- #1

- 1,789

- 4

Hi,

I have a particle physics exam tomorrow morning (in a few hours from now, in my time zone). I'm trying to figure out the whole reasoning behind pion-nucleon scattering. Please bear with me..

We write the scattering matrix as

[tex]S = 1 - iT[/tex]

where T is given by

[tex]T = f + i g \boldsymbol{\sigma}\cdot\hat{n}[/tex]

Here f and g are scalars, functions of the mandelstam variables s and p (total energy in CM frame squared and difference of momenta modulus). [itex]\hat{n}[/itex] is the normal vector perpendicular to the reaction plane. We reasoned that T has this form because the dependence on total momentum P and differential momentum p must vanish for T to be invariant under parity.

Now,

[tex]\rho^i = \frac{1}{2}(1 + \vec{\sigma}\cdot \vec{P}_{i})[/tex]

[tex]\rho^f = \frac{T\rho^{i}T^{\dagger}}{Tr[T\rho^{i}T^{\dagger}]}[/tex]

We also define a quantity

[tex]\rho_{m'm} = \langle s m'|\rho|s m\rangle[/tex]

and reason that since the state should be invariant under rotation about the z-axis, we must have

[tex]e^{-iS_{z}\pi}|m\rangle = (-1)^{-im}|m\rangle[/tex]

and hence

[tex]\rho_{m'm} = (-1)^{m-m'}\rho_{m'm}[/tex]

1. What is the motivation behind writing this density matrix?

2. How was it written in the first place?

3. What is [itex]\rho_{m'm}[/itex]? Its some kind of matrix element, but what does it signify? Does it signify a transition from state m to m'? (I'm having a bad day here :-|)

I never studied scattering theory this way, so I would appreciate if someone could give me a heads-up and point to the relevant text(s).

Thanks!

I have a particle physics exam tomorrow morning (in a few hours from now, in my time zone). I'm trying to figure out the whole reasoning behind pion-nucleon scattering. Please bear with me..

We write the scattering matrix as

[tex]S = 1 - iT[/tex]

where T is given by

[tex]T = f + i g \boldsymbol{\sigma}\cdot\hat{n}[/tex]

Here f and g are scalars, functions of the mandelstam variables s and p (total energy in CM frame squared and difference of momenta modulus). [itex]\hat{n}[/itex] is the normal vector perpendicular to the reaction plane. We reasoned that T has this form because the dependence on total momentum P and differential momentum p must vanish for T to be invariant under parity.

**So far so good**.Now,

**what I do not understand follows below**. We write the initial state density matrix as[tex]\rho^i = \frac{1}{2}(1 + \vec{\sigma}\cdot \vec{P}_{i})[/tex]

**How?**Then we write the final state density matrix as[tex]\rho^f = \frac{T\rho^{i}T^{\dagger}}{Tr[T\rho^{i}T^{\dagger}]}[/tex]

**How?**We also define a quantity

[tex]\rho_{m'm} = \langle s m'|\rho|s m\rangle[/tex]

and reason that since the state should be invariant under rotation about the z-axis, we must have

[tex]e^{-iS_{z}\pi}|m\rangle = (-1)^{-im}|m\rangle[/tex]

and hence

[tex]\rho_{m'm} = (-1)^{m-m'}\rho_{m'm}[/tex]

__My questions:__1. What is the motivation behind writing this density matrix?

2. How was it written in the first place?

3. What is [itex]\rho_{m'm}[/itex]? Its some kind of matrix element, but what does it signify? Does it signify a transition from state m to m'? (I'm having a bad day here :-|)

I never studied scattering theory this way, so I would appreciate if someone could give me a heads-up and point to the relevant text(s).

Thanks!

**EDIT**: I just read the section on projection operators and density matrices from Schiff's book. Am I correct in interpreting this as a way of specifying the initial state in terms of its spin? I get this if we have just one particle to begin with, but when we have two -- as is the case here (pion**and**nucleon) how do we write a composite density matrix?
Last edited: