Schrodinger equation magnetic field

The important outcome from this transformation is that now the non-adiabatic coupling term t is incorporated in the Schrodinger equation in the same way as a vector potential due to an external magnetic field. In other words, X behaves like a vector potential and therefore is expected to fulfill an equation of the kind [111a]... 

Schrodinger (IV) has tentatively advanced a form of the wave equation in which magnetic fields are considered. 

Just like any spectroscopic event EPR is a quantum-mechanical phenomenon, therefore its description requires formalisms from quantum mechanics. The energy levels of a static molecular system (e.g., a metalloprotein in a static magnetic field) are described by the time-independent Schrodinger wave equation,... 

In what follows, we present in this short review, the basic formalism of TDDFT of many-electron systems (1) for periodic TD scalar potentials, and also (2) for arbitrary TD electric and magnetic fields in a generalized manner. Practical schemes within the framework of quantum hydrodynamical approach as well as the orbital-based TD single-particle Schrodinger-like equations are presented. Also discussed is the linear response formalism within the framework of TDDFT along with a few miscellaneous aspects. 

This non-relativistic equation in terms of four-component spinors has been studied in detail by Levy-Leblond [44,45], who has shown that it results automatically from a study of the irreducible representations of the Gahlei group and that it gives a correct description of spin. It is easy to see that in the absence of an external magnetic field, equation (63) is equivalent to the Schrodinger equation in the sense that after elimination of the small component ... 

In the absence of a magnetic field, the Schrodinger equation of a particle moving in an external potential U(r) is... 

In cases where the classical energy, and hence the quantum Hamiltonian, do not contain terms that are explicitly time dependent (e.g., interactions with time varying external electric or magnetic fields would add to the above classical energy expression time dependent terms discussed later in this text), the separations of variables techniques can be used to reduce the Schrodinger equation to a time-independent equation. 

The quantum theory of molecular collisions in external fields described in this chapter is based on the solutions of the time-independent Schrodinger equation. The scattering formalism considered here can be used to calculate the collision properties of molecules in the presence of static electric or magnetic fields as well as in nonresonant AC fields. In the latter case, the time-dependent problem can be reduced to the time-independent one by means of the Floquet theory, discussed in the previous section. We will consider elastic or inelastic but chemically nonreac-tive collisions of molecules in an external field. The extension of the formalism to reactive scattering problems for molecules in external fields has been described in Ref. [12]. 

In a static magnetic field, the minimal prescription shows that the time-independent Schrodinger-Pauli equation of a fermion in a classical field is... 

The occurrence of the nonlinear Schrodinger equation is then a fairly generic result. For the A potential we have the magnetic field that is easily seen to be... 

Now we write the same Fourier of expansion for the electric field and write everything according to the magnetic field intensity H = B, and we find with the case that (e/H)Aq co the amplitude fixed to the wavelength as is the case for some solitons, for Gaussian packets, we arrive at the same cubic Schrodinger equation ... 

In Section 1.4, we discussed the history and foundations of MO theory by comparison with VB theory. One of the important principles mentioned was the orthogonality of molecular wave functions. For a given system, we can write down the Hamiltonian H as the sum of several terms, one for each of the interactions which will determine the energy E of the system the kinetic energies of the electrons, the electron-nucleus attraction, the electron-electron and nucleus-nucleus repulsion, plus sundry terms like spin-orbit coupling and, where appropriate, other perturbations such as an applied external magnetic or electric field. We now seek a set of wave functions P, W2,... which satisfy the Schrodinger equation ... 

The quantum number, m , originating from the 0(6) and Schrodinger wave equation, indicates how the orbital angular momentum is oriented relative to some fixed direction, particularly in a magnetic field. Thus, ml roughly characterizes the directions of maximum extension of the electron... 

The spinor that describes the spherical rotation satisfies Schrodinger s equation and specifies two orientations of the spin, colloquially known as up and down (j) and ( [), distinguished by the allowed values of the magnetic spin quantum number, ms = . The two-way splitting of a beam of silver ions in a Stern-Gerlach experiment is explained by the interaction of spin angular momentum with the magnetic field. 

In the absence of an electric or magnetic field all the Ylni( functions with t 0 arc (21 + l)-fold degenerate, which means that there are (21 + 1) functions, each having one of the (2t + I) possible values of m, with the same energy. It is a property of degenerate functions that linear combinations of them are also solutions of the Schrodinger equation. For example, just as //2/, ] and //2/, ] are solutions, so are... 

In the triplet model the spin polarization is with respect to the internal molecular states, TjJ>, Ty>, and T > of the triplet and evolves with time according to the time-dependent Schrodinger equation into a spin polarization with respect to the electron spin Zeeman levels Ti>, Tq>, and T i> in an external magnetic field Bq. Consider a simple case of axially symmetric zero-field splitting (i.e., D y 0 and E = 0 D and E are the usual zero-field parameters). Tx>, [Ty>, and TZ> are the eigenstates of the zero-field interaction Hzfs, where Z is the major principal axis. The initial polarization arising from the population differences among these states can be expressed as... 

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