Heisenberg formalism

From the Heisenberg formalism momentum should be represented by an operator that does not commute with x, i.e. [x,p] = ih. The momentum operator can therefore not also be multiplicative, but can be a differential operator. The representation p <----ih-J gives the correct form when operating... 

Heisenberg formalism. The Heisenberg view leads to an expression equivalent to the Schrodinger formalism that stresses the time evolution of quantum systems it has a clear correspondence with classical mechanics it is most conveniently expressed in terms of the dipole autocorrelation function (Gordon 1968). 

The theory of spectral moments and line shape is based on time-dependent perturbation theory, Eqs. 2.85 and 2.86, applied to ensembles of atoms, or equivalently on the Heisenberg formalism involving dipole autocorrelation functions, Eq. 2.90. 

A German physicist, Werner Heisenberg, formalized this idea with his uncertainly principle, which says that we cannot know the exact location or motion of an electron. Although we cannot describe the path of an electron s motion, we can speak of the probability of finding it in a given location at any time. In Figure 8.7, the darker areas represent a greater probability that the electron is present in this location. 

Before consideration is given to the experimental data, it is of interest to consider the connection between the Weiss field constants used by N6el and the Heisenberg exchange integrals. In the non-collinear models, the Heisenberg formalism is generally used. 

As we just saw in the de Broglie relation, the velocity of an electron is related to its wave nature. The position of an electron, however, is related to its particle nature. (Particles have well-defined position, but waves do not.) Consequently, our inability to observe the electron simultaneously as both a particle and a wave means that we cannot simultaneously measure its position and its velocity. Werner Heisenberg formalized this idea with the equation ... 

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... 

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. 

Apart from the operational, wave or action-based pictures of quantum mechanics provided by Heisenberg, Schrodinger, or Feynman, respectively, there is an additional, fully trajectory-based picture Bohmian mechanics [20,23]. Within this picture, the standard quantum formalism is understood in terms of trajectories defined... 

Gordon (1968) pointed out that the spectral profile, Eq. 5.3, presents the Bohr-Schrodinger form of spectroscopy, as transitions between stationary Bohr states, represented by time-dependent Schrodinger states, i), /). The Heisenberg form of quantum mechanics gives an equivalent expression that emphasizes the time evolution of the observables rather than that of the states. This formalism leads quite naturally to time-dependent... 

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... 

G.P. Thomson independently reached the same conclusions in 1927. The hypothesis that matter exhibits both corpuscular and wavelike characteristics served as a stimulus lor the formal development of quantum mechanics by E. Schrttdinger. M. Bom. W. Heisenberg, and others. Following the discovery, which eventually led to a Nobel Prize to Davisson and Thomson, electron diffraction was immediately utilized as a tool for the study of the structure of matter. 

Simply put, the Heisenberg uncertainty principle states that there are limits on knowing both where something is and how fast it is moving. Formally, we can write... 

It is interesting to note that the vibrational model of the nucleus predicts that each nucleus will be continuously undergoing zero-point motion in all of its modes. This zero-point motion of a quantum mechanical harmonic oscillator is a formal consequence of the Heisenberg uncertainty principle and can also be seen in the fact that the lowest energy state, N = 0, has the finite energy of h to/2. 

The plan of the paper is the following. In section 2 we introduce the elements at the basis of the Heisenberg group representation representation theory [12-14,20] that are needed to understand the alternative group-theoretical formulation of quantum mechanics. In section 3 the Heisenberg representation of quantum mechanics (with the time dependence transferred from the vectors of the Hilbert space to the operators) is used to introduce quantum observables and quantum Lie brackets within the group-theoretical formalism described in the previous section. In section 4 classical mechanics is obtained by taking the formal limit h —> 0 of quantum observables and brackets to obtain their classical counterparts. Section 5 is devoted to the derivation of... 

We described so far the mathematical apparutus which will be used to obtain a group-theoretical formulation of quantum mechanics (section 3), by means of the Heisenberg group, and to obtain the connection between quantum and classical mechanics (section 4) within the group-theoretical formal-... 

The group-theoretical formalism we have introduced so far is particularly suited to formulate quantum mechanics in the Heisenberg representation, where the time dependence is shifted from the wavefunctions to the operators. As we shall show in the next section, the formalism allows to show in a straightforward way that the Poisson brackets are obtained as a formal limit of the commutator when h —> 0. 

This section is devoted to the derivation of classical mechanics (observables and equations of motion) as a formal limit of quantum mechanics, within the group-theoretical formulation discussed in the previous sections. To this purpose, we need to recall here some properties of the Heisenberg group. 

This result concludes the presentation of the Heisenberg group approach as the powerful tool that allows to derive classical mechanics as a formal limit of quantum mechanics, for h —> 0. The most important ingredients that have been introduced to obtain this result are the Fourier-like representation of observables and equations of motion and the definition of the antiderivative operator. These elements will be used in section 5 to derive a similiar procedure for a mixed quantum-classical mechanics. An ansatz on the quantum-classical equations of motion will be necessary, but the subsequent application of Heisenberg group formalism will be a straightforward generalization of what has been done so far. 

The generalization of the Heisenberg group formalism to the group D" is not consistent. The comparison between the approaches proposed in section 3 and in section 5 can be summarized in the following scheme ... 

The derivation of a consistent mixed quantum-classical dynamics discussed in this paper was first proposed in Ref. [15] and commented and clarified in Ref. [1], This derivation is based on a group-theoretical formulation of quantum and classical mechanics, which introduces a very elegant and formally rigorous mathematical apparatus and allows to directly obtain classical mechanics as the limit for h —> 0 of quantum mechanics, in the Heisenberg representation of quantum dynamics. 

However, the Heisenberg group formalism is a very useful tool to represent quantum and classical dynamical quantities, such as observables and equations of motion, only when a prescription on the generator of the time evolution exists. The comparison with the fully quantum or fully classical dynamics allows us to deduce only the formal properties that the mixed quantum-classical brackets have to satisfy in order to generate a consistent evolution, but does... 

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