Quantum mechanics three-dimensional rotations

Gas molecules also rotate in three-dimensional space, and quantum mechanics says that rotational energies are also quantized. Therefore, we can also consider a part of the complete partition function of a molecule. Because this is the last kind of partition function we define, we will suggest that there is a complete molecular partition... 

Grinberg, H., Freed, K.F., and Williams, C.J. (1987). Three-dimensional analytical quantum mechanical theory for triatomic photodissociation Role of angle dependent dissociative surfaces on rotational and angular distributions in the rotational infinite order sudden limit, J. Chem. Phys. 86, 5456-5478. 

Three-dimensional quantum mechanical calculations [219] on a semi-empirical surface [220] predict that the product rotational distributions... 

Quantum mechanical and classical calculations have been performed [245] for H + Cl2 on a recently optimised extended LEPS surface [204]. The quantum mechanical results were transformed to three dimensions by the information theoretic procedure and are in good agreement with the distributions determined in the chemiluminescence experiments. However, three-dimensional trajectory calculations on the surface consistently underestimate (FR) at thermal energies and it is concluded that the LEPS surface which was optimised using one-dimensional calculations does not possess the angular dependence of the true three-dimensional surface. This appears to result from the lack of flexibility of the LEPS form. Trajectory studies [196] for H + Cl2 on another LEPS surface find a similar disposal of the enhanced reagent energy as was found for H + F2. The effect of vibrational excitation of the Cl2 on the detailed form of the product vibrational and rotational state distributions was described in Sect. 2.3. 

Many semi-classical and quantum mechanical calculations have been performed on the F + H2 reaction, mainly being restricted to one dimension [520, 521, 602]. The prediction of features due to quantum-mechanical interferences (resonances) dominates many of the calculations. In one semi-classical study [522], it was predicted that the rate coefficient for the reaction F (2P1/2) + H2 is about an order of magnitude smaller than that for F(2P3/2) 4- H2, which lends support to the conclusion [508] that the experimental studies relate solely to the reaction of ground state fluorine atoms. Information theory has been applied to many aspects of the reaction including the rotational energy disposal and branching ratios for F + HD [523, 524] and has been used for transformation of one-dimensional quantum results to three dimensions [150]. Linear surprisal plots occur for F 4- H2(i> = 0), as noted before, but non-linear surprisal plots are noted in calculations for F + H2 (v < 2) [524],... 

A set of numbers representing the states or observables is called a representation. In the geometrical space of three-dimensional vectors r a set of numbers representing r is the set of three coordinates x, y, z in a system of orthogonal axes. We may consider the system of unit vectors x, y, il in the directions of the axes as a basis for the representation of r. A coordinate is the scalar product of r with one of the unit vectors. A different basis is provided by a rotated set of axes. A vector is changed into a new vector by operating with a 3 x 3 matrix. This concept is easily extended to the spaces of quantum mechanics. 

In this paper, we will present a detailed analysis of the way In which resonances may affect the angular distribution of the products of reactive collisions. To do this, we have used an approximate three-dimensional (3D) quantum theory of reactive scattering (the Bending-Corrected Rotating Linear Model, or BCRLM) to generate the detailed scattering Information (S matrices) needed to compute the angular distribution of reaction products. We also employ a variety of tools, notably lifetime matrix analysis, to characterize the Importance of a resonance mechanism to the dynamics of reactions. 

In transition state theory it is assumed that a dynamical bottleneck in the interaction region controls chemical reactivity. Transition state theory relates the rate of a chemical reaction in a microcanonical ensemble to the number of energetically accessible vibrational-rotational levels of the interacting particles at the dynamical bottleneck. In spite of the success of transition state theory, direct evidence for a quantized spectrum of the transition state has been found only recently, and this evidence was found first in accurate quantum mechanical reactive scattering calculations. Quantized transition states have now been identified in accurate three-dimensional quantal calculations for 12 reactive atom-diatom systems. The systems are H + H2, D + H2, O + H2, Cl + H2, H + 02, F + H2, Cl + HC1, I + HI, I 4- DI, He + H2, Ne + H2, and O + HC1. 

As one moves up in dimensions in this generalization, common mathematical laws gradually get lost. Quarternions, for example, do not obey the commutative law (q qb qvq, while octonions (8-vectors) in addition do not obey the associative law q q qc a( b c)). Quarternions are encountered for example in relativistic (4-component) quantum mechanics, and they also form a more natural basis for parameterizing the rotation of a three-dimensional structure, rather than the traditional three Euler angles. The latter involve trigonometric functions that are both computationally expensive to evaluate and display singularities. Furthermore, the quartemion formulation treats all the coordinate axes as equivalent, while the Euler parameterization makes the z-axis a special direction. 

The mass variable is a strictly empirical assumption that only acquires meaning in non-Euclidean space-time on distortion of the Euclidean wave field defined by Eq. (2). The space-like Eq. (5), known as Schrodinger s time-independent equation, is not Lorentz invariant. It is satisfied by a non-local wave function which, in curved space, generates time-like matter-wave packets, characterized in terms of quantized energy and three-dimensional orbital angular momentum. The four-dimensional aspect of rotation, known as spin, is lost in the process and added on by assumption. For macroscopic systems, the wave-mechanical quantum condition ho) = E — V is replaced by Newtonian particle mechanics, in which E = mv +V. This condition, in turn, breaks down as v c. 

---