The Transformation Operator Evolution Equation

W.-S. Liu, X.-P Li

The time-dependent formulation of the Bogoliubov transformation (BT) for single-mode boson operators is derived in matrix multiplication. The combination of the BT differential equation with the time-evolution equation for time-dependent quantum oscillators yields general formulas to get the time-evolution operator of the systems. Examples are given.

Michele Cini

Xian-Geng Zhao

Donald H. Kobe, Kuo‐Ho Yang

The time-evolution operator is in general gauge dependent. Its gauge transformation property follows from the gauge transformation of the wave function and ensures gauge-invariant matrix elements. The same transformation property is shown here to follow from the formal solution of the Schr\"odinger equation for the time-evolution operator, which is a time-ordered exponential of the time integral of the Hamiltonian. The gauge transformation property of the time-evolution operator in the interaction picture is also obtained. The perturbation expansion of the time-evolution operator in one gauge can be transformed to give the perturbation expansion for the time-evolution operator in another gauge. The A\ensuremath{\cdot}p versus E\ensuremath{\cdot}r controversy in the electric dipole approximation is resolved by specifying the correct initial and final states.

V Aldaya, F Cossío, J Guerrero +1 more

Using a quantum version of the Arnold transformation of classical mechanics, all quantum dynamical systems whose classical equations of motion are non-homogeneous linear second-order ordinary differential equations, including systems with friction linear in velocity, can be related to the quantum free-particle dynamical system. This transformation provides a basic (Heisenberg–Weyl) algebra of quantum operators, along with well-defined Hermitian operators which can be chosen as evolution-like observables and complete the entire Schrödinger algebra. It also proves to be very helpful in performing certain computations quickly, to obtain, for example, wavefunctions and closed analytic expressions for time-evolution operators.

Gebhard Grübl, C. Leubner