Equation Klein-Gordon

According to the special theory of relativity, all physical laws are postulated to be invariant in any inertial frame of reference. Furthermore, the equations of motion must be invariant under a Lorentz transformation. 

Another requirement in satisfying the special theory of relativity is to have the spatial and temporal variables being treated on the same footing. The differential operators in Lorentz invariant form are QIQx, 6/0y, didz, and (l/ic)0/0t, giving the magnitude for spatial and temporal dimensions as x + y + z — c t. This means that the order of the differentials for the coordinate and time in the equation of motion must be the same. The time-dependent Schrodinger equation exhibits time derivatives in the first order and coordinate derivatives in the second order therefore it is not Lorentz invariant. 

The classical relativistic energy of a free particle moving in space is given as 

This equation is called the Klein-Gordon equation. The Klein-Gordon equation satisfies the order of the differentials to be in the same order for space and time, and thus is Lorentz invariant. This equation, however, does not give rise to a positive definite density due in part to the second-order differential in time. Furthermore, for a fermion system, the Klein-Gordon equation is 

Dirac was seeking an equation that is analogous to the Schrodinger equation, with a first-order derivative in time, and simultaneously satisfying the Lorentz invariance. He took a bold step If the relativistic energy, the expression under the radical in the energy expression (Eq. [27]) or in the square bracket (Eq. [28]), is factored into a perfect square such that 

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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 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... 

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... 

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

To an excellent approximation, the four Klein-Gordon equations (443) are d Alembert equations, which are locally gauge-invariant. 

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