Time-dependent invariants for dirac equation and Newton–Wigner position operator

Abstract

For Dirac equation, operator-invariants containing explicit time-dependence in parallel to known time-dependent invariants of nonrelativistic Schrödinger equation are introduced and discussed. As an example, a free Dirac particle is considered and new invariants are constructed for it. The integral of motion, which is initial Newton-Wigner position operator, is obtained explicitly for a free Dirac particle. For such particle with kick modeled by delta-function of time, the time-depending integral, which has physical meaning of initial momentum, is found.

Time-dependent integrals of motion for a precessing magnetic dipole

“…The new integrals of motion determine the solution of the Heisenberg equation of motion for the precessing spin. A relativistic free Dirac particle as shown in [12] has extra time-dependent integrals of motion. It is worth considering new integrals of motion for a relativistic free Dirac particle in a magnetic field, which is the relativistic generalization of the result of our study.…”

Invariant Quantum States of Quadratic Hamiltonians

“…Moreover, such kinds of specific linear combinations can be constructed for arbitrary time-dependent quadratic Hamiltonians, as was shown by Lewis and Riesenfeld [20]. Their method of time-dependent quantum operators and integrals of motion was generalized and further developed by Malkin and Man'ko with collaborators [6,[21][22][23][24][25][26][27][28][29][30][31] and other authors .…”

Lyapunov exponent in quantum mechanics. A phase-space approach

“…(45). For the kicked case in (46) the function ε (t) is obtained by establishing a matrix recurrence relation.…”

“…Let also τ = 1 and q 0 = p 0 = 0 in the initial condition (45). Then, one obtains the following recursion for the derivatives of G at µ = ν = 0.…”

“…For the calculation of the Lyapunov exponent, the function G (k, •, •, 0) is the one given in Eq. (45).…”