The Klein-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... 

This procedure leads to the Klein-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 ... 

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

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

The probability densities of the Klein-Gordon equation [46] in an 0(3) internal basis contains terms such as... 

Non-plane-wave solutions of the Klein-Gordon equation using unconventional basic functions and coupling ansatz ... 

Figure 4. Ionization yield as a function of laser intensity for a radiation pulse with a linear tum-on of 1007 //. The field frequency is an = 10 a.u.. These yields are computed at t = 11007 //. The line represents the results for the time-dapendent Schrodinger equation treatment while the dots are the results of the Klein-Gordon equation treatment,...

The simplest derivation , given in many books, e.g. in chapter 4, was in fact similar to that used by Schrodinger to obtain an equation which falls short of the relativistic Schrodinger equation only by the absence of spin, a concept which had not yet arisen [1], This first quantum-mechanical wave equation is now known as the Klein-Gordon equation, and applies to particles without spin. 

For the evolution of the scalar field, the most convenient is to use directly the perturbed version of the Klein-Gordon equation (which is in fact equivalent to the Euler equations), + d V = 0. This gives... 

Let us study the stability of the solution wq = Const found above. To do so, we perturb the Klein-Gordon equation around this solution. Here, we need only to consider the case 0 = 4>WQ=Const + where 6[Pg.142]

The Klein-Gordon equation for the free particle (

[Pg.151]

This important equation is known as the Klein-Gordon equation, and was proposed by various authors [6, 7, 8, 9] at much the same time. It is, however, an inconvenient equation to use, primarily because it involves a second-order differential operator with respect to time. Dirac therefore sought an equation linear in the momentum operator, whose solutions were also solutions of the Klein-Gordon equation. Dirac also required an equation which could more easily be generalised to take account of electromagnetic fields. The wave equation proposed by Dirac was [10]... 

Noting that the Klein-Gordon equation can be written in the form... 

The Klein-Gordon equation (Schrodinger s relativistic equation) has been used in the description of a relativistic particle with spin zero (see, e.g., Schiff, 1968) and can be treated using the so(2,1) algebraic methods (Barut, 1971 Cizek and Paldus, 1977, and references therein). It is obtained from the energy-momentum relationship... 

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