Klein–Gordon Equation

This is a compact lookup card. The mass-shell derivation, two-data Cauchy problem, conserved-current calculation, and one-particle interpretation boundary are owned by the Klein–Gordon Equation page.

Formula

With metric signature $\eta=\operatorname{diag}(1,-1,-1,-1)$ and

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

the free Klein–Gordon equation is

$$\left( \Box+\frac{m^2c^2}{\hbar^2} \right)\phi=0.$$

Equivalently,

$$\left( \frac{1}{c^2} \frac{\partial^2}{\partial t^2} - \nabla^2 + \frac{m^2c^2}{\hbar^2} \right)\phi=0.$$

For a plane wave

$$\phi \propto \exp \left[ -\frac{iEt}{\hbar} + \frac{i\mathbf p\cdot\mathbf x}{\hbar} \right],$$

the equation gives the relativistic dispersion relation

$$E^2 = \mathbf p^2c^2+m^2c^4.$$

Assumptions

• The field or wavefunction is scalar under Lorentz transformations.
• The metric signature and definition of $\Box$ are fixed.
• The displayed equation is free; external electromagnetic coupling requires a covariant-derivative prescription.
• A one-particle probability interpretation is not the same as the nonrelativistic Schrödinger interpretation.

Validity

The Klein–Gordon equation is the natural second-order relativistic equation associated with $E^2=\mathbf p^2c^2+m^2c^4$. As a fixed-particle wave equation it has interpretive limitations, including negative-frequency solutions and a conserved current whose time component is not positive definite for arbitrary solutions. In modern use, it is cleanest as the field equation for a scalar field or as a controlled relativistic-wave-equation bridge.

Common Mistakes

• Mixing metric signatures without changing the sign of $\Box$ or the mass term.
• Treating every solution as a nonrelativistic probability amplitude with density $\lvert\phi\rvert^2$.
• Forgetting that both positive- and negative-frequency branches appear.
• Calling the equation “spinless Schrödinger theory” without the relativistic interpretation caveats.

Quick Check

Insert a plane wave into the free equation and recover the dispersion relation.

Solution

For the plane wave, $\partial_t^2\phi=-E^2\phi/\hbar^2$ and $\nabla^2\phi=-\mathbf p^2\phi/\hbar^2$. Therefore

$$\left( -\frac{E^2}{\hbar^2c^2} + \frac{\mathbf p^2}{\hbar^2} + \frac{m^2c^2}{\hbar^2} \right)\phi=0.$$

Multiplying by $\hbar^2c^2$ gives $-E^2+\mathbf p^2c^2+m^2c^4=0$, or $E^2=\mathbf p^2c^2+m^2c^4$.

Canonical owner and released companions

• Klein–Gordon Equation
• Energy–Momentum Relation
• Minimal Coupling

References

• O. Klein, “Quantentheorie und fünfdimensionale Relativitätstheorie”, Zeitschrift für Physik 37, 895-906, 1926.
• W. Gordon, “Der Comptoneffekt nach der Schrödingerschen Theorie”, Zeitschrift für Physik 40, 117-133, 1926.
• J. J. Sakurai, Advanced Quantum Mechanics, Addison-Wesley, 1967.
• S. Weinberg, The Quantum Theory of Fields, Volume I, Cambridge University Press, 1995.