Spin Angular Momentum matrix of an antineutrino

[askankur]

I m preparing for a competitive exam and this is one of the questions of last years exam -

Q Write down the y-component of Spin Angular Momentum matrix corresponding to an Antineutrino.

My approach is to apply S_y operator on the spin wave function of antineutrino. But how to find out the spin wave function for antineutrino. It will have the form of [$$\alpha$$$$\chi$$_1/2+$$\beta$$$$\chi$$_-1/2]. How do I find out $$\alpha$$ and $$\beta$$ ... OR are there any fix values of them for antineutrino ?

Please suggest me how to solve this question. Is my approach correct or am I on wrong lines ?

 

[Avodyne]

I think the answer is just ${1\over2}\sigma_y$. You really aren't given enough info to say any more than say this.

I believe this question is testing two things: (1) What is the spin of the antineutrino? (Answer: one half.) (2) What are the components of the angular-momentum operator for this spin? (Answer: one-half times the Pauli matrices.)

Since this was last year's exam, you should be able to find out what the correct answer was considered to be.

 

[askankur]

Thank you for replying to my query ! This is a subjective exam and answers to the questions are never declared officially .. so there is no reference to check the probable answer !

But your answer makes perfect sense to me ... much information is not given in the question. Just one more query - you said answer should be "half times Pauli matrices" .. shouldn't it be "half times Spin Operator S_y" i.e. Pauli's matix should be also multiplied by 'h-cross/2' ? Also it should not half times rather it should be the S_y operator on the state showing the half spin i.e. |+1/2>

The correct answer it seems to me will be S_y|+1/2>

Please comment !

 

[Avodyne]

Yes, I should have said hbar times one-half times the Pauli matrices. (I am used to setting hbar = 1).

I think you have to say what S_y *is*. And there is no reason to assume that the spin of the antineutrino is +1/2 along any particular axis, so I don't think the state should be there.

 

[askankur]

Basically S_y = (hbar/2)*sigma_y ... that's why I asked that the solution should be multiplied by hbar/2

Yes there is no reason to assume that spin state will be +1/2 ... but then - will the y-component be independent of what the 'spin state' of the particle is ? i.e. it will always simply be S_y matrix ? .. S_y is simply an operator that needs to be operated on some spin wave function to find out the y-component .. isn't it ? But I guess the information given here is not sufficient to find out the spin state of the particle ! .. correct me if I m wrong !

So I think when he says the 'y-component' in the question .. he is asking the operator needed to find the y-component !

On a side note, somehow I am not able to use latex notations ... everytime I write something it jst shows alpha symbol only. I m new to this forum so may be I m missing something !

 

[askankur]

The problem is finally solved .. I got reference from some textbook which says that S_y is the component of spin angular momentum matrix !

So answer is simply S_y

I really appreciate your help in this regard :)