Schrodinger equation spherical harmonics

In summary, separation of variables has been used to solve the full r,0,( ) Schrodinger equation for one electron moving about a nucleus of charge Z. The 0 and (j) solutions are the spherical harmonics YL,m (0,(1>)- The bound-state radial solutions... 

A problem that arises in connection with the construction of the basis is that of finding what are the allowed values of the quantum numbers of the subalgebra G contained in a given representation of G. For example, what are the allowed values of Mj for a given J in Eq. (2.12). In this particular case, the answer is well known from the solution of the differential (Schrodinger) equation satisfied by the spherical harmonics (see Section 1.4), that is,... 

Fix an eigenvalue E. Suppose we have a solution to the eigenvalue equation for the Schrodinger operator in the given form. I.e, suppose we have a function a 1 and a spherical harmonic function such that... 

The sudden approximation is easy to implement. One solves the onedimensional Schrodinger equation (3.43) for several fixed orientation angles 7, evaluates the 7-dependent amplitudes (3.47), and determines the partial photodissociation amplitudes (3.46) by integration over 7. Because of the spherical harmonic Yjo(x, 0) on the right-hand side of (3.46), the integrand oscillates rapidly as a function of 7 if the rotational... 

This work introduced the concept of a vibronic R-matrix, defined on a hypersurface in the joint coordinate space of electrons and intemuclear coordinates. In considering the vibronic problem, it is assumed that a matrix representation of the Schrodinger equation for N+1 electrons has been partitioned to produce an equivalent set of multichannel one-electron equations coupled by a matrix array of nonlocal optical potential operators [270], In the body-fixed reference frame, partial wave functions in the separate channels have the form p(q xN)YL(0, radial channel orbital function i/(q r) and antisymmetrized in the electronic coordinates. Here 0 is a fixed-nuclei A-electron target state or pseudostate and Y] is a spherical harmonic function. Both and i r are parametric functions of the intemuclear coordinate q. It is assumed that the target states 0 for each value of q diagonalize the A-electron Hamiltonian matrix and are orthonormal. 

The wave functions (6.8) are known as atomic orbitals, for / = 0, 1,2, 3, etc., they are referred to as s, p, d, f, respectively, with the value of n as a prefix, i.e. Is, 2s, 2p, 3s, 3p, 3d, etc., From the explicit forms ofthe wave functions we can calculate both the sizes and shapes of the atomic orbitals, important properties when we come to consider molecule formation and structure. It is instructive to examine the angular parts of the hydrogen atom functions (the spherical harmonics) in a polar plot but noting from (6.9) that these are complex functions, we prefer to describe the angular wave functions by real linear combinations of the complex functions, which are also acceptable solutions of the Schrodinger equation. This procedure may be illustrated by considering the 2p orbitals. From equations (6.8) and (6.9) the complex wave functions are... 

We will now examine the angular dependence of the hydrogen-like orbitals. In terms of spherical harmonics the solution to the Schrodinger equation may be written as... 

The Hamiltonian of N identical atoms of mass m, confined in a trap approximated by a spherically symmetric harmonic oscillator of frequency p, is given by the Schrodinger equation... 

As an example of a propensity rule, transitions between two states of similar n are usually more intense than transitions between states of very different n. While the radial Schrodinger equation does not give rise to genuine selection rules in the same manner as those derived for spherical harmonics, radial matrix elements govern the intensities of allowed... 

The angular Schrodinger equation describes the quantisation of the angular momentum and its projection its solutions are represented by the set of spherical harmonic functions Yt m(, other atoms. The radial equation has solutions in the form of Laguerre polynomials... 

Let us assume that a nucleon moves around freely in the nuclear potratial well, which is spherically symmetric, and that the energy of the nucleon varies between potential and kinetic like a harmonic oscillator, i.e. the potential walls (see Figs. 11.1 and 11.2) are parabolic. For these conditions the solution of the Schrodinger equation yields ... 

Hooke molecules. If one has K pairs of particles (with the intrapair interaction of any kind) in a system of N particles and the other interactions are described by quadratic forms of coordinates satisfying some conditions, the separation of variables is possible and gives K one-particle Hamiltonians and N — K — I Hatnihonians describing spherical harmonic oscillators with properly defined coordinates (known as nomal coordinates see Chapter 7). Together with the one-particle Hamiltonian for the center-of-mass motion, we obtain N Schrodinger equations, each one for one particle" only. 

A Because the wavefunction does not depend upon the variables 6 and 0, the spherical harmonic functions are not involved, and the operator can be omitted from equation (6.10). With Z = I, the Schrodinger equation becomes ... 

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