Hamiltonian, single particle

The corresponding fiinctions i-, Xj etc. then define what are known as the normal coordinates of vibration, and the Hamiltonian can be written in tenns of these in precisely the fonn given by equation (AT 1.69). witli the caveat that each tenn refers not to the coordinates of a single particle, but rather to independent coordinates that involve the collective motion of many particles. An additional distinction is that treatment of the vibrational problem does not involve the complications of antisymmetry associated with identical fennions and the Pauli exclusion prmciple. Products of the nonnal coordinate fiinctions neveitlieless describe all vibrational states of the molecule (both ground and excited) in very much the same way that the product states of single-electron fiinctions describe the electronic states, although it must be emphasized that one model is based on independent motion and the other on collective motion, which are qualitatively very different. Neither model faithfully represents reality, but each serves as an extremely usefiil conceptual model and a basis for more accurate calculations. 

Schrddinger s equations are usually written in a more succinct manner by invoking the Hamiltonian operator H, so for example the time-dependent equation for a single particle... 

The continuum model with the Hamiltonian equal to the sum of Eq. (3.10) and Eq. (3.12), describing the interaction of electrons close to the Fermi surface with the optical phonons, is called the Takayama-Lin-Liu-Maki (TLM) model [5, 6], The Hamiltonian of the continuum model retains the important symmetries of the discrete Hamiltonian Eq. (3.2). In particular, the spectrum of the single-particle states of the TLM model is a symmetric function of energy. 

Assume the impurity particle C to be harmonically bound to a main system of oscillators numbered by i = 0, 1, 2,. .. through a single particle with the number i = 0. Fig. A1.5 shows particles labeled by i at cubic lattice sites. The complete Hamiltonian function of the system under discussion is represented as follows ... 

In an equivalent classical equation, the variable Ik cancels to give the Hamiltonian function, which for a single particle of mass m,... 

The above observation suggests an intriguing relationship between a bulk property of infinite nuclear matter and a surface property of finite systems. Here we want to point out that this correlation can be understood naturally in terms of the Landau-Migdal approach. To this end we consider a simple mean-field model (see, e.g., ref.[16]) with the Hamiltonian consisting of the single-particle mean field part Hq and the residual particle-hole interaction Hph-... 

To understand this more clearly, consider a simpler model where A consists of single excitations, only single-particle operators are retained in the effective Hamiltonian, and we choose the reference function iho to be a single determinant. Then, from a cumulant decomposition of the two-particle terms, the effective Hamiltonian becomes... 

Finally, we note that if we retain two-particle operators in the effective Hamiltonian, but restrict A to single-particle form, we recover exactly the orbital rotation formalism of the multiconfigurational self-consistent field. Indeed, this is the way in which we obtain the CASSCF wavefunctions used in this work. 

According to standard NMR theory, the spin-lattice relaxation is proportional to the spectral density of the relevant spin Hamiltonian fluctuations at the transition frequencies coi. The spectral density is given by the Fourier transform of the auto-correlation fimction of the single particle fluctuations. For an exponentially decaying auto-correlation function with auto-correlation time Tc, the well-known formula for the spectral density reads as ... 

Free single particle partition sum. The one-particle sum over states of the ideal gas is easily evaluated if the free-particle Hamiltonian (with V (R) = 0) is used,... 

Here ( )is the single particle energy and v(k,k ) the interaction. The symmetry of the superconducting state can be derived from that of the Hamiltonian. In general, the symmetry group Q is the direct product... 

Using the Dirac notations a) = ipa( ) and assuming that ipa( ) are or-thonormal functions (a (3) = 5ap we can write the single-particle matrix (tight-binding ) Hamiltonian in the Hilbert space formed by 4>a ( )... 

The solution of single-particle quantum problems, formulated with the help of a matrix Hamiltonian, is possible along the usual line of finding the wave-functions on a lattice, solving the Schrodinger equation (6). The other method, namely matrix Green functions, considered in this section, was found to be more convenient for transport calculations, especially when interactions are included. 

Now we see clear the problem while the new dot Hamiltonian (154) is very simple and exactly solvable, the new tunneling Hamiltonian (162) is complicated. Moreover, instead of one linear electron-vibron interaction term, the exponent in (162) produces all powers of vibronic operators. Actually, we simply remove the complexity from one place to the other. This approach works well, if the tunneling can be considered as a perturbation, we consider it in the next section. In the general case the problem is quite difficult, but in the single-particle approximation it can be solved exactly [98-101]. 

Apart from the demands of the Pauli principle, the motion of electrons described by the wavefunction P° attached to the Hamiltonian H° is independent. This situation is called the independent particle or single-particle picture. Examples of single-particle wavefunctions are the hydrogenic functions (pfr,ms) introduced above, and also wavefunctions from a Hartree-Fock (HF) approach (see Section 7.3). HF wavefunctions follow from a self-consistent procedure, i.e., they are derived from an ab initio calculation without any adjustable parameters. Therefore, they represent the best wavefunctions within the independent particle model. As mentioned above, the description of the Z-electron system by independent particle functions then leads to the shell model. However, if the Coulomb interaction between the electrons is taken more accurately into account (not by a mean-field approach), this simplified picture changes and the electrons are subject to a correlated motion which is not described by the shell model. This correlated motion will be explained for the simplest correlated system, the ground state of helium. 

The Schrodinger equation also leads to a continuity equation that can be interpreted as a local conservation of probability density (here for a single particle with the Hamiltonian H = —ft2/(2m)V2 + V) ... 

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