Schrodingers Picture

We shall take the Heisenberg and Schrodinger pictures to coincide at time 7 = 0. We next correlate with the complete hamiltonian 27(0) (i.e., the total hamiltonian in the Heisenberg picture at time 1 = 0, which by the above convention is also the total hamiltonian in the Schrodinger picture) an unperturbed hamiltonian 27o(0). We shall write... 

The inclusion of (nonrelativistic) property operators, in combination with relativistic approximation schemes, bears some complications known as the picture-change error (PCE) [67,190,191] as it completely neglects the unitary transformation of that property operator from the original Dirac to the Schrodinger picture. Such PCEs are especially large for properties where the inner (core) part of the valence orbital is probed, for example, nuclear electric field gradients (EEG), which are an important... 

In the Schrodinger picture operators in the case of a closed system do not depend explicitly on the time, but the state vector is time dependent. However, the expectation values are generally functions of the time. The commutator of the Hamiltonian operator H= —(h/2iri)(d/dt) and another operator A, is defined by... 

In the functional Schrodinger-picture, wave functionals carry all information of quantum states in real time (K. Freese et.al., 1985 1988). The wave functionals satisfy the functional Schrodinger equation... 

In the previous chapter we discussed the usual realization of many-body quantum mechanics in terms of differential operators (Schrodinger picture). As in the case of the two-body problem, it is possible to formulate many-body quantum mechanics in terms of algebraic operators. This is done by introducing, for each coordinate, r2,... and momentum p, p2,..., boson creation and annihilation operators, b ia, bia. The index i runs over the number of relevant degrees of freedom, while the index a runs from 1 to n + 1, where n is the number of space dimensions (see note 3 of Chapter 2). The boson operators satisfy the usual commutation relations, which are for i j,... 

The formulation of the preceding section is very general. We are interested, however, in rotations and vibrations of polyatomic molecules. We therefore discuss now specific applications of the algebraic method beginning with the simple case of one-dimensional coupled oscillators, presented in Section 3.3 in the Schrodinger picture. In the algebraic theory, as mentioned, one associates to each coordinate, x, and related momentum, px = — iti d/dx, an algebra. For... 

In the previous sections the correspondence between the Schrodinger picture and the algebraic picture was briefly reviewed for some special cases (dynamical symmetries). In general the situation is much more complex, and one needs more elaborate methods to construct the potential functions. These methods are particularly important in the case of coupled problems. This leads to the general question of what is the geometric interpretation of algebraic models. 

Following the common approach in relativistic field theory, which aims at a manifestly covariant representation of the dynamics inherent in the field operators, so far all quantities have been introduced in the Heisenberg picture. To develop the framework of relativistic DFT, however, it is common practice to transform to the Schrodinger picture, so that the relativistic theory can be formulated in close analogy to its nonrelativistic limit. As usual we choose the two pictures to coincide at = 0. Once the field operators in the Schroodinger-picture have been identified via j/5 (x) = tj/(x, = 0), etc, the Hamiltonians He,s, Hy s and are immediately obtained in terms of the Schrodinger-picture field operators. 

Eq. (6.26) is the TDSE in the Schrodinger picture. In general, it proves more convenient to discuss the time evolution of the driven system in a rotating frame, such as the frame rotating with the laser carrier frequency q- After transformation into the carrier frequency picture and application of the rotating wave approximation (RWA), the TDSE takes the form [92]... 

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. 

The quantity p — QA is called a generalised momentum. It appears in both classical electromagnetism and quantum mechanics. In the Schrodinger picture, we make the substitution... 

We recall some basic results of quantum dynamics [3], First, the state of the system and the time evolution can be expressed in a generalized (Dirac) notation, which is often very convenient. The state at time t is specified by x(t)) with the representations x(-Rjf) = (R x t)) and x P,t) = (P x(t)) in coordinate and momentum space, respectively. Probability is a concept that is inherent in quantum mechanics. (R x(t)) 2 is the probability density in coordinate space, and (-P x(f) 2 is H e same quantity in momentum space. The time evolution (in the Schrodinger picture) can be expressed as... 

The procedure, known as second quantization, developed as an essential first step in the formulation of quantum statistical mechanics, which, as in the Boltzmann version, is based on the interaction between particles. In the Schrodinger picture the only particle-like structures are associated with waves in 3N-dimensional configuration space. In the Heisenberg picture particles appear by assumption. Recall, that in order to substantiate the reality of photons, it was necessary to quantize the electromagnetic field as an infinite number of harmonic oscillators. By the same device, quantization of the scalar r/>-field, defined in configuration space, produces an equivalent description of an infinite number of particles in 3-dimensional space [35, 36]. The assumed symmetry of the sub-space in three dimensions decides whether these particles are bosons or fermions. The crucial point is that, with their number indeterminate, the particles cannot be considered individuals [37], but rather as intuitively understandable 3-dimensional waves - (Born) -with a continuous density of energy and momentum - (Heisenberg). 

A typical problem of interest at Los Alamos is the solution of the infrared multiple photon excitation dynamics of sulfur hexafluoride. This very problem has been quite popular in the literature in the past few years. (7) The solution of this problem is modeled by a molecular Hamiltonian which explicitly treats the asymmetric stretch ladder of the molecule coupled implicitly to the other molecular degrees of freedom. (See Fig. 12.) We consider the the first seven vibrational states of the mode of SF (6v ) the octahedral symmetry of the SF molecule makes these vibrational levels degenerate, and coupling between vibrational and rotational motion splits these degeneracies slightly. Furthermore, there is a rotational manifold of states associated with each vibrational level. Even to describe the zeroth-order level states of this molecule is itself a fairly complicated problem. Now if we were to include collisions in our model of multiple photon excitation of SF, e wou d have to solve a matrix Bloch equation with a minimum of 84 x 84 elements. Clearly such a problem is beyond our current abilities, so in fact we neglect collisional effects in order to stay with a Schrodinger picture of the excitation dynamics. 

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... 

Here, Equation (50) was used. Now, we see that these equations are equivalent to those of the macroscopic Onsagerian equations (Equation (21) or (22)). In the Schrodinger picture, the density operator is time dependent, but the observables of the oscillator are time independent. We define this density operator as... 

According to this requirement, we could give the actual form of the equation of motion for the density operator in the Schrodinger picture. We will see that this equation corresponds to the Liouville-von Neumann equation in the case of dissipative processes. From Equations (113) and (119), it follows that the density operator in the Schrodinger picture could be written by a Hermitian operator in the form... 

From the general evolution equation (Equation (34)) of the Hermitian operator, the equation of motion of the density operator in Schrodinger picture could be derived as follows ... 

From the above-presented theory, we can conclude that an ensemble from a pure state always proceeds to a mixed state a consequence of irreversibility. Thus, it is impossible to describe the evolution of the pure state of a damped oscillator in the Schrodinger picture. Consequently, it is impossible to construct a linear Schrodinger equation in which the position and the momentum operator are time independent. 

In the case of s linearly damped oscillator, the transformation of the Heisenberg picture into the Schrodinger picture by the method applied in classical quantum theory is impossible because the operator has a time-dependent part due to the dissipative process. Thus, a new way must be found to construct the wave equation of the oscillator. Kostin introduced a supplementary dissipation potential into his wave equation and constructed this dissipation potential by an assumption that the energy eigenvalues of the oscillator decay exponentially over time [39]. In Kostin s version of the wave equation, the operators are time independent, but the dissipation potential is nonlinear with respect to the wave function. In our theory, it is assumed that the abstract wave equation of the linearly damped oscillator has the form... 

Note that the invariance of quantum observables under unitary transformations has enabled us to represent quantum time evolutions either as an evolution of the wavefunction with the operator fixed, or as an evolution of the operator with constant wavefunctions. Equation (2.1) describes the time evolution of wavefunctions in the Schrodinger picture. In the Heisenberg picture the wavefunctions do not evolve in time. Instead we have a time evolution equation for the Heisenberg operators ... 

Equations (10.126) and (10.127) represent the quantum master equation in the interaction representation. We now transform it to the Schrodinger picture using... 

On transforming Eq. (55) into the Schrodinger picture, the master equation of the system takes the form... 

With the parameters (101), the master equation of two atoms in a broadband squeezed vacuum, written in the Schrodinger picture, reads as... 

Now let us turn to the photon distribution function (PDF)/(n) = (n pm(t) n), where n) is the multimode Fock state, n = (n, m,...), and pm(t) is the time-dependent density matrix of the mth field mode in the Schrodinger picture. Note that all the calculations in the preceding sections were performed in the... 

At this point we may abandon the Heisenberg picture and proceed to the Schrodinger representation. Of course, both representations are equivalent, as soon as the field problem has been reduced to studying a finite-dimensional quantum system. However, the most of numerous investigations of the time-dependent quantum oscillator, since Husimi s paper [285], were performed in the Schrodinger picture. So it is natural to use the known results. According to several studies [279,285,286], all the characteristics of the quantum oscillator are determined completely by the complex solution of the classical oscillator equation of motion... 

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