# Quantum mechanics question

1. Jun 2, 2010

### eoghan

1. The problem statement, all variables and given/known data
A system in a state with L=1, S=1, J=2, Mj=2 and odd parity decay in a particle $$\Delta^-$$ (even parity and spin 1/2) and in a proton (even parity).

1) Find the possible momenta of S and L of the final system
2) Find the most general final system
3) Find the angular distribution of probability to find both particles with spin parallel to z axis.

3. The attempt at a solution
1)
Final S=0,1
Final L=2 (S=0) or L=1,2,3(S=1)
but L=1 and L=3 is not admissible due to the parity conservation

So there is only: S=0, L=2
or S=1, L=2

2)The two final states are
$$|2022>=|2020>$$
$$|2122>=\frac{\sqrt{2}|2120>-|2111>}{\sqrt{3}}$$

where the notation is: $$|L,S,J,Mj>=|L,S,m_L, m_S>$$

I don't know in which of this two states the particle will decay, so I have to consider the general state:
$$\alpha|2020>+\beta\frac{\sqrt{2}|2120>-|2111>}{\sqrt{3}}$$
with $$|\alpha|^2+|\beta|^2=1$$

3) The probability is the $$|\frac{\beta}{3}Y_2^2|^2$$
But I don't know how to find alpha and beta

Last edited: Jun 2, 2010