[Physics] Why isn’t the wave equation $\nabla^2 \psi – 1/c^2 \partial_{tt} \psi = (\frac{mc}{\hbar})^2\psi$

klein-gordon-equationquantum mechanicsschroedinger equation

Special relativity was well established by the time the schrodinger equation came out. Using the correspondence of classical energy with frequency and momentum with wave number, this is the equation that comes out, and looks sensible because this is of the form of a wave equation, like the one for sound etc, except with an inhomogeneous term

$$\nabla^2 \psi – 1/c^2 \partial_{tt} \psi = (\frac{mc}{\hbar})^2\psi$$

Instead, we have schrodinger's equation which reminds of the heat equation as it is first order in time.

EDIT Some thoughts after seeing the first posted answer

What is wrong with negative energy solutions? When we do freshman physics and solve some equation quadratic in, say, time… we reject the negative time solution as unphysical for our case. We do have a reason why we get them, like, for the same initial conditions and acceleration, this equation tells usthat given these iniial velocity and acceleration, the particle would have been at the place before we started our clock, since we're interested only in what happens after the negative times don't concern us. Another example is if we have areflected wave on a rope, we get these solutions of plane progressive waves travelling at opposite velocities unbounded by our wall and rope extent. We say there is a wall, an infinite energy barrier and whatever progressive wave is travelling beyond that is unphysical, not to be considered, etc.

Now the same thing could be said about $E=\pm\sqrt{p^2c^2+(mc^2)^2}$ that negative solution can't be true because a particle can't have an energy lower than its rest mass energy $(mc^2)$ and reject that.

If we have a divergent series, and because answers must be finite, we say.. ahh! These are not real numbers in the ordinary sense, they are p-adics! And in this interpretation we get rid of divergence. IIRC Casimirt effect was the relevant phenomenon here.

My question boils down to this. I guess the general perception is that mathematics is only as convenient as long as it gives us the answers for physical phenomenon. I feel this is sensible because nature can possibly be more absolute than any formal framework we can construct to analyze it. How and when is it OK to sidestep maths and not miss out a crucial mystery in physics.

Best Answer

That's the Klein-Gordon equation, which applies to scalar fields. For fermionic fields, the appropriate relativistic equation is the Dirac equation, but that was only discovered by Dirac years after Schrödinger discovered his nonrelativistic equation. The nonrelativistic Schrödinger equation is a lot easier to solve too.

The relativistic equations admit negative energy solutions. For fermions, that was only resolved by Dirac much later with his theory of the Dirac sea. For bosons, the issue was resolved by "second quantization".

The problem with negative energy solutions is the lack of stability. A positive energy electron can radiate photons, and decay into a negative energy state, if negative energy states do exist.