Hamiltonian commutator

The electronic Hamiltonian commutes with both the square of the angular momentum operator r and its z-component and so the three operators have simultaneous eigenfunctions. Solution of the electronic Schrddinger problem gives the well-known hydrogenic atomic orbitals... 

It is important to distinguish between mmetiy properties of wave functions on one hand and those of density matrices and densities on the other. The symmetry properties of wave functions are derived from those of the Hamiltonian. The "normal" situation is that the Hamiltonian commutes with a set of symmetry operations which form a group. The eigenfunctions of that Hamiltonian must then transform according to the irreducible representations of the group. Approximate wave functions with the same symmetry properties can be constructed, and they make it possible to simplify the calculations. 

In the case of a perfect crystal the Hamiltonian commutes with the elements of a certain space group and the wave functions therefore transform under the space group operations accorc g to the irreducible representations of the space group. Primarily this means that the wave functions are Bloch functions labeled by a wave vector k in the first Brillouin zone. Under pure translations they transform as follows... 

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. 

In the absence of an electric field, the non-BO Hamiltonian commutes with the square of the angular momentum operator, [H, P] = 0, and so the eigenfunctions of the Hamiltonian also have to be eigenfunctions of J. This condition is met, for example, by functions such as... 

Because V depends only on r, one finds that this Hamiltonian commutes with the orbital angular-momentum operators L2 and Lz. Hence the... 

Since Pc commutes with P2 and with P[Pg.110]

Those systems for which spin is conserved are those systems which are well described by a spin-free Hamiltonian. The spin-free Hamiltonian commutes with the symmetric group 5 F of permutations on electronic spatial indices. It follows that irreducible representations of this symmetric group are good quantum numbers. Certain irreducible representations of S F will be found to correspond to spin quantum numbers. 

Since the unperturbed Hamiltonian commutes with the unperturbed projection operator we obtain for terms linear in A ... 

Of course, the time derivative involved in this equation is zero, since the density operator and the corresponding Hamiltonian commute ... 

In the case of periodic boundary conditions the chain Hamiltonian commutes with the operator that displaces all electrons by one unit cell cyclically. Therefore, its eigenfunctions must be characterized by the hole quasi-... 

Since the Heisenberg Hamiltonian commutes with the total-spin operators, the 22iV-dimensional spin space can be separated in subspaces with... 

This Hamiltonian commutes with rj2 but does not commute with S2. Therefore, the eigenfunctions of the Hamiltonian (102) can be described by quantum numbers T] and r]z. For the cyclic model the states with three different values of 77 have zero energy [17] [as it was for the model (98)]. They include one state with rj — 0 (101), all states with rj = N/2 ... 

The second line of this equation follows from (7.102) above. We note that the awkward sin 0 factors in (7.89) have now disappeared. As Hougen points out, the eigenfunctions of the true Hamiltonian involve one less variable and so one less quantum number than the eigenfunctions of the artificial Hamiltonian and consequently the two operators cannot be completely isomorphic. However, a simple restriction on the extra quantum number in the artificial problem identifies that part of the full artifical Hamiltonian which is isomorphic with the true operator. Since the isomorphic Hamiltonian commutes with (Jz — W-), the two operators have a set of simultaneous eigenfunctions. Equation (7.102) states that only those eigenfunctions of the isomorphic Hamiltonian which have an eigenvalue of zero for (Jz - Wz) are eigenfunctions of the true Hamiltonian. 

These relations turn out to be quite useftd. For example, when spins are weakly coupled, the Zeeman and scalar coupling portions of the Hamiltonian commute, so that these effects may be treated separately and in any order. 

We have seen that the 180° 13C pulse reordered the 13C populations, but transfer of proton polarization to the 13C system occurred only when the and 13C 90° pulses were applied at time 4. We chose to use a matrix that represented the concurrent application of both pulses. However, these are independent, their Hamiltonians commute, and we could have considered instead the application of the two pulses separately—in either order. Here we examine the effect of applying the H pulse. 

The electrons of a linear molecule move in an electrostatic field which has cylindrical symmetry. Therefore, the electronic Hamiltonian commutes with the projection of the sum of the orbital and spin angular momenta on the symmetry axis... 

Since the XXZ Hamiltonian commutes with the z-component of total spin... 

H and have been formulated in the commutator form, which ensures the correct behaviour under charge conjugation (Kalian 1958), although we will not dwell on this point in the following. The Hamiltonian commutes with the charge operator... 

We can take advantage of the fact that the sum magnetization and the difference magnetization of the S-spin subspace of the Hamiltonian commute in the absence of r.f. irradiation and write our Hamiltonian as... 

Since the Hamiltonian commutes with Pa(0), no oscillations of the coherences are observed and the solution of the equation for Pa(t) is simply given by an exponential decay... 

The Hamiltonian commutes with the angular momentum operator as well as that for the square of the angul momentum I-, The wavefunctions above are also eigenfunctions of these operators, with eigenvalues miti ( )and f + 11 (t ). It should be emphasized that the total angular momentum is L = + )h,... 

At this point the reader may feel that we have done little in the way of explaining molecular symmetry. All we have done is to state basic results, normally treated in introductory courses on quantum mechanics, coimected with the fact that it is possible to find a complete set of simultaneous eigenfunctions for two or more commuting operators. However, as we shall see in section A 1.4.3,2. the fact that the molecular Hamiltonian //commutes with and is intimately connected to the fact that //commutes with (or, equivalently, is invariant to) any rotation of the molecule about a space-fixed axis passing through the centre of mass of the molecule. As stated above, an operation that leaves the Hamiltonian invariant is a symmetry operation of the Hamiltonian. The infinite set of all possible rotations of the ... 

The translational linear momentum is conserved for an isolated molecule in field free space and, as we see below, this is closely related to the fact that the molecular Hamiltonian commutes with all... 

Finally, we consider the complete molecular Hamiltonian which contains not only terms depending on the electron spin, but also terms depending on the nuclear spin /(see chapter 7 of [1]). This Hamiltonian commutes with the components of Pgiven in (equation A1.4.H. The diagonalization of the matrix representation of the complete molecular Hamiltonian proceeds as described in section A 1.4. 11. The theory of rotational symmetry is an extensive subject and we have only scratched the surface here. A relatively new book, which is concerned with molecules, is by Zare [6] (see [7] for the solutions to all the problems in [6] and a list of the errors). This book describes, for example, the method for obtaining the functions from... 

Many molecular hamiltonians commute with the total spin angular momentum operator, a fact that leads to the consideration of transformation properties of electron field operators under rotations in spin space. Basis functions, natural for such studies, are... 

A model hamiltonian should have the structure of the full hamiltonian, but could in principle have terms consisting of higher order products of annihilation and creation operators. Here we limit considerations to such operators that contain a one-electron part and an electron-electron interaction part. The number of independent matrix elements can be considerably reduced by symmetry considerations and by requiring compatibility with other operator representatives. It is clear that the form of the spectral density requires that the hamiltonian commutes with the total orbital angular momentum and with various spin operators. These are given in the limited basis as... 

As in the subshell hamiltonian in Eq. (4.150), there should be an interaction parameter V(7Z/5) corresponding to the electron pair operators. The possibility exists to introduce parameters that couple pair states of equal L and S but different 7 s. We require that the two-particle part of the hamiltonian commutes with the parity operator... 

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