# Solutions of the free one-particle Klein Gordon equation

• silverwhale
In summary, the KG equation for relativistic quantum mechanics is introduced in "Wachter, relativistic quantum mechanics" as -\hbar^2 \frac{\partial^2 \phi(x)}{\partial t^2} = (-c^2 \hbar^2 \nabla^2 + m^2_0 c^4) \phi(x). The separation ansatz is used to solve this equation, resulting in four combinations of solutions. The book only lists two solutions, but the other two can be found by taking the complex conjugates of the given solutions.
silverwhale
In the book "Wachter, relativistic quantum mechanics", in page 5, the KG eq. is introduced as follows:

$$-\hbar^2 \frac{\partial^2 \phi(x)}{\partial t^2} = (-c^2 \hbar^2 \nabla^2 + m^2_0 c^4) \phi(x).$$

Now I tried to solve this equation using the separation ansatz (product ansatz).
I get:

$$\phi (ct) = exp [\pm \frac{ i p_0 ct}{\hbar} ],$$
and
$$\psi (x) = exp[\pm \frac{i \vec{p} \cdot \vec{x}}{\hbar}].$$
Where the usual relativistic four-momentum conservation equation holds.
Now, this amounts to four combinations of solutions. But in the book, only two are written, namely:

$$\phi^{(1)}_p= e^{-i (c p_0 t - \vec{p} \cdot \vec{x})/ \hbar},$$

$$\phi^{(2)}_p= e^{+i (c p_0 t - \vec{p} \cdot \vec{x})/ \hbar}.$$

I really, really want to solve the equation; I feel a little bit frustrated because I don't get only the two solutions. Basically I am halfway there but I don't know what to do next.

So any help would be greatly appreciated!

The other two solutions you are missing are the complex conjugates of the ones you have written down. So the full set of solutions is: \phi^{(1)}_p= e^{-i (c p_0 t - \vec{p} \cdot \vec{x})/ \hbar},\phi^{(2)}_p= e^{+i (c p_0 t - \vec{p} \cdot \vec{x})/ \hbar},\phi^{(3)}_p= e^{-i (c p_0 t + \vec{p} \cdot \vec{x})/ \hbar},\phi^{(4)}_p= e^{+i (c p_0 t + \vec{p} \cdot \vec{x})/ \hbar}.

## 1. What is the free one-particle Klein Gordon equation?

The free one-particle Klein Gordon equation is a relativistic wave equation that describes the motion of a single particle with zero spin. It takes into account both the particle's mass and its energy, and is commonly used in quantum field theory.

## 2. What are solutions to the free one-particle Klein Gordon equation?

The solutions to the free one-particle Klein Gordon equation are complex-valued functions that describe the probability amplitude of the particle at different points in space and time. These solutions can be interpreted as waves, with the amplitude and frequency determining the energy and momentum of the particle.

## 3. How is the free one-particle Klein Gordon equation derived?

The free one-particle Klein Gordon equation can be derived using relativistic energy-momentum relations and the principle of least action. It is also a special case of the more general Klein-Gordon equation, which describes particles with non-zero spin.

## 4. What are the physical implications of solutions to the free one-particle Klein Gordon equation?

The solutions to the free one-particle Klein Gordon equation have several important physical implications. They are used to calculate the probabilities of different outcomes in particle interactions, and can also be used to study the behavior of particles in different environments, such as in the presence of external fields.

## 5. How is the free one-particle Klein Gordon equation used in scientific research?

The free one-particle Klein Gordon equation is used in a variety of scientific research, particularly in quantum field theory and particle physics. It is also used in cosmology and astrophysics to study the behavior of particles in extreme environments, such as in the early universe or around black holes.

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