Heisenberg operators

We next introduce the configuration space Heisenberg operator... 

This operator is a SchrSdinger picture operator. The corresponding Heisenberg operator is... 

Equation (10-22) will be a (formal) solution of (10-1) if the Heisenberg operator in(x) satisfies the free field equation... 

In a manner analogous to the above we can define the out Heisenberg operators rout(x) by the formal solution... 

To facilitate the derivation we shall assume that we are in the Heisenberg picture and dealing with a time-independent hamiltonian, i.e., H(t) — 27(0) = 27, in which case Heisenberg operators at different times are related by the equation... 

Summarizing, we have noted that the Heisenberg operators Q+(t) obey field free equations i.e., that their time derivatives are given by the commutator of the operator with Ha+(t) = Ho+(0) and that this operator H0+(t) is equal to H(t) = H(0). The eigenstates of H0+ are, therefore, just the eigenstates of H. We can, therefore, identify the states Tn>+ with the previously defined >ln and the operator [Pg.602]

The operator V+(t) transforms Heisenberg operators into in operators... 

The Dirac Equation in a Central Field.—The previous sections have indicated that at times it is useful to have an explicit representation of the matrix element <0 (a ) n> where tfi(x) is the Heisenberg operator satisfying Eq. (10-1). Of particular interest is the case when the external field A (x) is time-independent, Ae = Ae(x), so that the states > can be assumed to be eigenstates of the then... 

Precisely the result (11-91) would have been obtained if, instead of working in the Coulomb gauge, we had adopted the Lorentz gauge. The theory is then described by Heisenberg operators satisfying the following equations of motion2... 

Assume that there exists a unitary operator U(it) which maps the Heisenberg operator Q(t) at time t into the operator (—<). Assume further that this mapping has the property of leaving the hamiltonian invariant, i.e., that U(it)SU(it)" 1 = H. Consider then the equation satisfied by the transformed operator... 

Conditional probability, 267 density function, 152 Condon, E. U., 404 Configuration space amplitude, 501 Heisenberg operator, 507 operators, 507, 514, 543 Conservation laws for light particles (leptons), 539 for heavy particles (baryons), 539 Continuous memoryless channels, 239 Contraction symbol for two time-labelled operators, 608 Control of flow, 265 Converse to coding theorem, 215 Convex downward function, 210 Convex upward function, 209 Cook, L. F 724... 

Heisenberg operators Q+ t), 600 Heisenberg picture in quantum theory, 665... 

The stochastic problem is to describe properly the time evolution of the Heisenberg operator d(t) averaged over all the realizations of collisional process in the interval (0,t). The averaging, performed in the impact theory, results in the phenomenological kinetic equation [170, 158]... 

In the Heisenberg picture the operators themselves depend explicitly on the time and the time evolution of the system is determined by a differential equation for the operators. The time-dependent Heisenberg operator AH(t) is obtained from the corresponding Schrodinger operator As by the unitary transformation... 

As a simple model, we confine our attention just to a single mode Ha(t) of the Hamiltonian (23). Note that neither any instantaneous eigenstate of Ha(t) is an exact quantum state nor e-/3ii W is a density operator. To calculate the thermal expectation value of an operator A, one needs either the Heisenberg operator Ah or the density operator pa(t) = UapaUa Now we use the time-dependent creation and annihilation operators (24), invariant operators, to construct the Fock space. 

Exercise. For any observable A one defines the time-dependent or Heisenberg operator A(t) by... 

I use temporarily roman n, m to include zero.) Show that the Heisenberg operators an(t), a (t) obey the same rules provided they are taken at the same f. The commutation relations of two of these operators taken at different times are not simple they involve the solution of the equations of motion. 

Exercise. The Schrodinger-Langevin equation (5.3) implies that the average of any Heisenberg operator (1.24) obeys... 

Here q and p are Heisenberg operators, y is the usual damping coefficient, and (t) is a random force, which is also an operator. Not only does one have to characterize the stochastic behavior of g(t), but also its commutation relations, in such a way that the canonical commutation relation [q(t), p(t)] = i is preserved at all times and the fluctuation-dissipation theorem is obeyed. ) Moreover it appears impossible to maintain the delta correlation in time in view of the fact that quantum theory necessarily cuts off the high frequencies. ) We conclude that no quantum Langevin equation can be obtained without invoking explicitly the equation of motion of the bath that causes the fluctuations.1 That is the reason why this type of equation has so much less practical use than its classical counterpart. 

In accordance with this definition the Heisenberg operators ca(t), cjj(f), etc. are equal to the time-independent Schrodinger operators at some initial time to. ca(to) = ca, etc. Density matrix of the system is assumed to be equilibrium at this time p(to) = peq. Usually we can take to = 0 for simplicity, but if we want to use to 0 the transformation to Heisenberg operators should be written as... 

The free-particle Hamiltonian has equivalent form if one uses Schrodinger or Heisenberg operators... 

We consider also the other method, based on the equations of motion for operators. From Liouville - von Neumann equation we find (all c-operators are Heisenberg operators in the formula below, (t) is omitted for shortness)... 

Substituting these expressions into (201) we obtain again (212). Note also that if we take to / 0, then Heisenberg operators for free fermions are... 

Now let us consider again free fermions. Heisenberg operators for free fermions... 

Finally, we need the expression of a Heisenberg operator, defined by the full Hamiltonian H = Ho + V(t), through an operator in the interaction representation. The transformation, corresponding to (258), is given by... 

We start from the general definition of a Green function as the average of two Heisenberg operators A(t) and B(t), denoted as... 

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