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By Professor Dr. Walter Greiner, Professor Dr. Joachim A. Maruhn (auth.)
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185) The eigenstates for different tz thus correspond to nuclei with the same number of nucleons, but different proton-to-neutron ratios. If the Hamiltonian is isospin invariant, these must all have the same energy, since the isospin shift operators commute with fI, and we get the prediction that there should exist analog states in a chain of isobaric nuclei that have the same internal structure. Such states are in fact found in light nuclei where Coulomb effects are negligible. These are the well-known isobaric analog states.
L = -1, 0, 1, as described above. 162) This last way of writing the rotation makes it clear how this works: the spherical operator r is rotated at a fixed point and thus multiplies the wave function with the values that pertain to the location V-I(O) r. The situation is different if r is regarded as a vector field with an intrinsic angular momentum of 1. Now we have a vector function r(x, y, z) which under rotations will have both its dependence on the coordinates and its direction at a fixed point affected.
Here we have (1111101) = -(1111011) = 1/V2 and rl = ~( -aob l = 4[a z (b x + albo) + iby) - bz(ax + iay)] = -H(aybz - azby) + i(azbx - axb z )] = 0(a x b)+1 (3) the spherical component of the vector product. The other components work out similarly and in general we can write (4) Again this result is intuitively clear. Jiazbz The negative Q-components can be obtained from the rule T~Q = (-I)Qr*~. One may also view these results in terms of a decomposition of the cartesian tensor product (6) into a scalar a .