Representation Schrodinger

For in the Schrodinger representation w is J dr 2 if is the sharp state associated with a, then... 

Compare this with Eq. (7-59) and observe the order of the factors on the right Equation (7-83) is the equation of motion of the statistical matrix in the Schrodinger representation p is constant, of course, in the Heisenberg representation. 

The reason for file choice of this specific algdira can be justified by the correspondence between the Heisenberg and SchrOdinger representations (129). Applying this algebra to the moment expansion yields the px gators in the resolvent form,... 

The time evolution in the Schrodinger representation may be described in terms of a development operator T(t, t0) by the equation... 

Hartree-Fock wavefunction as a sin e Slater determinant of one-electron functions and then considered the evolution operator, which generates the time variation of these functions. For the present chapter, we believe it is more transparent to deal with the one-electron functions themselves, rather than the c-operators or the evolution operator. Consequently, we work entirely in the Schrodinger representation. 

Equations 2.85 and 2.86 may be considered the Schrodinger representation of the absorption of radiation by quantum systems in terms of spectroscopic transitions between states i) and /). In the Schrodinger picture, the time evolution of a system is described as a change of the state of the system, as implemented here in the form of the time-dependent perturbation theory. The results hardly resemble the classical relationships outlined above, compare Eqs. 2.68 and 2.86, even if we rewrite Eq. 2.86 in terms of an emission profile. Alternatively, one may choose to describe the time evolution in terms of time-dependent observables, the Heisenberg picture . In that case, expressions result that have great similarity with the classical expressions quoted above as we will see next. 

Thus one has formally transferred the time dependence from the probability distribution onto the observed quantity - in analogy with the quantum mechanical transformation from the Schrodinger representation to the Heisenberg representation. Accordingly one may define a time-dependent vector Q(t) by setting... 

Tang et al. [20] have analyzed a reduction procedure using the Schrodinger representation of the dynamics of the n-state system. The molecular wave function of the n-state system can be written as a superposition... 

Time evolution in quantum mechanics is described, in the Schrodinger representation, by the Schrodinger time-dependent equation... 

The average value of any operator O can be written as (O) = (t Os t) in the Schrodinger representation or (O) = (0 Off(t) 0) in the Heisenberg representation, where 0) is some initial state. This initial state is in principle arbitrary, but in many-particle problems it is convenient to take this state as an equilibrium state, consequently without time-dependent perturbation we obtain usual equilibrium Green functions. 

The preceding discussion has been completely based on the Heisenberg representation. The foundations of DFT, on the other hand, are usually formulated within the framework of the Schrodinger picture, so that one might ask in how far this field theoretical procedure can be useful. It is, however, possible to go over to an appropriately chosen Schrodinger representation as long as one does not try to eliminate the quantised photon fields (compare Sections 7d, lOg of Ref. [34]). The Hamiltonian then reads... 

This would be the Schrodinger representation. Alternatively, we could use the equivalent Heisenberg form, as in ... 

Before pursuing the variation of the atomic action integral, it is helpful to first recover the statement of the principle of stationary action in the Schrodinger representation for the total system. If one sets the boundary of the region Cl at infinity in eqn (8.118) to obtain the variation of the total system action integral 2 [ ]> and restricts the variation so that ST vanishes at the time end-points and the end-points themselves are not varied, then only the terms multiplied by the variations in the first integral on the right-hand side remain. The Euler equation obtained by the requirement that this restricted... 

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... 

Now it is the essence of the principle of stationary action that the total change in action is equal to the difference in the values of the generator /( ) evaluated at the two time end-points (eqn (8.79)). In the Schrodinger representation this corresponds to equalling the difference in the... 

Obviously, is simply 4 (Z = 0) and is by definition time independent. Equation (2.62) is a unitary transformation on the wavefunctions and the operators at time t. The original representation in which the wavefunctions are time dependent while the operators are not, is transformed to another representation in which the operators depend on time while the wavefunctions do not. The original fonnulation is referred to as the Schrodinger representation, while the one obtained using (2.62) is called the Heisenberg representation. We sometimes use the subscript S to emphasize the Schrodinger representation nature of a wavefunction or an operator, that is. 

To see the physical significance of these results consider the Hamiltonian that describes the radiation field, a single two-level molecule located at the origin, and the interaction between them, using for the latter the fully quantum analog of Eq. (3.1) in the Schrodinger representation... 

Note that Eqs (9.46) are completely identical to the set of equations (9.6) and (9.7). The problem of a single oscillator coupled linearly to a set of other oscillators that are otherwise independent is found to be isomorphic, in the rotating wave approximation, to the problem of a quantum level coupled to a manifold of other levels. There is one important difference between these problems though. Equations (9.6) and (9.7) were solved for the initial conditions Co(Z = 0) = 1, Cz(Z = 0) = 0, while here a t = 0) and h/(Z = 0) are the Schrodinger representation counterparts of fl(Z) and Still, Eqs (9.46) can be solved by Laplace transform following the route used to solve (9.6) and (9.7). 

Note that an equation similar to (10.19a) that relates the interaction representation of any other operator to the Schrodinger representation... 

Schrodinger representation Heisenberg representation Interaction representation... 

Using this reduction operation we may obtain interesting relationships by taking traces over bath states of the equations of motion (10.15) and (10.21). Consider for example the Liouville equation in the Schrodinger representation, Eq. (10.15). (Note below, an operator A in the interaction representation is denoted Ai while in the Schrodinger representation it carries no label. Labels S and 5 denote system and bath.)... 

In the second equality we have regained the Schrodinger representation of o (r). A final approximation, valid for times longer than the relaxation time of M t), is to take the upper limit of the time integral in (10.149) to infinity, leading to... 

Here and A denote respectively the Heisenberg and Schrodinger representations of the operator. Equation (11.83) implies that. 4 n = dAy /dt and2l are respectively time derivatives of A in the Heisenberg and the Schrodinger representations. Eq. (11.81) is a relationship between these representations in which t is replaced by —ihk. 

The notation is kept for p in order to distinguish between p(t) in the Schrodinger representation and p/ (i) in the interaction representation. Any other operator A will appear as such in the Schrodinger representation, and as A(t) in the interaction representation. 

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