Geodesic
Eigenfunctions of quadratic Hamiltonians in the Wigner representation
Teoretičeskaâ i matematičeskaâ fizika, Tome 60 (1984) no. 3, pp. 413-422 
È. A. Akhundova; V. V. Dodonov; V. I. Man'ko. Eigenfunctions of quadratic Hamiltonians in the Wigner representation. Teoretičeskaâ i matematičeskaâ fizika, Tome 60 (1984) no. 3, pp. 413-422. http://geodesic.mathdoc.fr/item/TMF_1984_60_3_a7/
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     title = {Eigenfunctions of quadratic {Hamiltonians} in the {Wigner} representation},
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Voir la notice de l'article provenant de la source Math-Net.Ru

Exact solutions of the Schrödinger equation in the Wigner representation are obtained for an arbitrary time-dependent $N$-dimensional quadratic Hamiltonian. It is shown that a complete system of solutions can always be chosen in the form of products of $N$ Laguerre polynomials having arguments that are quadratic integrals of the motion of the corresponding classical problem. The generating function found for the transition probabilities between the Foek states is a multidimensional generalization of Husimi's well-known expression for an oscillator with variable frequency. The motion of a charged particle in a uniform time-dependent electromagnetic field is considered in detail as an example.

[1] Wigner E., Phys. Rev., 40:5 (1932), 749–759 | DOI | MR

[2] Tatarskii V. I., UFN, 139:4 (1983), 587–619 | DOI | MR

[3] Chernikov N. A., ZhETF, 53:3(9) (1967), 1006–1017 | Zbl

[4] Malkin I. A., Manko V. I., Trifonov D. A., Kratkie soobscheniya po fizike FIAN, 1971, no. 5, 20–26 | MR

[5] Dodonov V. V., Manko V. I., Tr. FIAN, 152, 1983, 145–193 | MR

[6] Takabayasi T., Progr. Theor. Phys., 11:4–5 (1954), 341–373 | DOI | MR | Zbl

[7] Gorshenkov V. N., Izv. VUZov, ser. fiz., 1974, no. 5, 123–124

[8] Akhundova E. A., Dodonov V. V., Man'ko V. I., Physica, 115A:2 (1982), 215–231 | DOI | MR

[9] Lewis H. R., Riesenfeld W. B., Jr., J. Math. Phys., 10:8 (1969), 1458–1473 | DOI | MR | Zbl

[10] Malkin I. A., Manko V. I., Dinamicheskie simmetrii i kogerentnye sostoyaniya kvantovykh sistem, Nauka, M., 1979, 320 pp. | MR

[11] Dodonov V. V., Malkin I. A., Manko V. I., Kratkie soobscheniya po fizike FIAN, 1976, no. 1, 3–6

[12] Lee H. W., Scully M. O., J. Chem. Phys., 73:5 (1980), 2238–2242 | DOI | MR

[13] O'Connell R. F., Wigner E. P., Phys. Lett., 83A:4 (1981), 145–148 | DOI | MR

[14] Husimi K., Progr. Theor. Phys., 9:4 (1953), 381–402 | DOI | MR | Zbl

[15] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii, t. 2, Nauka, M., 1965, 190–191 | MR