# Solutions of 1D-Dirac equation for free particle only have positive energy solutions?

Im studying the (stationary) free 1D-Dirac equation

$$H\Psi(x) =(mc^2\sigma_1-i\hbar c\sigma_3\frac{\partial}{\partial x})\Psi(x) = E\Psi(x),$$

where $$\sigma_1$$ and $$\sigma_3$$ are the pauli matrices.

I was able to determine the (unnormalized) solutions

$$\Psi_1(x) = \binom{1}{\frac{mc^2}{E-pc}}e^{-ipx/\hbar} \quad \text{and} \quad \Psi_2(x) = \binom{\frac{mc^2}{pc+E}}{1}e^{-ipx/\hbar},$$

where $$E=\sqrt{m^2c^4+c^2p^2}$$.

I'm wondering, why do I get $$H\Psi_1 = +E\Psi_1$$ and $$H\Psi_2 = +E\Psi_2$$? Shouldn't one of the two eigenstates have a negative energy eigenvalue, i.e. $$H\Psi = -E\Psi$$?

• Your equation only has 2 components. The Dirac equation is a 4-component equation. May 20, 2022 at 16:19
• yes this is due to the fact that this is the one dimensional equation. the gamma matrices are in the 1d case the 2x2 pauli metrices. Therfore the Eigenstates are 2 dimensional vectors. May 20, 2022 at 16:22
• I was also wondering, why there are only 2 components. I think this is because in the 1d case there is no angular momentum, and therefore there is no such thing as spin. So i thought the 2 orthogonal vectors represent each the positive and negative energy case. May 20, 2022 at 17:31
• Isn't this more like Weyl equation? May 20, 2022 at 22:56

Your solutions are actually identical up to normalization. If you multiply your $$\Psi_1$$ by $$\frac{mc^2}{pc+E}$$ you get what you are calling $$\Psi_2$$.

In this solution $$E$$ and $$p$$ are only fixed by the condition $$E^2= p^2c^2+m^2c^4$$, so for any given $$p$$ there are two solutions for $$E$$ of differing sign. This is the sense in which there are two solutions.

I have an Idea on how to get the negative energy states. When I calculate the Energy, I get the condition $$E^2 = p^2c^2 + m^2c^4$$.

So I can define $$E = \pm \sqrt{p^2c^2+m^2c^4} = \pm E_p$$ and therefore I have two different differential equations:

$$H\Psi_+ = +E_p\Psi_+$$ and

$$H\Psi_- = -E_p\Psi_-$$

the second equation should give me the negative energy states.