While studying the Dirac equation my professor defined the helicity operator as
$$\hat{\lambda}=\dfrac{\vec S \cdot \vec{p}}{|\vec p|}$$ where $\vec S$ is the spin matrix and $\vec{p}$ is the momentum operator.
My question is: what is the object in the denominator? If it's intended as an operator, then how is it defined the division by an operator? And what operator would it be? (There would be the square root of a sum of derivatives I think). If it's not an operator, then what is it? If I want to apply this operator to a wave function which is not eigenstate of the momentum operator what do I put in the denominator?
My professor did not treat it as an operator so I think that It's a constant, but which constant is it?
Best Answer
The helicity operator you've written is not an operator in the sense you're thinking. That is, an operator on the Hilbert space of states. So this is actually not what you want to do:
The helicity operator acts on the left- and right-handed components, $\psi_L(p),\psi_R(p)$ of the Dirac spinor, which are certainly not states (they furnish a non-unitary rep. of the Poincaré group and, upon quantization, multiply the creation and annihilation operators in the field expansion). So really $\vec{p}$ is not an operator at all (nor is $\vec{S}$), it is just the plain 3-momentum of a wave component.