Solutions of the free one-particle Klein Gordon equation

[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!

 

[azntoon]

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}.