Gordon equation

In going from the Schrodinger equation to the Klein-Gordon equation, we obtain the neeessary symmetry between spaee and time by having seeond-order derivatives throughout. It is usually written in a form that brings out its relativistic invarianee by using what is ealled/our-vector notation. We define a four-vector X to have components... 

Don t confuse this with my earlier use of x for a space-spin variable the notation is common usage in both applications.) The Klein-Gordon equation is therefore... 

It turns out that the Klein-Gordon equation cannot describe electron spin in the limit of small kinetic energy, it can be shown to reduce to the familiar Schrodinger equation. 

The Schrodinger equation and the Klein-Gordon equation both involve second order partial derivatives, and to recover such an equation from the Dirac equation we can operate on equation 18.12 with the operator... 

A little operator algebra shows that this gives exactly the Klein-Gordon equation if the y s satisfy the relationship... 

Spin 0 Particles.—The covariant wave equation describing a spin 0, mass m particle is the Klein-Gordon equation ... 

Although the Klein-Gordon equation is of second order in the time derivative, for a positive energy particle the knowledge of at some given time is sufficient to determine the subsequent evolution of the particle since 8ldt is then given by Eq. (9-85). Alternatively Eq. (9-85) can be adopted as the equation of motion for a free spin zero particle of mass m. We shall do so here. 

This scalar product is conserved in time if and 2 obey the Klein-Gordon equation. It furthermore possesses all the properties usually required of a scalar product, namely... 

Since in (9-150), k2 = m2, i > x) also satisfies the Klein-Gordon equation... 

Klein-Gordon amplitude relation to Schrodinger amplitude, 500 Klein-Gordon equation for destruction operator, 507... 

This procedure leads to the Klein-Gordon equation... 

The Hamiltonian, Eq. (25), can be minimized exactly. The ground state satisfies the time-independent sine Gordon equation ... 

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 spin-independent part of these equations is identical to the Klein-Gordon equation. If the singularity of V is not stronger than 1/r then,... 

The magnetic fluxes F and G obey the Klein-Gordon equation for a massless particle in the vacuum ... 

One more important property of the self-dual Yang-Mills equations is that they are equivalent to the compatibility conditions of some overdetermined system of linear partial differential equations [11,12]. In other words, the selfdual Yang-Mills equations admit the Lax representation and, in this sense, are integrable. For this very reason it is possible to reduce Eq. (2) to the widely studied solitonic equations, such as the Euler-Amold, Burgers, and Devy-Stuardson equations [13,14] and Liouville and sine-Gordon equations [15] by use of the symmetry reduction method. 

The Aharonov-Bohm effect requires topological consideration [1], (i.e., a structured vacuum), and there exist conservation laws of topological origin, the simplest one is given by the sine-Gordon equation, which also appears in the discussion of 0(3) electrodynamics by Evans and Crowell [5]. 

With the Gordon equations (275) one can easily extend Langley s analysis to the Scanlan criterion for active strands bounded by chemical crosslinks. With the same generality as that of Eq.(7.20) ... 

Using Eqs. (115) and (221), this Lagrangian gives the Klein-Gordon equations... 

This additional effective mass is introduced from spontaneous symmetry breaking of the vacuum. The two Klein-Gordon equations therefore take on the form... 

Conventional single particle quantization is based on the quantum ansatz (399) applied to the Einstein equation (415) to produce the Klein-Gordon equation... 

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