Abstract
The use of hydrodynamics in the description of quantum systems has a long and diverse history. Perhaps one of the earliest applications [1] was to the stopping power problem which concerns the excitation phenomena taking place when a charged particle passes through matter. At an atomistic level, the moving ion suffers a loss of energy as a result of collisions with atomic electrons. However, individual Rutherford scattering events cannot in fact take place due to the long-range nature of the Coulomb potential which implies that the ion interacts at once with many electrons. Furthermore, the electrons which respond to the passage of the ion, interact with each other, so that the field experienced by a given electron is comprised of the total field due to the ion and the dynamic screening charge of all other electrons. This situation is exceedingly complex and indicates that a quantitative understanding can only be achieved by a consideration of the collective response of the electrons. Presumably this is what Bohr [2] had in mind when he imagined electrons as forming a trailing wake behind the ion. This picture has a natural realization in Bloch’s hydrodynamic theory [1] which treats the electrons as a charged fluid, analogous to an ordinary fluid, whose dynamical state is specified in terms of local variables such as density, velocity and pressure. What is far from obvious is the extent to which this picture is meaningful for a degenerate quantum system.
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References
F. Bloch, Z. Phys. 81, 363 (1933).
N. Bohr, K. Dan. Vidensk. Selsk. Mat.-Fys. Medd. 18, no. 8 (1948).
S. A. Rice and P. Gray, Statistical Mechanics of Simple Fluids (Interscience, New York, 1965).
L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics (Benjamin, New York, 1962).
R. H. Ritchie, Progr. Theor. Phys. (Kyoto) 29, 607 (1963).
A. J. Bennett, Phys. Rev. B 1, 203 (1970).
J. Harris, Phys. Rev. B 4, 1022 (1971).
F. Forstmann and R. R. Gerhardts, Metal Optics Near the Plasma Frequency (Springer-Verlag, Berlin, 1986).
C. Schwartz and W.L. Schaich, Phys. Rev. B 26, 7008 (1982).
S. C. Ying, Nuovo Cimento B 23, 270 (1974).
A. Eguiluz, S. C. Ying, and J. J. Quinn, Phys. Rev. B 11, 2118 (1975).
E. Zaremba and H. C. Tso, Phys. Rev. B 49, 8147 (1994).
E. K. U. Gross, J. F. Dobson and M. Petersilka, in Density Functional Theory II edited by R. F. Nalewajski, Springer Series on Topics in Current Chemistry (Springer, Berlin, 1996), p.81.
P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).
R.M. Dreizler and E. K. U. Gross, Density Functional Theory (Springer-Verlag, Berlin, 1990).
W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).
M. J. Stott and E. Zaremba, Phys. Rev. A 21, 12 (1980); 22, 2293 (1980).
T. Ando, Solid State Commun. 21, 133 (1977).
A. Zangwill and P. Soven, Phys. Rev. A 21, 1561 (1980).
E. K. U. Gross and W. Kohn, Phys. Rev. Lett. 55, 2850 (1980).
E. K. U. Gross and W. Kohn, Adv. Quantum Chem. 21, 255 (1990).
K. Nuroh, M. J. Stott and E. Zaremba, Phys. Rev. Lett. 49, 862 (1982).
W. Ekardt, Phys. Rev. B 31, 6360 (1985).
K. Sturm, E. Zaremba and K. Nuroh, Phys. Rev. B 42, 6973 (1990).
A. Liebsch, Phys. Rev. B 36, 7378 (1987).
J. F. Dobson and G. H. Harris, J. Phys. C21, L729 (1988).
D. A. Broido, K. Kempa and P. Bakshi, Phys. Rev. B 42, 11400 (1990)
V. Gudmundsson and R. Gerhardts, Phys. Rev. B 43, 12098 (1991).
J. F. Dobson, Phys. Rev. B 46, 11163 (1992)
J. Dempsey and B. I. Halperin, Phys. Rev. B 47, 4662 (1993).
D. E. Beck, Phys. Rev. B 35, 7325 (1987).
W. L. Schaich and J. F. Dobson, Phys. Rev. B 49, 14700 (1994).
E. Zaremba, Phys. Rev. B 53, 10512 (1996).
A. V. Chaplik, Sov. Phys. JETP 33, 947 (1971).
F. Stern, Phys. Rev. Lett. 30, 278 (1973).
C. F. von Weizsäcker, Z. Phys. 96, 431 (1935).
Y. Tomishima and K. Yonei, J. Phys. Soc. Jpn., 21, 142 (1966).
W. Stich et al, Z. Phys. A 309, 5 (1982).
A. Chizmeshya and E. Zaremba, Phys. Rev. B 37, 2805 (1988).
P. Nozières and D. Pines, Theory of Quantum Fluids, Vol. II (Addison-Wesley, Redwood City, 1990).
M. Sundaram, A. C. Gossard, J. H. English and R. M. Westervelt, Superlatt. Microstruct. 4, 683 (1988).
M. Shayegan, T. Sajoto, M. Santos and C. Silvestre, Appl. Phys. Lett. 53, 791 (1988).
H. C. Tso and E. Zaremba, to be published.
L. Brey, N.F. Johnson and B.I. Halperin, Phys. Rev. B 40, 10647 (1989).
S. K. Yip, Phys. Rev. B 43, 1707 (1991).
J. F. Dobson, Phys. Rev. Lett. 73, 2244 (1994).
K. D. Tsuei, et al, Surf. Sci. 247, 302 (1991).
A. E. DePristo and J. D. Kress, Phys. Rev. A 35, 438 (1987).
M. Levy, J. P. Perdew and V. Sahni, Phys. Rev. A 30, 2745 (1984).
P. M. Echenique, F. Flores and R. H. Ritchie, Solid State Physics 41, 229 (1990).
R. Kronig and J. Korringa, Physica 10, 406 (1943).
A. Arnau and E. Zaremba, Nucl. Instr. and Meth. in Phys. Res. B 90, 32 (1994).
P. Nozières and D. Pines, Theory of Quantum Fluids, Vol. I (Addison-Wesley, Redwood City, 1989).
G. Vignale and W. Kohn, Phys. Rev. Lett. 77, 2037 (1996): see also the chapter by Vignale and Kohn in the present volume.
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Zaremba, E., Tso, H.C. (1998). Hydrodynamics in the Thomas-Fermi-Dirac-von Weizsäcker Approximation. In: Dobson, J.F., Vignale, G., Das, M.P. (eds) Electronic Density Functional Theory. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-0316-7_16
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