✨ Equation of the Week ✨
This equation is the Klein-Gordon equation, a relativistic wave equation that describes the propagation of spin-0 particles (scalar particles) such as the Higgs boson. It is a fundamental equation in relativistic quantum field theory.
- : The d'Alembertian operator, representing the relativistic generalization of the Laplacian. It involves second-order partial derivatives with respect to time and spatial coordinates.
- : The mass of the particle (in natural units where ).
- : The scalar field, a function of spacetime coordinates that represents the particle.
The Klein-Gordon equation can be derived from the relativistic energy-momentum relation by using the quantum mechanical substitutions and .
Example Problem
Consider a plane wave solution to the Klein-Gordon equation of the form , where is a constant amplitude and is the four-momentum. Substitute this plane wave into the Klein-Gordon equation and find the relationship between the energy , momentum , and mass .
Solution
Step 1: Compute the second derivatives of the plane wave
Step 2: Substitute into the Klein-Gordon equation
Step 3: Derive the energy-momentum relation
Since is generally non-zero, we must have:
Step 4: Conclusion
This result shows that the plane wave solution is a valid solution to the Klein-Gordon equation if and only if the energy and momentum of the particle satisfy the relativistic energy-momentum relation.
Contributor: Oskar Klein & Walter Gordon
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