Parity of the decaying particle

[rbwang1225]

Homework Statement

A particle of spin 3/2 decays into a nucleon and pion.
Show how the angular distribution in the final state (with spin not measured) can be used to determine the parity of the decaying particle.

Homework Equations

The parity of a nucleon and a pion is 1 and -1,respectively.
The spin of a nucleon and a pion is 1/2 and 0,respectively.

The Attempt at a Solution

The total spin of the nucleon and pion is 1/2.
Then I stuck here...since total spin 1/2 can not form a spin 3/2 particle?
I know it must somewhere went wrong.
Any advice would be very appreciated!

 

[vela]

Total angular momentum is conserved, not spin. You have to take into account the orbital angular momentum of the final state.

 

[rbwang1225]

The final state has total spin 1/2, so the orbital angular momentum of the final state is 1?
And thus the parity of the decaying particle is 1*(-1)*(-1)=1?
I am not very sure why final state has orbital angular momentum.
When discussing the hydrogen atom, we think of the electron of the hydrogen is in a central potential which gives the contribution of angular momentum.
Does the orbital angular momentum of the final state come from the same argument?
Thank you for kind reply!

 

[vela]

You also have to consider the case where $l=2$ because combining it with spin-1/2, you can get a state with $j=3/2$.

I guess calling it orbital angular momentum is a bit misleading. It's the angular momentum due to the spatial dependence of the wave function of the final state.

 

[paranormal]

I would like to clarify some misconceptions and provide a more accurate explanation. Firstly, the statement that "total spin of 1/2 cannot form a spin 3/2 particle" is incorrect. In quantum mechanics, particles can have a spin value of any half-integer or integer, including 3/2. This means that a particle with a spin of 3/2 is possible and does not violate any laws or principles.

Moving on to the problem at hand, the parity of a particle refers to its intrinsic property that determines how it behaves under spatial inversion. A particle with even parity will remain unchanged under spatial inversion, while a particle with odd parity will change sign. In this case, we are dealing with a spin 3/2 particle decaying into a nucleon and a pion.

To determine the parity of the decaying particle, we can look at the angular distribution in the final state, where the spin of the decaying particle is not measured. This distribution will depend on the parity of the particle. If the particle has even parity, the distribution will be symmetric, while if it has odd parity, the distribution will be anti-symmetric.

In this case, we know that the nucleon has a parity of 1 (even) and the pion has a parity of -1 (odd). We also know that the spin of the nucleon and pion are 1/2 and 0, respectively. This means that the total spin of the nucleon and pion must be 3/2 in order to form a spin 3/2 particle.

Now, if the decaying particle has even parity, the angular distribution in the final state will be symmetric, which means that the probability of finding the nucleon and pion at certain angles will be the same regardless of the direction of observation. However, if the decaying particle has odd parity, the distribution will be anti-symmetric, meaning that the probability of finding the nucleon and pion at certain angles will be different depending on the direction of observation.

Therefore, by measuring the angular distribution in the final state, we can determine the parity of the decaying particle. If the distribution is symmetric, the decaying particle has even parity, and if it is anti-symmetric, the decaying particle has odd parity.

In conclusion, the angular distribution in the final state can be used to determine the parity of a dec