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✨ Equation of the Week ✨

(+m2)ϕ=0\left( \Box + m^2 \right) \phi = 0

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.

  • =t22=μμ\Box = \partial_t^2 - \nabla^2 = \partial_\mu \partial^\mu: The d'Alembertian operator, representing the relativistic generalization of the Laplacian. It involves second-order partial derivatives with respect to time and spatial coordinates.
  • mm: The mass of the particle (in natural units where =c=1\hbar = c = 1).
  • ϕ\phi: 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 E2=p2+m2E^2 = p^2 + m^2 by using the quantum mechanical substitutions EitE \rightarrow i\partial_t and pi\mathbf{p} \rightarrow -i\nabla.

Example Problem

Consider a plane wave solution to the Klein-Gordon equation of the form ϕ(x)=Aeipμxμ=Aei(Etpx)\phi(x) = A e^{-ip_\mu x^\mu} = A e^{-i(Et - \mathbf{p} \cdot \mathbf{x})}, where AA is a constant amplitude and pμ=(E,p)p_\mu = (E, \mathbf{p}) is the four-momentum. Substitute this plane wave into the Klein-Gordon equation and find the relationship between the energy EE, momentum p\mathbf{p}, and mass mm.

Solution

Step 1: Compute the second derivatives of the plane wave

t2ϕ(x)=(iE)2Aei(Etpx)=E2ϕ(x)\partial_t^2 \phi(x) = (-iE)^2 A e^{-i(Et - \mathbf{p} \cdot \mathbf{x})} = -E^2 \phi(x)
2ϕ(x)=(ip)2Aei(Etpx)=p2ϕ(x)=p2ϕ(x)\nabla^2 \phi(x) = (-\mathbf{i} \mathbf{p})^2 A e^{-i(Et - \mathbf{p} \cdot \mathbf{x})} = -|\mathbf{p}|^2 \phi(x) = -p^2 \phi(x)

Step 2: Substitute into the Klein-Gordon equation

(t22+m2)ϕ(x)=(E2(p2)+m2)ϕ(x)=0(\partial_t^2 - \nabla^2 + m^2) \phi(x) = (-E^2 - (-p^2) + m^2) \phi(x) = 0
(E2+p2+m2)ϕ(x)=0(-E^2 + p^2 + m^2) \phi(x) = 0

Step 3: Derive the energy-momentum relation

Since ϕ(x)\phi(x) is generally non-zero, we must have:

E2p2m2=0E^2 - p^2 - m^2 = 0
E2=p2+m2E^2 = p^2 + m^2

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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