Klein-Gordon equation

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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. It was named after Oskar Klein and Walter Gordon.

1 Details
2 Relativistic free particle solution
3 See also
4 References
5 External links

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Details

The Schrödinger equation for a free particle is

<math>

\frac{\mathbf{p}^2}{2m} \psi = i \hbar \frac{\partial}{\partial t}\psi </math> where

<math>\mathbf{p} = -i \hbar \mathbf{\nabla}</math> is the momentum operator (<math>\nabla</math> 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

<math>E = \sqrt{\mathbf{p}^2 c^2 + m^2 c^4}</math>

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

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

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

<math>

(\Box^2 + \mu^2) \psi = 0. </math> where

<math> \mu = \frac{mc}{\hbar} \,</math> and

<math> \Box^2 = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2\,</math>. 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 allegedly first found by Schrödinger, before he made the discovery of the equation that now bears his name. He rejected it because he couldn't make it include the spin of the electron. The way Schrödinger found his equation was by making simplifications in the Klein-Gordon equation.

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.

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Relativistic free particle solution

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

<math>

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

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

<math>

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

except with the constraint

<math>

-k^2+\frac{\omega^2}{c^2}=\frac{m^2c^2}{\hbar^2} </math>

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

<math>

\langle\mathbf{p}\rangle=\langle \psi |-i\hbar\mathbf{\nabla}|\psi\rangle = \hbar\mathbf{k} </math>

<math>

\langle E\rangle=\langle \psi |i\hbar\frac{\partial}{\partial t}|\psi\rangle = \hbar\omega </math>

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:

<math>\left.\right.

\langle E \rangle^2=m^2c^4+\langle \mathbf{p} \rangle^2c^2 </math>

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

<math>\left.\right.

\langle E \rangle=\langle |\mathbf{p}| \rangle c </math>

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