Asymptotic behavior of wave function in scattering of spin 1/2 particles against spin 0 target

Question

I need a little hand here. My professor left me as homework to study the chapter VII of the Ref. [1]. Everything was going well until the section "Particle with spin". In Eq.(7.1), is given the asymptotic wave function of the system composed by a pair of particles with spin 0 and 1/2 respectively as

$$ \psi \approx e^{ikz}\left\vert \alpha \right\rangle +F\left( \theta ,\phi \right) \frac{e^{ikr}}{r}\left\vert \alpha \right\rangle $$

where $\left\vert \alpha \right\rangle$ is initial state of the spin 1/2 particle an $F\left( \theta ,\phi\right)$ the scattering amplitude, given by a 2x2 matrix in this representation. In the case involving spinless particles, the scattering amplitude is a simple function of the angular coordinates, naturally appearing whit the solution of the Schrodinger equation. Concerning of this, I have some questions:

• What is the differences in the approach that makes the scattering amplitude to be a matrix?

• There would exists influence of the spin in the scattering even when the Hamiltonian don't depends on it?

• How can I prove that the differential cross section would be $\frac{d\sigma }{d\Omega }=\left\vert \left\langle \left. \beta \right. \left\vert F\left( \theta ,\phi \right) \right\vert \left. \alpha \right. \right\rangle \right\vert ^{2}$ (with $\left\vert \beta \right\rangle $ being the observed spin state) instead of $\frac{d\sigma }{d\Omega }=\left\vert f\left( \theta ,\phi \right) \right\vert ^{2}$ ?

Someone can help me with this? Thanks in advance

[1] Omnes, Roland. "INTRODUCTION TO PARTICLE PHYSICS." (1971).

Answer 1

First, I think it is more exactly if you write: $$|\psi\rangle \approx e^{ikz}\left\vert \alpha \right\rangle +F\left( \theta ,\phi \right) \frac{e^{ikr}}{r}\left\vert \alpha \right\rangle $$ where $|\psi\rangle$ is a state vector, not a wave function.

In the case of spinless particles, wave functions are scalar functions, and state vectors are one-component vector. So the scattering amplitude is a $1\times1$ matrix, or a simple function. However, for the case of spin particles, state vectors are 2-component vectors (in Pauli's (non-relativistic) theory), whose corresponding wave functions are called "spinor"; or 4-component vectors (in Dirac's (relativistic) theory). Your case is of the Pauli's theory, so the scattering amplitude here is a $2\times2$ matrix. This is for your first question.

When you calculate the differential cross section, first you should find the transition amplitude. Transition means that, by scattering, the initial state $|\alpha\rangle$ evolves to a certain final state. So, to find the transition amplitude, you should make a projection on the desired final state (namely, $|\beta\rangle$), or: $\langle\beta|\psi\rangle$. The first term appearing when you do this projection is the non-scattering term, the second one is scattering term, which we are interested in. This is, I think, the answer for your third question.

And, what do you mean by "Hamiltonian of a spin particle does not depend on spin"? Actually, I cannot catch the idea of your second question.