Reference: References: Robert D. Klauber, Student Friendly Quantum Field Theory, (Sandtrove Press, 2013) – Chapter 3, Problem 3.1.
The Klein-Gordon equation was one of the first attempts at producing a relativistic quantum theory. In natural units, the equation is
This equation also results from the Euler-Lagrange equation for a scalar field with Lagrangian
This is the Lagrangian for zero potential .
To write solutions to equation 1, we can introduce some new notation. In natural units, the four-momentum is
The scalar product of four-momentum with a spacetime vector is therefore
For a plane wave with angular frequency , Planck’s relation is , and the wave vector has components in the three spatial directions of , where is the component of the wavelength in direction . For example, a wave moving in the direction has , so . The four-vector is
and since in natural units, we have
A plane wave solution to 1 turns out to be
We can see this by direct substitution. Consider one term from the sum. Then
However, using the invariant scalar from relativity:
so 1 is true for a single component . Since the solution 8 is a linear combination of such solutions, and the original differential equation is linear, then 8 is also a solution. [The normalization factor is irrelevant in proving that 8 is a solution; it’s just there to make future calculations easier.]
Note that the first term (involving ) is also a solution of the free-particle Schrödinger equation, which is
If we take
For a free particle, the energy , so the Schrödinger equation is satisfied. The second term (with ) does not satisfy the Schrödinger equation, since in that case we get
The extra minus sign means the two sides don’t match.