Klein-Gordon equation: Difference between revisions

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where <math>\rho = \frac {i\hbar}{2mc^2}(\psi^{*}\frac{\partial\psi}{\partial t}-\psi\frac{\partial\psi^{*}}{\partial t})\qquad \qquad \qquad  \qquad \qquad \qquad(9.7)</math>
where <math>\rho = \frac {i\hbar}{2mc^2}(\psi^{*}\frac{\partial\psi}{\partial t}-\psi\frac{\partial\psi^{*}}{\partial t})\qquad \qquad \qquad  \qquad \qquad \qquad(9.7)</math>


<math>\bold j = \frac {\hbar}{2mi}(\psi^{*}\nabla\psi-\psi\nabla\psi^{*})\qquad \qquad \qquad \qquad \qquad \qquad  \qquad(9.8)</math>
<math>\bold j = \frac {\hbar}{2mi}(\psi^{*}\nabla\psi-\psi\nabla\psi^{*})\qquad \qquad \qquad \qquad \qquad \qquad  \qquad \; \;(9.8)</math>


From (9.6) we can see that the integral of the density <math>\bold \rho</math> over all space is conserved. However, <math>\bold\rho</math> is not positively definite. Therefore, we can neither interpret <math>\bold \rho</math> as the particle probability density nor can we interpret <math>\bold j</math> as the particle current. The appropriate interpretation are charge density for <math>e\rho(\bold r,t)</math> and electric current for <math>e\bold j(\bold r, t)</math> since charge density and electric current can be either positive or negative.
From (9.6) we can see that the integral of the density <math>\bold \rho</math> over all space is conserved. However, <math>\bold\rho</math> is not positively definite. Therefore, we can neither interpret <math>\bold \rho</math> as the particle probability density nor can we interpret <math>\bold j</math> as the particle current. The appropriate interpretation are charge density for <math>e\rho(\bold r,t)</math> and electric current for <math>e\bold j(\bold r, t)</math> since charge density and electric current can be either positive or negative.

Revision as of 16:57, 18 April 2009

How to construct

Starting from the relativistic connection between energy and momentum:

Substituting and , we get Klein-Gordon equation for free particles as follows:

Klein-Gordon can also be written as the following:

where is d'Alembert operator and .

Equation (9.3) looks like a classical wave equation with an extra term .

For a charged particle couple with electromagnetic field, Klein-Gordon equation is as follows:

Klein-Gordon is second order in time. Therefore, to see how the states of a system evolve in time we need to know both and at a certain time. While in nonrelativistic quantum mechanics, we only need

Also because the Klein-Gordon equation is second order in time, it has the solutions with either sign of energy . The negative energy solution of Klein-Gordon equation has a strange property that the energy decreases as the magnitude of the momentum increases. We will see that the negative energy solutions of Klein-Gordon equation describe antiparticles, while the positive energy solutions describe particles.

Continuity equation

Multiplying (9.2) by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bold \psi^{*}} from the left, we get:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{\hbar^2}{c^2}\psi^{*} \frac{\partial ^2\psi(\bold r, t)}{\partial t^2}=\psi^{*}(-\hbar^2\nabla^2+m^2c^2)\psi(\bold r, t)\qquad \qquad \qquad (9.4)}

Multiplying the complex conjugate form of (9.2) by from the left, we get:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{\hbar^2}{c^2}\psi \frac{\partial ^2\psi^{*}(\bold r, t)}{\partial t^2}=\psi(-\hbar^2\nabla^2+m^2c^2)\psi^{*}(\bold r, t)\qquad \qquad \qquad (9.5)}

Subtracting (9.5) from (9.4), we get:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{\hbar^2}{c^2}(\psi^{*} \frac{\partial ^2\psi}{\partial t^2}-\psi \frac{\partial ^2\psi^{*}}{\partial t^2})=\hbar^2(\psi\nabla^2\psi^{*}-\psi^{*}\nabla^2\psi)}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Rightarrow \frac {\partial}{\partial t}[\frac {i\hbar}{2mc^2}(\psi^{*}\frac{\partial\psi}{\partial t}-\psi\frac{\partial\psi^{*}}{\partial t})]+\nabla [\frac {\hbar}{2mi}(\psi^{*}\nabla\psi-\psi\nabla\psi^{*})]=0}

this give us the continuity equation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac {\partial \rho}{\partial t}+\nabla \bold j = 0 \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad(9.6)}

where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bold j = \frac {\hbar}{2mi}(\psi^{*}\nabla\psi-\psi\nabla\psi^{*})\qquad \qquad \qquad \qquad \qquad \qquad \qquad \; \;(9.8)}

From (9.6) we can see that the integral of the density Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bold \rho} over all space is conserved. However, is not positively definite. Therefore, we can neither interpret Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bold \rho} as the particle probability density nor can we interpret Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bold j} as the particle current. The appropriate interpretation are charge density for and electric current for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e\bold j(\bold r, t)} since charge density and electric current can be either positive or negative.

Nonrelativistic limit

In nonrelativistic limit when or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p \ll mc} , we have:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E=[(pc)^2+(mc^2)^2]^{1/2}=mc^2[1+(\frac{p}{mc})^2]^{1/2}\approx mc^2(1+\frac{1}{2}(\frac{p}{mc})^2)=mc^2+\frac{p^2}{2m}}

So, the relativistic energy is different from classical energy by , therefore, we can expect that if we write the solution of Klein-Gordon equation as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi e^{-imc^2t/\hbar}} and substitute it into Klein-Gordon equation, we will get Schrodinger equation for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bold \psi} .

Indeed, doing so we get:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Rightarrow -\frac {\hbar ^2}{2mc^2} \frac{\partial ^2 \psi}{\partial t^2}+i\hbar \frac {\partial \psi}{\partial t}=-\frac {\hbar ^2}{2m}\nabla ^2 \psi}

In the nonrelativistic limit the first term is considered negligibly small. As a result, for free particles in this limit we get back the Schrodinger equation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i\hbar \frac {\partial \psi}{\partial t}=-\frac {\hbar ^2}{2m} \nabla ^2 \psi}

Negative energy states and antiparticles

Klein-Gordon equation with Coulomb potential