschluss dieser Arbeit kaum m¨oglich gewesen w¨are.
Vor drei Jahren habe ich als ambitionierter Handwerker angefangen, dieses Projekt zu bearbeiten. Frank Großmann hat wesentlich dazu beigetragen, dass ich es als Physiker zu Ende bringen konnte. Er hat immer Zeit gefunden, Fragen zu beantworten, interessante Vorschl¨age zu machen und generell Probleme aus dem Weg zu r¨aumen. Dar¨uber hinaus hat er den Besuch von Konferenzen und Workshops m¨oglich gemacht und durch seinen umfangreichen Bekanntenkreis bei diesen Veranstaltungen immer wieder neue Kontakte hergestellt. Schließlich w¨are ohne seine Erfahrung das Schreiben von Ver¨offentlichungen und nicht zuletzt dieser Arbeit bedeutend komplizierter geworden.
F¨ur die Einf¨uhrung in die Thematik und umfangreiche Diskussionen ¨uber Probleme, neue Ideen und sich ergebende Resultate m¨ochte ich J¨urgen Stockburger, Joachim Anker- hold und David Tannor danken. Die pers¨onlichen Unterhaltungen und l¨angeren email Ko- rrespondenzen haben sich aus meiner Sicht immer gelohnt. Auch f¨ur die Einladungen nach Ulm und Rehovot bin ich dankbar.
Professor R¨udiger Schmidt hat mir in seiner Arbeitsgruppe schon zum Diplom einen Platz gew¨ahrt. Ich bin sehr froh, dass er sich bereit erkl¨art hat, mir auch als Doktorvater zur Verf¨ugung zu stehen. Die Arbeit hier in der Gruppe habe ich immer als angenehm empfunden. Dass ich dabei nicht an dem Umgang mit den b¨urokratischen Notwendigkeiten gescheitert bin, liegt wohl auch daran dass Gundula Sch¨adlich meiner Naivit¨at den Wirren der institutionalisierten Forschung gegen¨uber immer mit einem L¨acheln begegnete. Dar¨uber hinaus haben die Arbeitsgruppenausfl¨uge, erhitzte Diskussionen beim Kaffeetrinken und andere nicht-physikalische Aktivit¨aten ihren Teil dazu beigetragen, dass der Kopf wieder frei wurde, um ihn sich ¨uber den zu l¨osenden Problemen zerbrechen zu k¨onnen. Außerdem m¨ochte ich mich bei allen f¨ur die Unterst¨utzung bei der Verbesserung des Stils dieser Arbeit bedanken.
In den vergangenen acht Jahren, w¨ahrend des Studiums und der Promotion habe ich an der Universit¨at und vor allem auch außerhalb viele neue Menschen kennengelernt. Einige davon haben mich fast die gesamte Zeit begleitet. Es ist beruhigend zu wissen, dass man nach einem Tag lang Gleichungsjonglieren nach Hause kommen kann und immer jemanden findet, der willens ist, sich ¨uber etwas anderes zu unterhalten, gemeinsam zu essen oder irgend etwas zu unternehmen. Ohne meine Freunde w¨aren die letzten Jahre viel weniger farbenfroh und viel einseitiger geworden. Ich bitte um Vergebung f¨ur den ¨Arger den ich stellenweise verursacht habe. Im Speziellen sch¨atze ich mich ausgesprochen gl¨ucklich, dass ich immer mit netten und, meinen teilweise weltfremden Gedanken gegen¨uber, toleranten Mitbewohnern zusammenleben durfte.
Noch viel l¨anger als Freunde habe ich eine Familie. Und auch dabei habe ich Gl¨uck gehabt. Eine Schwester zu haben ist was tolles. Man kann sich ¨uber Gott und die Welt unterhalten, sich ¨uber die Vergangenheit und die Zukunft auseinandersetzen. Egal wer gerade das Land verlassen hat, danach sind wir um Erfahrungen reicher aber auch weiterhin immer noch Geschwister. Meine Schwester hat mir nicht nur erkl¨art, dass es vielleicht keine so gute Idee ist, als Deutscher einen englischen Text verfassen zu wollen, sondern mir auch dabei geholfen, die Konsequenzen davon weitestgehend zu verbergen.
Tja und schließlich bin ich erst seit ein paar Jahren Physiker, aber seit fast drei Jahrzehn- ten Sohn. Im Prinzip habe ich alles was ich kann und weiß, in irgendeiner Form meinen Eltern zu verdanken. Sie haben mir beigebracht nach den Sternen zu greifen. Dass ich mich heute glaube Wissenschaftler nennen zu k¨onnen ist ein Resultat davon, dass sie nie ein kindliches Warum oder Wieso unbeantwortet gelassen haben und dass sie mir sp¨ater beigebracht haben, Antworten selber zu suchen. Sie haben mich immer unterst¨utzt und M¨oglichkeiten geschaffen, weiter zu kommen; vom Besuch der Manos ¨uber das Austausch- jahr in den USA bis zum Studium und dar¨uber hinaus. Egal wohin es mich in der Zukunft