Hamiltonian equation symmetry properties

Podolsky method, Renner-Teller effect, triatomic molecules, Hamiltonian equations, 612—615 Poincare sphere, phase properties, 206 Point group symmetry ... 

The set of variables in Eqs. (11.4.2) and (11.4.3) must include all of the slowly relaxing variables. When the Hamiltonian has certain symmetry properties, the set of Eqs. (11. 3.26) and (11.4.2) can be separated into groups of uncoupled equations. Since, in general, we do not know how to compute the time-correlation functions (F(r), F+(0)), the elements of f should be regarded as quantities to be determined from a comparison between theory and experiment. However, symmetry can be used to relate the off-diagonal elements of V to each other and thereby to reduce the number of independent quantities. 

This last equation shows that if e + ev, then C ft) = 0, thus proving the theorem. This means that in a system such that the Hamiltonian has inversion symmetry, properties of different parity are totally uncorrelated for all time. The initial value of Eq. 

We will discuss the Dirac equation for the one-electron atom in more detail in chapter 7. Here we are only interested in the symmetry properties of the Hamiltonian for such systems. We know that for the corresponding nonrelativistic case, angular momentum and spin are normal constants of motion, represented by operators that commute with the Hamiltonian. In particular... 

Although the Fock operator appears to be a Hamiltonian there is an important difference, namely the fact that F itself is a function of the m.o. s and the set of equations must be solved iteratively. The equations are clearly the same as for atoms, but without the simplifying property of spherical symmetry that allows numerical solution. 

Since the Hamiltonian is symmetric in space coordinates the time-dependent Schrodinger equation prevents a system of identical particles in a symmetric state from passing into an anti-symmetric state. The symmetry character of the eigenfunctions therefore is a property of the particles themselves. Only one eigenfunction corresponds to each eigenfunction and hence there is no exchange degeneracy. 

The time evolution of the probability density is induced by Hamiltonian dynamics so that it has its properties—in particular, the time-reversal symmetry. However, the solutions of Liouville s equation can also break this symmetry as it is the case for Newton s equations. This is the case if each trajectory (43) has a different probability weight than its time reversal (44) and that both are physically distinct (45). 

In this chapter we introduce the SchrSdinger equation this equation is fundamental to all applications of quantum mechanics to chemical problems. For molecules of chemical interest it is an equation which is exceedingly difficult to solve and any possible simplifications due to the symmetry of the system concerned are very welcome. We are able to introduce symmetry, and thereby the results of the previous chapters, by proving one single but immensely valuable fact the transformation operators Om commute with the Hamiltonian operator, Jf. It is by this subtle thread that we can then deduce some of the properties of the solutions of the Schrodinger equation without even solving it. 

The variational method for Schrodinger equation solution was applied for p-type electron trial wave function of the defect at the surface, while it transforms into s-state for the defect in the bulk. The choice was due to the strong anisotropy of the Hamiltonian at the surface so that the s-type spherical symmetry is forbidden from the symmetry considerations. For sufficiently high barrier between the solid and vacuum the hydrogen-Uke p-state wave function with zero value at the surface has been taken as the trial function with two variational parameters. Due to the dependence of the variational parameters on the defect distance from the solid surface, the p-type wave function reveals the correct transition to the s-type function for the defect in the bulk. Such wave function describes successfully the most important physical properties of solids related to the surface influence [41]. 

The first step in the application of symmetry to molecular properties is therefore to recognize and organize all of the symmetry elements that the molecule possesses. A symmetry element is an imaginary point, line, or plane in the molecule about which a symmetry operation is performed. An operator is a symbol that tells you to do something to whatever follows it. Thus, for example, the Hamiltonian operator is the sum of the partial differential equations relating to the kinetic and... 

The most obvious new feature of the Dirac equation as compared with the standard nonrelativistic Schrodinger equation is the explicit appearance of spin through the term a p. Any spin operator trivially commutes with a spin-free Hamiltonian, but the introduction of spin-dependent terms may change this property, as demonstrated in the case of ji/-coupling. A further scrutiny of spin symmetry is therefore a natural first step in discussing the symmetry of the Dirac Hamiltonian. This requires a basis of spin functions on which to carry out the various operations, and a convenient choice is the familiar eigenfunctions of the operator, i, i) and j, — ), also called the a and spin functions. 

Since the related Hamiltonian needs to remain invariant under all the symmetry operations of the molecular symmetry (point) group, the potential energy expansion, see equation (5), may contain only those terms which are totally symmetric under all symmetry operations. Consequently, a simple group theoretical approach, based principally on properties of the permutation groups can be devised, " which yields the number and symmetry classification of anharmonic force constants. The burgeoning number of force constants at higher orders can be appreciated from the entries given in Table 4. 

The principal properties of the energy as a function of k are as follows. Within the Brillouin zone E(k) is a continuous function. It is, of course, a multiple-valued function in the reduced zone scheme. At a Brillouin zone plane the gradient of (k) must be in the plane, except in certain exceptional cases. Finally, the band structure must be symmetric under inversion, k — k. This is usually referred to as time-reversal symmetry, but for simple Hamiltonians without spin-orbit coupling it follows simply from complex conjugation of Schrodinger s equation. 

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