Klein-Gordon equation

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Quantum mechanics
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The Klein-Gordon equation (Klein-Fock-Gordon equation or sometimes Klein-Gordon-Fock equation) is the relativistic version of the Schrödinger equation, which is used to describe spinless particles. It was named after Oskar Klein and Walter Gordon.

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The Schrödinger equation for a free particle is

\frac{\mathbf{p}^2}{2m} \psi = i \hbar \frac{\partial}{\partial t}\psi

where

\mathbf{p} = -i \hbar \mathbf{\nabla} is the momentum operator (\nabla being the del operator).

The Schrödinger equation suffers from not being relativistically covariant, meaning it does not take into account Einstein's special relativity.

It is natural to try to use the identity from special relativity

E = \sqrt{\mathbf{p}^2 c^2 + m^2 c^4}

for the energy; then, just inserting the quantum mechanical momentum operator, yields the equation

\sqrt{(-i\hbar\mathbf{\nabla})^2 c^2 + m^2 c^4} \psi= i \hbar \frac{\partial}{\partial t}\psi.

This, however, is a cumbersome expression to work with because of the square root. In addition, this equation, as it stands, is nonlocal.

Klein and Gordon instead worked with the more general square of this equation (the Klein-Gordon equation for a free particle), which in covariant notation reads

(\Box^2 + \mu^2) \psi = 0,

where

\mu = \frac{mc}{\hbar} \,

and

\Box^2 = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2.\,

This operator is called the d'Alembert operator. Today this form is interpreted as the relativistic field equation for a scalar (i.e. spin-0) particle.

The Klein-Gordon equation was first considered as a quantum wave equation by Schrödinger in his search for an equation describing de Broglie waves. The equation is found in his notebooks from late 1925, and he appears to have prepared a manuscript applying it to the hydrogen atom. Yet, without taking into account the electron's spin, the Klein-Gordon equation predicts the hydrogen atom's fine structure incorrectly, including overestimating the overall magnitude of the splitting pattern by a factor of 4n/(2n-1) for the n-th energy level. In January 1926, Schrödinger submitted for publication instead his equation, a non-relativistic approximation that predicts the Bohr energy levels of hydrogen without fine structure.

In 1926, soon after the Schrödinger equation was introduced, Fock wrote an article about its generalization for the case of magnetic fields, where forces were dependent on velocity, and independently derived this equation. Both Klein and Fock used Kaluza and Klein's method. Fock also determined the gauge theory for the wave equation. The Klein-Gordon equation for a free particle has a simple plane wave solution.

The Klein-Gordon equation for a free particle can be written as

\mathbf{\nabla}^2\psi-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\psi = \frac{m^2c^2}{\hbar^2}\psi

with the same solution as in the non-relativistic case:

\psi(\mathbf{r}, t) = e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}

except with the constraint

-k^2+\frac{\omega^2}{c^2}=\frac{m^2c^2}{\hbar^2}.

Just as with the non-relativistic particle, we have for energy and momentum:

\langle\mathbf{p}\rangle=\langle \psi |-i\hbar\mathbf{\nabla}|\psi\rangle = \hbar\mathbf{k},
\langle E\rangle=\langle \psi |i\hbar\frac{\partial}{\partial t}|\psi\rangle = \hbar\omega.

Except that now when we solve for k and ω and substitute into the constraint equation, we recover the relationship between energy and momentum for relativistic massive particles:

\left.\right. \langle E \rangle^2=m^2c^4+\langle \mathbf{p} \rangle^2c^2.

For massless particles, we may set m = 0 in the above equations. We then recover the relationship between energy and momentum for massless particles:

\left.\right. \langle E \rangle=\langle |\mathbf{p}| \rangle c.

The Klein-Gordon equation can be derived from the following action

\mathcal{S}=\int \mathrm{d}^4x \left(\frac{1}{2}\partial_{\mu}\phi \partial^{\mu}\phi - \frac{1}{2}\frac{m^2 c^2}{\hbar^2} \phi^2 \right)

where φ is the Klein-Gordon field and m is its mass.

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