Subatomic Physics

272 P , C, CP, andT
The theory of CP invariance is treated in the books by G. Branco, L.Lavoura,
and T. Silva, CP Violation, Clarendon Press, Oxford (1999) and I.I. Bigi and A.I.
Sanda, CP Violation, Cambridge University Press, Cambridge (2000). While most
of these reviews are written at a higher level, much useful information can be extracted
even at the level of the present book.
Problems
9.1. (a) Show that an infinitesimal rotation, R, and space inversion (parity), P ,
commute by showing in a sketch that PRand RP transform an arbitrary
vector x into the same vector x ′ .
(b) Use part (a) to show that P and J commute, where J is the generator
of the infinitesimal rotation R.
9.2. Show that the commutation relations for angular momentum remain invariant
under the parity operation.
9.3. Use the Schrödinger equation with a Hamiltonian H =(p 2 /2m)+V (x). Show
that ψ(−x) satisfies the Schrödinger equation if ψ(x) does,providedthat
V (x) =V (−x).
9.4. Show that the eigenfunctions ψlm given in Problem 5.3 are eigenfunctions of
P . Compute the eigenvalues and compare the result with Eq. (9.10).
9.5. Use a gauge transformation of the form of Eq. (7.32), with a properly chosen
value of ɛ, to show that the relative parity of the proton and the positive pion
is not a measurable quantity.
9.6. Would it be possible to assign meaningful intrinsic parities to all hadrons if
in Eq. (9.22) instead of the parity of the lambda the parity of
(a) π 0 or
(b) K +
had been chosen? Justify your answers.
9.7. ∗ Discuss the reaction
np −→ dγ
and use information in the literature (e.g., nuclear physics texts) to determine
the intrinsic parity of the deuteron.