- #1
binbagsss
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1. Homework Statement
The question is to determine which decays are possible for:
i) ##P^0 ->\prod^+ \prod^-##
ii)##P^0 ->\prod^0 \prod^0##
where ##J^p = 0^-, 1^- ## respectively for ##\prod^+, \prod^- , \prod^0## and ##P^0## respectively.
For part i, the LHS has odd parity. ##P=(-1)^l##, so on the RHS we require ##l## to be odd.
Also need to conserve total angular momentum ##J=(l+s)+(l+s-1)+...+ | l-s |## *
On LHS ##J=1.##
##s=0##, so conservation gives ##l=-1## , which is consistent with an odd parity , so the decay is allowed.
part ii) We have the same J and P arguments, so I would have concluded the decay is possible.
The solution however is that is not because the RHS now has 2 identical bosons so the final wavefunction must be symmetric under the exchange of the two neutral pions. However this requires that the orbital angular momentum is even, so we have inconsistency.
So here's what I know :
If you swap 2 bosons the wave function has to be unchanged, but if you swap 2 fermions the wave function changes sign.
So , with this, I now don't see why we can't apply the argument to the decay in part i) - unless this property is only true for a system of identical particles??
More importantly, I don't follow the argument completely: The angular momentum being odd or even, i.e- as far as I can see the only way for ##l## to creep in, comes from the parity being odd or even- ##P=(-1)^l##, but parity is describing how the wave function behaves under a change from ##\vec r ## to ##\vec -r## So what has this got to do with swapping 2 bosons? The only possible argument I can think of would be along the lines of considering the particular case were one of the particles is situated at ##\vec r ## and the other at ##\vec -r## when we swap the bosons position??Thanks in advance !
The question is to determine which decays are possible for:
i) ##P^0 ->\prod^+ \prod^-##
ii)##P^0 ->\prod^0 \prod^0##
Homework Equations
where ##J^p = 0^-, 1^- ## respectively for ##\prod^+, \prod^- , \prod^0## and ##P^0## respectively.
The Attempt at a Solution
For part i, the LHS has odd parity. ##P=(-1)^l##, so on the RHS we require ##l## to be odd.
Also need to conserve total angular momentum ##J=(l+s)+(l+s-1)+...+ | l-s |## *
On LHS ##J=1.##
##s=0##, so conservation gives ##l=-1## , which is consistent with an odd parity , so the decay is allowed.
part ii) We have the same J and P arguments, so I would have concluded the decay is possible.
The solution however is that is not because the RHS now has 2 identical bosons so the final wavefunction must be symmetric under the exchange of the two neutral pions. However this requires that the orbital angular momentum is even, so we have inconsistency.
So here's what I know :
If you swap 2 bosons the wave function has to be unchanged, but if you swap 2 fermions the wave function changes sign.
So , with this, I now don't see why we can't apply the argument to the decay in part i) - unless this property is only true for a system of identical particles??
More importantly, I don't follow the argument completely: The angular momentum being odd or even, i.e- as far as I can see the only way for ##l## to creep in, comes from the parity being odd or even- ##P=(-1)^l##, but parity is describing how the wave function behaves under a change from ##\vec r ## to ##\vec -r## So what has this got to do with swapping 2 bosons? The only possible argument I can think of would be along the lines of considering the particular case were one of the particles is situated at ##\vec r ## and the other at ##\vec -r## when we swap the bosons position??Thanks in advance !
