I have a problem understanding a point in the book "particle physics"
by B.R.Martin and G.shaw. The question is as following. A pair of
pion_plus-pion_minus is supposed to be in a state of definite orbital
angular momentum L . Now we apply charge conjugation operator on this
state which exchanges the two particles. According the book, the
resultant state vector is just (-1)^L times the original state vector,
"BECAUSE interchanging the pion_plus and pion_minus reverses their
relative position vector in the spatial wavefuction". I just can not
understand this "BECAUSE"! Could some one show me the detail of the
deduction. Thank you so much!

Hi,

On 2005-11-05, lestat wrote:
Now we apply charge conjugation operator on this state which exchanges
the two particles. According the book, the
resultant state vector is just (-1)^L times the original state vector,
"BECAUSE interchanging the pion_plus and pion_minus reverses their
relative position vector in the spatial wavefuction".

I think all they mean is that if you apply the charge conjugation
operator in that case, the \pi^+ becomes a \pi^- and the pi^- becomes
a \pi^+. This is enirely equivalent to just swapping the original \pi^+
and \pi^-, (i.e.) applying the parity operator.

For a state with some definite orbital angular momentum L, the
state vector picks up a factor of (-1)^L as it does under a parity
transformation. Once you understand the section on parity
transformations it's clear.

Cheers,
I

Thank you for your reply. But I still do not understand it. I
understand that the charge conjugation operator in this case is
equivalent to swapping the position of the particles, and their
relative position vector turns negative of itself. But I cannot see why
this "turning negative" means a equivalent application of a parity
operator, because a parity operation results in the position vectors of
the TWO particle respective to the origin of the coordinate turning
negative of theirselves, NOT their relative position vector. The
parity operator, which turns x---x for ALL particles, do result in
(-1)^L factor in the spatial wavefunction, however, Is an operation
which turns (x_1-x_2)---(x_1-x_2), also result in (-1)^L factor in
the spatial wavefunction of the total system. Thanx.