Hamiltonian invariant under

For Hamiltonians invariant under rotational and time-reversal transformations the corresponding ensemble of matrices is called the Gaussian orthogonal ensemble (GOE). It was established that GOE describes the statistical fluctuation properties of a quantum system whose classical analog is completely chaotic. 

In this Appendix, Htickel theory calculations are used to demonstrate that the polynomials of Appendix 1 can be applied as basis functions for the irreducible subspaces of a Hamiltonian invariant under icosahedral point symmetry, while extended Htickel theory calculations on cubium cages of cubic point symmetry are used to demonstrate the same result for the kubic harmonics, since single bond-length regular orbits are not possible in all cases. 

The eleetrostatie potential is not invariant under rotations of the eleetron about the x or y axes (those perpendieular to the moleeular axis), so Lx and Ly do not eommute with the Hamiltonian. Therefore, only Lz provides a "good quantum number" in the sense that the operator Lz eommutes with the Hamiltonian. 

We recall, from elementary classical mechanics, that symmetry properties of the Lagrangian (or Hamiltonian) generally imply the existence of conserved quantities. If the Lagrangian is invariant under time displacement, for example, then the energy is conserved similarly, translation invariance implies momentum conservation. More generally, Noether s Theorem states that for each continuous N-dimensional group of transformations that commutes with the dynamics, there exist N conserved quantities. 

This result holds equally well, of course, when R happens to be the operator representing the entropy of an ensemble. Both Tr Wx In Wx and Tr WN In WN are invariant under unitary transformations, and so have no time dependence arising from the Schrodinger equation. This implies a paradox with the second law of thermodynamics in that apparently no increase in entropy can occur in an equilibrium isolated system. This paradox has been resolved by observing that no real laboratory system can in fact be conceived in which the hamiltonian is truly independent of time the uncertainty principle allows virtual fluctuations of the hamiltonian with time at all boundaries that are used to define the configuration and isolate the system, and it is easy to prove that such fluctuations necessarily increase the entropy.30... 

The hamiltonian is, therefore, invariant under U(is). Similarly the -matrix is invariant under U(is)... 

The above formulation can be generalized to a general multidimensional case in the form invariant under any coordinate transformation, as was done before for the ground-state case. We consider the general Hamiltonian given by Eq. (32). The formulation can be carried out in the same way as before. The equation for the additional term w is given by... 

It is also of interest to study the "inverse" problem. If something is known about the symmetry properties of the density or the (first order) density matrix, what can be said about the symmetry properties of the corresponding wave functions In a one electron problem the effective Hamiltonian is constructed either from the density [in density functional theories] or from the full first order density matrix [in Hartree-Fock type theories]. If the density or density matrix is invariant under all the operations of a space CToup, the effective one electron Hamiltonian commutes with all those elements. Consequently the eigenfunctions of the Hamiltonian transform under these operations according to the irreducible representations of the space group. We have a scheme which is selfconsistent with respect to symmetty. 

The classical dynamics of the FPC is governed by the Hamiltonian (1) for F = 0 and is regular as evident from the Poincare surface of section in Fig. 1(a) (D. Wintgen et.al., 1992 P. Schlagheck, 1992), where position and momentum of the outer electron are represented by a point each time when the inner electron collides with the nucleus. Due to the homogeneity of the Hamiltonian (1), the dynamics remain invariant under scaling transformations (P. Schlagheck et.al., 2003 J. Madronero, 2004)... 

Thus the Casimir operator of SO(3) is the familiar square of the angular momentum (a constant of the motion when the Hamiltonian is invariant under rotation). One can show that SO(3) has only one Casimir operator, and it is thus an algebra of rank one. Multiplication of C by a constant a, which obviously satisfies (2.7), does not count as an independent Casimir operator, nor do powers of C (i.e., C2,...) count. Casimir operators can be constructed directly from the algebra. This construction has been done for the large majority of algebras used in physics. 

The traditional treatment of molecules relies upon a molecular Hamiltonian that is invariant under inversion of all particle coordinates through the center of mass. For such a molecular Hamiltonian, the energy levels possess a well-defined parity. Time-dependent states conserve their parity in time provided that the parity is well defined initially. Such states cannot be chiral. Nevertheless, chiral states can be defined as time-dependent states that change so slowly, owing to tunneling processes, that they are stationary on the time scale of normal chemical events. [22] The discovery of parity violation in weak nuclear interactions drastically changes this simple picture, [14, 23-28] For a recent review, see Bouchiat and Bouchiat. [29]... 

In turn, taking into account that in case of He2Br2 the Hamiltonian is also invariant under Ri R2 inversion, then a well-defined parity,pi2, basis set is built up as follows ... 

From our point of view the most significant thing about the Hamiltonian operators H9l and f/nno is that they both commute with the operators Og, we say that i/el and f/nuc are invariant under all symmetry transformation operators of the point group of the molecular framework... 

The electronic Hamiltonian (4.124) is invariant under inversion of electronic coordinates with respect to the molecule-fixed axes if and only if the molecule is homonuclear. However, (4.124) is invariant under inversion of electronic and nuclear coordinates with respect to space-fixed axes, for both homonuclear and heteronuclear diatomic molecules. 

An important case is where F in (9.183) is replaced by the Hamiltonian operator H. The Hamiltonian is invariant under all symmetry operations, so that the function belongs to the irreducible representation Tp. For... 

Now if the Hamiltonian is invariant under inversion, that is, if the potential is symmetric,... 

Here x) stands for the positional coordinates of all the particles in the system, E is the energy of the system, and 77 is the Hamiltonian operator. Since a symmetry operator merely rearranges indistinguishable particles so as to leave the system in an indistinguishable configuration, the Hamiltonian is invariant under any spatial symmetry operator R. Let tpi denote a set of eigenfunctions of H so that... 

Starting from n)> we notice that it is neither an eigenstate of a Heisenberg Hamiltonian nor an eigenstate of the total spin operator S2. For instance, the Neel state is not invariant under the action of the nearest-neighbor XY spin terms of the Heisenberg Hamiltonian, which may produce a spin-flip on two nearest-neighbor sites at a time with respect to n). These XY excitation operators from the vacuum 4>n > can be written as... 

The Coulombic Hamiltonian (4.1) is invariant under translations and rotations, and it is hence convenient to separate the motion of the "center of mass" % and to study the new Hamiltonian ... 

Hamiltonian dynamics show that classical mechanics is invariant to ( t) and (t). In a macroscopic description of dissipative systems, we use collective variables of temperature, pressure, concentration, and convection velocity to define an instantaneous state. The evolution equations of the collective variables are not invariant under time reversal... 

Since the Laplacian V2 is invariant under orthogonal transformations of the coordinate system [i.e. under the 3D rotation-inversion group 0/(3)], the symmetry of the Hamiltonian is essentially governed by the symmetry of the potential function V. Thus, if V refers to an electron in a hydrogen atom H would be invariant under the group 0/(3) if it refers to an electron in a crystal, H would be invariant under the symmetry transformations of the space group of the crystal. 

For a Hamiltonian which is invariant under the space inversion operator P it has already been shown that... 

The behaviour of a composite quantum system under space inversion may be affected if its constituent particles have intrinsic parity. Consider a composite bound system with a Hamiltonian which is invariant under space inversion. Let ma be the mass of constituent particle a with internal wave function ipa, -e. 

The Hamiltonian must be invariant under translations along the lattice by any integral multiple (n) of the lattice spacing a,... 

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