Exploring Quantum Physics - Week 7 Question 9

Question:

How to construct another wave-function from one such that it overall gain a negative sign in the Dirac equation?

Solution:

The Dirac equation is:

$i\hbar{\frac{\partial \Psi}{\partial t}} = (\hat{\alpha}\hat{p} + mc^2\hat{\beta})\Psi$.

We know $\{\alpha, \beta\} = \alpha\beta + \beta\alpha = 0$, so we have $\alpha\beta = -\beta\alpha$, that how a negative sign is introduced.

So all we need to do to make sure it gain an overall negative sign is simply make sure we have odd number of  $\alpha \beta$.

That explains the existence of anti matter!