On the use of spacetime transformations in path integration

Exact Wave Functions and Coherent States of a Damped Driven Harmonic Oscillator

“…(2-1)), we assume that there exists a Hermitian operator I(x,p,t) which is explicitly time-dependent and satisfies the invariant condition the solution of which is where E(t) and £ 3 (t) have been defined in Eqs. (2)(3)(4)(5)(6)(7)(8)(9)(10). Again, substituting…”

“…We define a coherent state Ia,t> as the eigenstate of a(t) corresponding to the eigenvalue a, a(t)la,t>-Qlat> , where a is a complex number and a(t) has been defined in Eq. (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12) or (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17). Equation (3-1) can thus be rewritten in the coordinate representation…”

“…Through the normalization procedure aa -a a ea -a* l /2 (3-6) e -e e e, equation (3)(4)(5) can be rewritten as e-xal)2-/2 " a n 2/2 a xl)n! (at)n 0(xt)…”

“…(2-1), (2)(3) and (2)(3)(4) we obtain three first-order differential equations for the coefficients in the propagator (2)(3). The solutions of those differential equations yield the explicit form of the (12,!13) propagator,…”

“…(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21). The eigenstate n (x,t) can be obtained by applying A0(t) n times successively to 0 (x,t).…”