Klein-Gordon Hamiltonian

For w = 1 or 2 they have the general form of a radial eigenvalue problem arising from some Hamiltonian. In fact, the radial parts of the nonrelativistic hydrogenic Hamiltonian, Klein-Gordon, and second-order iterated Dirac Hamiltonians with 1/r potential can all be expressed in this form for w = 1 and suitable choices of the parameters , rj, x. Similarly, the three-dimensional isotropic harmonic oscillator radial equation has this form for w = 2. 

The Dirac equation is invariant to Lorentz transformations [8], a necessary requirement of a relativistic equation. In the limit of large quantum numbers the Dirac equation reduces to the Klein-Gordon equation [9,10]. The time-independent form of Dirac s Hamiltonian is given by... 

The present account has been published before, see Refs. [7, 82, 83], and references therein. Using our preference for complex symmetric forms we will proceed directly to derive a Klein-Gordon-like equation as follows. Consider the nonpositive square root, cf. the Dirac equation, from the simple ansatz of the Hamiltonian H... 

We will start by setting up a simple 2x2 matrix that (without interaction) displays perfect symmetry between the particle and its antiparticle image. Note that it is well known that the Klein-Gordon and the Dirac equation can be written formally as a standard self-adjoint secular problem (see e.g. [11,12]), based on the simple Hamiltonian matrix (in mass units)... 

The Schrodinger equation with this Hamiltonian is called the square-root Klein-Gordon equation because of its formal similarity to a square-root of the Klein-Gordon equation (Oscar Klein and Walter Gordon in fact had little to do with the square-root equation). Unfortunately, the meaning of the square-root Klein-Gordon equation is obscured by the following points. 

Considering the derivation of DKH Hamiltonians so far, we are facing the problem to express all operators in momemtum space, which is somewhat unpleasant for most molecular quantum chemical calculations which employ atom-centered position-space basis functions of the Gaussian type as explained in section 10.3. The origin of the momentum-space presentation of the DKH method is traced back to the square-root operator in Sq of Eq. (12.54). This square root requires the evaluation of the square root of the momentum operator as already discussed in the context of the Klein-Gordon equation in chapter 5. Such a square-root expression can hardly be evaluated in a position-space formulation with linear momentum operators as differential operators. In a momentum-space formulation, however, the momentum operator takes a... 

We will not re-derive this formulation, see e.g. [12] and references therein, except make a brief outline of the main results. Rather than going through the construction through apposite complex symmetric forms, we will here proceed directly via the observation that the classical-quantum equations of relativity cf. the Klein-Gordon equation, is a quadratic form in the actual observables. Considering the non-positive square root from the simple ansatz of the Hamiltonian below... 

The general Hamiltonian for the Klein-Gordon equation for order parameter iy at nth site is written as ... 

In Chapter 11, Bandyopadhyay et al. have reported on the non-linear Klein-Gordon equation that is based on their discrete Hamiltonian in a typical array of ferroelectric domains. The effect of second quantization, in a particular environment, toward the nonlinearity has been described. This is considered useful for a future study in this new field of investigation of quantum breathers in ferroelectrics. 

The treatment of the kinetic energy term differs in the relativistic and the nonrelativistic Hamiltonians. As Klein and Gordon did, one could start with Einstein s relativistic energy expression and insert the appropriate quantum mechanical operators for the three components of the angular momenta to arrive at a second-order differential equation in spatial and time coordinates. The equation thus arrived at, known as the Klien-Gordon equation, although it treats space and time equivalently and is Lorentz invariant, leads to difficulties in... 

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