Volume 102A, number 7 PHYSICS LETTERS 28 may 1984 WIGNER FUNCTIONS OF A PARTICLE IN A TIME-DEPENDENT UNIFORM FIELD V.V. DODONOV, V.I. MAN'KO and O.V. SHAKHMISTOVA P.N. Lebedev Physical lnstitute, Moscow, USSR Received 16 January 1984 Wigner functions and exact transition amplitudes between energy eigenstates for a particle moving in a time-dependent uniform field are calculated. Although the problem of the motion of a quantum particle in a uniform potential field is considered in text-books on quantum mechanics, none the less there exist yet unexplored aspects of this problem. The aim of our article is to obtain the solutions in the Wigner- Weyl representation and transition amplitudes between energy eigenstates in the case of a time-dependent force. Earlier these items were not studied, as far as we know. We consider the system with the hamiltonian /~(t) =/~ 2/2rn -- f(t)2 . (1) The solutions of the Schr6dinger equation with this hamiltonian in the special case off(t) = F = const are well known both in the momentum and coordi- nate representation [ 1 ] : ~E(P, t) = (2nh ]FI) -1/2 X exp[(i/hF)(Ep - p3/6m) - iEt/h] , (2) ~E(x, t) = [(2m)l/3/Trl/2[y{a/6h 2/31 X Ai(-(x + E/F)(2mF/h2)l/3)e-iEt/l~, (3) Ai(z) is the Airy function. The corresponding solu- tion in the Wigner-Weyl representation (its proper- ties were discussed in detail, e.g., in refs. [2,3])could be obtained from eqs. (2), (3) with the aid of the Fourier transformation: co 1 , + ~u)eipU/hdu WE(q, P) = f t~E(q -- ~U)~E( q 1 * lu)eiqu/hdu = f ~oE( p + ~u)~OE( p -- . (4) We prefer, however, a more direct approach. It is well known that the Schr6dinger equation for the wave function ihaf,/at =.0q, is equivalent to the Liouville equation for the Wigner function, provided that the hamiltonian is a quadratic form of coordinates and momenta. Thus in the case under study one has: aw/at + (p/m)aW/aq + f(t)aw/ap = o . (5) Consequently, every solution of the Schr6dinger equa- tion in the Wigner-Weyl representation does not de- pend on q, p, t separately, but only on two indepen- dent variables - the integrals of motion qo(q, P, t) and PO(q, P, t): qo(q, P, t) = q - pt/rn + 7(t), t •(t) = f rf<T)dr/m, (6) 0 po(q, p, t) = p - ~ (t), t 8(t) = ff(r)dr. 0 (7) 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) 295