# Question about Spinors and Probability Densities

So I was toying around attempting to simulate some relativistic wave equations for a recreational project. Now I have never studied spinors in dept and the knowledge I have is from reading online (apologies in advance if the question may be too obvious).

To keep things simple and have less terms to deal with, I simplified the Dirac Equation by assuming that the particle is massless to get the Weyl Equation $$\sigma^\mu \partial_\mu \psi = 0$$ where $$\psi=(\psi_L,\psi_R)\in\mathbb{C}^2$$ seems to be what is called a "spinor" with $$L$$ and $$R$$ describing spin direction, and $$\sigma^\mu$$ the Pauli matrices. Then I got two separate PDEs by multiplying out the matrices: $$\partial_t\psi_R + \partial_x\psi_L -i \partial_y\psi_L + \partial_z\psi_R = 0$$ $$\partial_t\psi_L + \partial_x\psi_R +i \partial_y\psi_R - \partial_z\psi_L = 0$$

Next, because most programming languages do not have built in support for complex numbers, I handwavingly took $$\psi_k = u_k + i v_k$$ and further broke down the 2 PDEs into 4 PDEs: $$\partial_tu_R + \partial_xu_L + \partial_yu_L + \partial_zu_R = 0$$ $$\partial_tv_R + \partial_xv_L - \partial_yu_L + \partial_zv_R = 0$$ $$\partial_tu_L + \partial_xu_R - \partial_yv_R - \partial_zu_L = 0$$ $$\partial_tv_L + \partial_xv_R + \partial_yu_R - \partial_zv_L = 0$$

Now my question is this:

1. Firstly, is such an approach considered to be correct? Just double checking.

2. Now for my real question: In QM, we have that $$\mathbb{P}(x,t) = \psi \psi^* = |\psi(x,y)|^2$$ as out probability density function. By extrapolating this, I am assuming that with spinors $$\mathbb{P}_k(x,t) = \psi_k \psi_k^* = |\psi_k(x,y)|^2$$ where $$k$$ is the particle spin direction. Now assuming that I don't care about the spin and just want the probability of finding a particle with any spin, would it be as simple as $$\mathbb{P}(x,t) = \mathbb{P}_L(x,t) + \mathbb{P}_R(x,t) = \psi.\psi^* = \langle\psi|\psi\rangle$$?