The Heisenberg equations of motion

An important advantage of formulating harmonic oscillators problems in terms of raising and lowering operators is that these operators evolve very simply in time. Using the Heisenberg equations of motion (2.66), the expression (2.155) and the commutation relations for a and leads to 

Consequently, the Heisenberg representations of the position and momentum operators are 

As an example for the use of this formulation let us calculate the (in-principle time-dependent) variance, (Ax(Z)2), defined by Eq. (2.149) for a Harmonic oscillator in its ground state. Using the expression for position operator in the Heisenberg representation from Eq. (2.166) and the fact that (0 Ax(Z)2 0) = 0 x(Z) 0 for an oscillator centered at the origin, this can be written in the from 

Problems involving harmonic oscillators that are shifted in their equilibrium positions relative to some preset origin are ubiquitous in simple models of quantum dynamical processes. We consider a few examples in this section. 

1 Harmonic oscillator under an additional constant force 

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Using the Heisenberg equation of motion, (AS,2,40). the connnutator in the last expression may be replaced by the time-derivative operator... 

As long as the system can be described by the rate constant - this rules out the localization as well as the coherent tunneling case - it can with a reasonable accuracy be considered in the imaginary-time framework. For this reason we rely on the Im F approach in the main part of this section. In a separate subsection the TLS real-time dynamics is analyzed, however on a simpler but less rigorous basis of the Heisenberg equations of motion. A systematic and exhaustive discussion of this problem may be found in the review [Leggett et al. 1987]. 

Solving now the Heisenberg equations of motion for the a operators perturbatively in the same way as in the weak-coupling case, one arrives (at = 0) at the celebrated non-interacting blip approximation [Dekker 1987b Aslangul et al. 1985]... 

The u and v representations are sometimes distinguished as the Schrodinger and the Heisenberg representation. For stationary operators P, then, the Heisenberg equation of motion is... 

We shall again postulate commutation rules which have the property that the equations of motion of the matter field and of the electromagnetic field are consequences of the Heisenberg equation of motion ... 

Since the original operator Q(t) obeyed the Heisenberg equation of motion... 

In this volume dedicated to Yngve Ohm we feel it is particularly appropriate to extend his ideas and merge them with the powerful practical and conceptual tools of Density Functional Theory (6). We extend the formalism used in the TDVP to mixed states and consider the states to be labeled by the densities of electronic space and spin coordinates. (In the treatment presented here we do not explicitly consider the nuclei but consider them to be fixed. Elsewhere we shall show that it is indeed straightforward to extend our treatment in the same way as Ohm et al. and obtain equations that avoid the Bom-Oppenheimer Approximation.) In this article we obtain a formulation of exact equations for the evolution of electronic space-spin densities, which are equivalent to the Heisenberg equation of motion for the electtons in the system. Using the observation that densities can be expressed as quadratic expansions of functions, we also obtain exact equations for Aese one-particle functions. 

The equations of motion for NGF are obtained from the Heisenberg equation of motion for operators... 

In the following, we indicate the time derivative of a hermitian operator B with the symbol B. In the Heisenberg representation of quantum mechanics, it obeys the Heisenberg equation of motion... 

Dirac s development of TDHF theory invokes the Heisenberg equation of motion for the density operator as a basic postulate,... 

Equations (8.126) and (8.127) are identical in form to the corresponding results obtained for the variation of the Lagrange function operator in eqns (8.97) and (8.98). They are variational statements of the Heisenberg equation of motion for the observable F in the Schrodinger representation. When T describes a stationary state... 

To re-express this result in the form given in eqn (8.143) for the field-free case, we need the Heisenberg equation of motion for F(Q, t) for a system in the presence of a magnetic field. This is obtained in the same manner as eqn (8.141) in the field-free case, using eqn (8.211) for the Schrodinger equation of motion to give... 

The Heisenberg equation of motion of the linearly damped oscillator... 

A similar calculation may be carried out for the time evolution of an observable. Starting with the Heisenberg equation of motion for a dynamical variable B,... 

Equation (2.66) is referred to as the Heisenberg equation of motion. Note that it should be solved as an initial value problem, given that H(f = 0) = A. In fact, Eq. (2.62b) can be regarded as the formal solution of the Heisenberg equation (2.66) in the same way that the expression (Z) = = 0) is a formal solution... 

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