Quantum mechanics alternative representations

Physicist P. A. M. Dirac suggested an inspired notation for the Hilbert space of quantum mechanics [essentially, the Euclidean space of (9.20a, b) for / — oo, which introduces some subtleties not required for the finite-dimensional thermodynamic geometry]. Dirac s notation applies equally well to matrix equations [such as (9.7)-(9.19)] and to differential equations [such as Schrodinger s equation] that relate operators (mathematical objects that change functions or vectors of the space) and wavefunctions in quantum theory. Dirac s notation shows explicitly that the disparate-looking matrix mechanical vs. wave mechanical representations of quantum theory are actually equivalent, by exhibiting them in unified symbols that are free of the extraneous details of a particular mathematical representation. Dirac s notation can also help us to recognize such commonality in alternative mathematical representations of equilibrium thermodynamics. 

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... 

Approximate techniques for alternative representations of quantum mechanics... 

Similar to quantum mechanics, which can be formulated in terms of different quantities in addition to the traditional wave function formulation, in quantum chemistry a number of alternative tools are developed for this purpose, which may be useful in the context of the present book. We have already described different approximate models of representing the electronic structure using (many-electronic) wave functions. The coordinate and second quantization representations were employed to get this. However, the entire amount of information contained in the many-electron wave function taken in whatever representation is enormously large. In fact it is mostly excessive for the purpose of describing the properties of any molecular system due to the specific structure of the operators to be averaged to obtain physically relevant information and for the symmetry properties of the wave functions the expectation values have to be calculated over. Thus some reduced descriptions are possible, which will be presented here for reference. 

Wavelets are a set of basis functions that are alternatives to the complex exponential functions of Fourier transforms which appear naturally in the momentum-space representation of quantum mechanics. Pure Fourier transforms suffer from the infinite scale applicable to sine and cosine functions. A desirable transform would allow for localization (within the bounds of the Heisenberg Uncertainty Principle). A common way to localize is to left-multiply the complex exponential function with a translatable Gaussian window , in order to obtain a better transform. However, it is not suitable when <1) varies rapidly. Therefore, an even better way is to multiply with a normalized translatable and dilatable window, v /yj,(x) = a vl/([x - b]/a), called the analysing function, where b is related to position and 1/a is related to the complex momentum. vl/(x) is the continuous wavelet mother function. The transform itself is now... 

The methods most frequently used to predict the properties of mixtures for over 100 years have inevitably undergone only minor additions and corrections to, it is claimed, improve the representation of experimental data for specific categories of substances. It is, however, possible that completely different alternatives to these traditional approaches are required, particularly for a method to be both predictive and applicable over a wide range of fluids and conditions. Such methods might arise from future research and methods based on statistical mechanics and quantum-mechanical calculations" are ultimately sought rather than empiricism. 

The profound physical meaning [42], the capacity to gain an accurate and deep understanding of the phenomenology and the philosophical implications [43,44] of the representation of quantum mechanics proposed by Madelung [29], Landau [32], and London [33] (MLL) cannot be overstressed. Bohm showed that the hydrodynamical quantum mechanics is deterministic and provides an interpretation of physical reality alternative to that of the Copenhagen School [34, 35]. 

Before we can start with the discussion of time-dependent perturbation theory in the form of response theory, we need to introduce an alternative formulation of quantum mechanics, called the interaction or Dirac representation. In general, several representations of the wavefunctions or state vectors and of the operators of quantum mechanics are equivalent, i.e. valid, as long as the expectation values of operators ( 0 I d I o) or inner products of the wavefunctions ( o n) are always the same. Measurable quantities and thus the physics are contained in the expectation values or inner products, whereas operators and wavefunctions are mathematical constructs used in a particular formulation of the theory. One example of this was already discussed in Section 2.9 on gauge transformations of the vector and scalar potentials. In the present section we want to look at a transformation that is related to the time dependence of the wavefunctions and operators. 

As in any molecular level simulation one of the first decisions to make is what inter- and intramolecular force field to use. We have basically two choices. Firstly, we can set about bringing together as much information as possible from experiment and quantum mechanical calculations to develop good force fields and in this way to aim for quantitatively accurate modeling. With this approach there is usually little alternative, but to employ a fully atomic representation with, for instance, the GROMOS force field. ° It must be remembered however that no force field will be completely accurate and all of them have limitations. 

Classical shape representations of molecules are based on assumed analogies between quantum mechanical molecules and macroscopic, classical objects. Since most of the mass of molecules is concentrated in the nuclei, it has appeared natural to place emphasis on nuclear arrangements, and the chemically more relevant shapes of electron densities have become the focus of molecular shapes analysis only recently. It is important to distinguish stereochemistry and molecular shape analysis. The term stereochemistry is commonly used for the 3D pattern of formal bonds, whereas molecular shape often refers to the shape of the fuzzy electron density cloud, or to simpler representations of large-scale molecular features, such as an a-helix or a -sheet of a protein. Several alternative shape representations are also used, such as fused sphere van der Waals (VDW) surfaces, Connolly surfaces, solvent accessibility surfaces, or molecular electrostatic potential (MEP) surfaces. 

The conceptual framework for the - semiclassical simulation of ultrafast spectroscopic observables is provided by the Wigner representation of quantum mechanics [2, 3]. Specifically, for the ultrafast pump-probe spectroscopy using classical trajectories, methods based on the semiclassical limit of the Liouville-von Neumann equation for the time evolution of the vibronic density matrix have been developed [4-8]. Our approach [4,6-8] is related to the Liouville space theory of nonlinear spectroscopy developed by Mukamel et al. [9]. It is characterized by the ability to approximately describe quantum phenomena such as optical transitions by averaging over the ensemble of classical trajectories. Moreover, quantum corrections for the nuclear dynamics can be introduced in a systematic manner, e.g. in the framework of the entangled trajectory method [10,11]. Alternatively, these effects can be also accounted for in the framework of the multiple spawning method [12]. In general, trajectory-based methods require drastically less computational effort than full quantum mechanical calculations and provide physical insight in ultrafast processes. Additionally, they can be combined directly with quantum chemistry methods for the electronic structure calculations. 

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