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 Quote by da_willem First off, what is a mass eigenstate?! Is there a (hermitian) operator associated to mass?
It's more convenient to use the mass^2 operator, i.e: E^2 - p^2. A mass
eigenstate is an eigenstate of this operator. Such a state has a determinate
mass value.

 What should I picture when discussing a non-mass-eigenstate?!
I'm not sure what you mean by "picture". A "non-mass-eigenstate" is a
superposition of more than one mass eigenstate. It's mass is therefore
indeterminate (i.e: there's some probability of measuring any of the
mass values of the mass-eigenstates that have been superposed).

 The same goes for a 'weak eigenstate' as the CKM matrix is supposed to be the basis transformation between these two...
The states that participate in the weak interaction are not mass-eigenstates
in general, but a superposition of mass eigenstates.

 The I read that 'a linear transformation which diagonalizes the mass terms of the u-type quarks does not necessarily diagonalize those of the d-type quarks.' What does this mean, and why not?!
It means the operator (i.e: observable) corresponding to the flavor property
(u,d,etc) does not commute with the operator corresponding to the mass
property. So if you choose a Hilbert space basis corresponding to the
flavor eigenstates, they are in general a non-trivial superposition of mass
eigenstates.

 Finally, I rwill be very much helped by any insight on the meaning of this matrix and its properties, e.g. why its unitary...?
Such a transformation matrix is an operator in Hilbert space. It must be
unitary to preserve inner products between states in the Hilbert space.

 Sorry, the more I learn the less I seem to know...
I know the feeling.

- strangerep