Coordinate operator, eigenfunction

The eigenfunctions of a quantum mechanical operator depend on the coordinates upon which the operator acts these functions are called wavefunetions... 

A. If the two operators act on different coordinates (or, more generally, on different sets of coordinates), then they obviously commute. Moreover, in this case, it is straightforward to find the complete set of eigenfunctions of both operators one simply forms a product of any eigenfunction (say fk) of R and any eigenfunction (say gn) of S. The function fk gn is an eigenfunction of both R and S ... 

Our next objective is to find the analytical forms for these simultaneous eigenfunctions. For that purpose, it is more convenient to express the operators Lx, Ly, Zz, and P in spherical polar coordinates r, 6, q> rather than in cartesian coordinates x, y, z. The relationships between r, 6, q> and x, y, z are shown in Figure 5.1. The transformation equations are... 

The N-electron Hamiltonian Ti. corresponding to the expectation value in Eq. (1) does not contain any spin coordinates hence both operators and commute with Ti.. Consequently, the eigenfunctions of the Hamiltonian are also eigenfunctions of these two spin operators. In particular, according to Lowdin s expressions [82], we have... 

The px, Py and pz orbitals are eigenfunctions of the if angular momentum operator with eigenvalues equal to L(L+1 )4f = 2 hf. Since if andLz commute and act on the same (angle) coordinates, they possess a complete set of simultaneous eigenfunctions. 

Here t = r, a] stands for all spatial and spin coordinates. If two operators A and B have the same eigenfunctions, then this means that they commute with each other... 

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... 

The Fourier method is best suited to cartesian coordinates because the expansion functions QtkR/LR etlr Lr are just the eigenfunctions of the kinetic energy operator. For problems including the rotational degree of freedom other propagation methods have been developed (Mowrey, Sun, and Kouri 1989 Le Quere and Leforestier 1990 Dateo, Engel, Almeida, and Metiu 1991 Dateo and Metiu 1991). 

Similarly, for the operator, we introduce unit operators on both sides of the Heaviside step function, but using the coordinate eigenfunctions in this case ... 

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