Il presente articolo è una rassegna di alcuni aspetti matematici fondamentali della formulazione Wigneriana della meccanica quantistica. A partire dagli assiomi della meccanica quantistica e della meccanica statistica quantistica viene motivata l'introduzione della trasformazione di Wigner e viene infine dedotta l'equazione di Wigner.
Referenze Bibliografiche
[1]
A. ARNOLD-
H. LANGE-
P. F. ZWEIFEL,
A discrete-velocity stationary Wigner equation,
J. Math. Phys.,
41 (
2000), 7167-7180. |
MR 1788568 |
Zbl 1019.82020[2]
A. ARNOLD-
H. STEINRÜCK,
The "electromagnetic" Wigner equation for an electron with spin,
Z. Angew. Math. Phys.,
40 (
1989), 793-815. |
MR 1027576 |
Zbl 0701.35130[3]
J. BANASIAK-
L. BARLETTI,
On the existence of propagators in stationary Wigner equation without velocity cut-off,
Transport Theory Stat. Phys.,
30 (
2001), 659-672. |
MR 1865352 |
Zbl 0990.82019[4]
L. BARLETTI-
P. F. ZWEIFEL,
Parity-decomposition method for the stationary Wigner equation with inflow boundary conditions,
Transport Theory Stat. Phys.,
30 (
2001), 507-520. |
MR 1866627 |
Zbl 1006.82032[5]
N. BEN ABDALLAH-
P. DEGOND-
I. GAMBA,
Inflow boundary conditions for the time dependent one-dimensional Schrödinger equation,
C. R. Acad. Sci. Paris, Sér. I Math.,
331 (
2000), 1023-1028. |
MR 1809447 |
Zbl 1158.35344[7] P. BORDONE-M. PASCOLI-R. BRUNETTI-A. BERTONI-C. JACOBONI, Quantum transport of electrons in open nanostructures with the Wigner-function formalism, Phys. Rev. B, 59 (1999), 3060-3069.
[8]
P. CARRUTHERS-
F. ZACHARIASEN,
Quantum collision theory with phase-space distributions,
Rev. Mod. Phys.,
55 (
1983), 245-285. |
MR 698046[9]
T. A. CLAASEN-
W. F. MECKLENBRÄUKER,
The Wigner distribution - a tool for time-frequency signal analysis,
Philips J. Res.,
35 (
1980), 217-250. |
MR 590577 |
Zbl 0474.94007[10] S. R. DE GROOT-L. G. SUTTORP, Foundations of Electrodynamics, North-Holland, 1972.
[11] D. K. FERRY-S. M. GOODNICK, Transport in Nanostructures, Cambridge University Press, 1997.
[12]
R. P. FEYNMAN,
Statistical Mechanics,
W. A. Benjamin Inc.,
1972. |
Zbl 0997.82500[13]
G. B. FOLLAND,
Harmonic Analysis in Phase Space,
Princeton University Press,
1989. |
MR 983366 |
Zbl 0682.43001[14] W. R. FRENSLEY, Boundary conditions for open quantum systems driven far from equilibrium, Rev. Modern Phys., 62 (1990), 745-791.
[15] F. FROMMLET, Time irreversibility in quantum mechanical systems, PhD thesis, Technischen Universität Berlin, 2000.
[16]
P. GÉRARD-
P. A. MARKOWICH-
N. J. MAUSER-
F. POUPAUD,
Homogenization limits and Wigner transforms,
Comm. Pure Appl. Math.,
50 (
1997), 323-379. |
MR 1438151 |
Zbl 0881.35099[17] R. L. LIBOFF, Kinetic Theory: Classical, Quantum and Relativistic Descriptions, Wiley, 1998.
[18]
P. L. LIONS-
T. PAUL,
Sur les mesures de Wigner,
Rev. Matematica Iberoamericana,
9 (
1993), 553-618. |
MR 1251718 |
Zbl 0801.35117[19]
G. W. MACKEY,
The Mathematical Foundations of Quantum Mechanics,
W. A. Benjamin Inc.,
1963. |
Zbl 0114.44002[20]
P. A. MARKOWICH,
On the equivalence of the Schrödinger and the quantum Liouville equations,
Math. Meth. Appl. Sci.,
11 (
1989), 459-469. |
MR 1001097 |
Zbl 0696.47042[21]
P. A. MARKOWICH-
N. J. MAUSER-
F. POUPAUD,
A Wigner function approach to (semi)classical limits: Electrons in a periodic potential,
J. Math. Phys.,
35 (
1994), 1066-1094. |
MR 1262733 |
Zbl 0805.35106[22]
P. A. MARKOWICH-
C. A. RINGHOFER-
C. SCHMEISER,
Semiconductor Equations,
Springer Verlag,
1990. |
MR 1063852 |
Zbl 0765.35001[23]
J. VON NEUMANN,
Mathematical Foundations of Quantum Mechanics,
Princeton University Press,
1955. |
MR 66944 |
Zbl 0064.21503[24]
M. REED-
B. SIMON,
Methods of Modern Mathematical Physics, I - Functional Analysis,
Academic Press,
1972. |
MR 493419 |
Zbl 0242.46001[25]
V. I. TATARSKIĬ,
The Wigner representation of quantum mechanics,
Sov. Phys. Usp.,
26 (
1983), 311-327. |
MR 730012[27]
E. WIGNER,
On the quantum correction for thermodynamic equilibrium,
Phys. Rev.,
40 (
1932), 749-759. |
Jbk 58.0948.07[28] P. ZHAO-H. L. CUI-D. L. WOOLARD-K. L. JENSEN-F. A. BUOT, Equivalent circuit parameters of resonant tunneling diodes extracted from self-consistent Wigner-Poisson simulation, IEEE Transactions on Electron Devices, 48 (2001), 614-626.