Klein-Gordon equation

How to construct

Starting from the relativistic connection between energy and momentum:

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^2=\bold p^2c^2+m^2c^4}

Substituting 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 \rightarrow i\hbar \frac{\partial}{\partial t}} and 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 p \rightarrow -i\hbar \nabla} , we get Klein-Gordon equation for free particles as follows:

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 -\hbar^2 \frac{\partial ^2\psi(\bold r, t)}{\partial t^2}=(-\hbar^2c^2\nabla^2+m^2c^4)\psi(\bold r, t)\qquad \qquad \qquad \qquad \qquad (1)}

Klein-Gordon can also be written as the following:

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 (\square-K^2)\psi(\bold r, t)=0\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\;\;\;(2)}

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 \square=\nabla^2-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}} is d'Alembert operator and 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 K=\frac{mc}{\hbar}} .

Equation (2) looks like a classical wave equation with an extra term 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 K^2} .

Potentials couple to the Klein-Gordon equation as four-vectors. A good example of this is the potential four-vector:

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 \Phi ^{\mu} = ( \phi ; \bold{A} )<\math> Coupling this to the momentum four-vector, <math>p^{\mu} = ( \frac{E}{c} ; \bold{p} )<\math> where <math>E \rightarrow i\hbar \frac{\partial}{\partial t}} and 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{p} \rightarrow -i\hbar \bold{\nabla}} in the quantum limit, we find the conjugate momentum four vector:

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^{\mu} = p^{\mu}-\frac{e}{c}\Phi^{\mu} = ( \frac{E}{c}-\frac{e}{c}\phi ; \bold{p}-\frac{e}{c}\bold{A} )}

Squaring the conjugate momentum four vector and multiplying 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 c^2} , we obtain the Klein-Gordon equation in an electromagnetic field:

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 c^2 P^{\mu}P_{\mu} = m^2 c^4 = (E – e\phi)^2 – (c\bold{p} – e\bold{A})^2}

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\left[ i\hbar \frac {\partial}{\partial t}-e\phi(\bold r, t) \right] ^2\psi(\bold r, t)=\left( \left[ -i\hbar\nabla-\frac{e}{c}\bold A(\bold r, t)\right] ^2c^2+m^2c^4\right) \psi(\bold r, t)}

Klein-Gordon is second order in time. Therefore, to see how the states of a system evolve in time we need to know both 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(\bold r, t)} and 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\psi(\bold r, t)}{\partial t}} at a certain time. While in nonrelativistic quantum mechanics, we only need 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(\bold r, t)}

Also because the Klein-Gordon equation is second order in time, it has the solutions 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(\bold r, t)=e^{i(\bold p \bold r - Et)/\hbar}} with either sign of energy $E=\pm c{\sqrt {{\mathbf {p}}^{2}+m^{2}c^{2}}}$. 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 (1) by ${\mathbf {\psi }}^{*}$ from the left, we get:

$-{\frac {\hbar ^{2}}{c^{2}}}\psi ^{*}{\frac {\partial ^{2}\psi ({\mathbf {r}},t)}{\partial t^{2}}}=\psi ^{*}(-\hbar ^{2}\nabla ^{2}+m^{2}c^{2})\psi ({\mathbf {r}},t)\qquad \qquad \qquad (3)$

Multiplying the complex conjugate form of (1) by ${\mathbf {\psi }}$ from the left, we get:

$-{\frac {\hbar ^{2}}{c^{2}}}\psi {\frac {\partial ^{2}\psi ^{*}({\mathbf {r}},t)}{\partial t^{2}}}=\psi (-\hbar ^{2}\nabla ^{2}+m^{2}c^{2})\psi ^{*}({\mathbf {r}},t)\qquad \qquad \qquad (4)$

Subtracting (4) from (3), we get:

$-{\frac {\hbar ^{2}}{c^{2}}}\left(\psi ^{*}{\frac {\partial ^{2}\psi }{\partial t^{2}}}-\psi {\frac {\partial ^{2}\psi ^{*}}{\partial t^{2}}}\right)=\hbar ^{2}\left(\psi \nabla ^{2}\psi ^{*}-\psi ^{*}\nabla ^{2}\psi \right)$

$\Rightarrow -{\frac {\hbar ^{2}}{c^{2}}}{\frac {\partial }{\partial t}}\left(\psi ^{*}{\frac {\partial \psi }{\partial t}}-\psi {\frac {\partial \psi ^{*}}{\partial t}}\right)+\hbar ^{2}\nabla \left(\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*}\right)=0$

$\Rightarrow {\frac {\partial }{\partial t}}\left[{\frac {i\hbar }{2mc^{2}}}\left(\psi ^{*}{\frac {\partial \psi }{\partial t}}-\psi {\frac {\partial \psi ^{*}}{\partial t}}\right)\right]+\nabla \left[{\frac {\hbar }{2mi}}\left(\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*}\right)\right]=0$

this give us the continuity equation:

${\frac {\partial \rho }{\partial t}}+\nabla {\mathbf {j}}=0\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (5)$

where $\rho ={\frac {i\hbar }{2mc^{2}}}\left(\psi ^{*}{\frac {\partial \psi }{\partial t}}-\psi {\frac {\partial \psi ^{*}}{\partial t}}\right)\qquad \qquad \qquad \qquad \qquad \;\;\;\;(6)$

${\mathbf {j}}={\frac {\hbar }{2mi}}(\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*})\qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\;(7)$

From (5) 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, 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} 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 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\rho(\bold r,t)} 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 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 v \ll c} 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\left[ 1+(\frac{p}{mc})^2\right] ^{1/2}\approx mc^2\left( 1+\frac{1}{2}\left( \frac{p}{mc}\right) ^2\right) =mc^2+\frac{p^2}{2m}}

So, the relativistic energy is different from classical energy 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 mc^2} , 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 -\hbar ^2\frac {\partial ^2 \psi}{\partial t^2}e^{-imc^2t/\hbar}+2\frac {\partial \psi}{\partial t}imc^2 \hbar e^{-imc^2t/\hbar}+\psi m^2c^4 e^{-imc^2t/\hbar}=-\hbar ^2 c^2\nabla ^2 \psi e^{-imc^2t/\hbar}+m^2c^4 \psi e^{-imc^2t/\hbar}}

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}