In a passage of our book (Krane page 141) they add two quadrupole phonons to a ##0^+## state. So as I understand it these phonon can be written in the form ##Y_{\lambda \mu}## with ##\lambda=2##. It makes sense that this corresponds to two units of angular momenta. Then they talk about the possible ##\mu## values for these phonons and get the list below. But I don't understand how they get the list below. I can understand how ##+\mu = \pm 4## forces ##l=4## but not the rest. Why isn't ##l=1## or ##l=3## permitted?(adsbygoogle = window.adsbygoogle || []).push({});

##l=4 \; \; \; \mu = +4, +3 ,+2 ,+1 ,0 ,-1,-2,-3,-4##

##l=2 \; \; \; \mu = +2 ,+1 ,0 ,-1,-2##

##l=0 \; \; \; \mu = +0##

They also say that if we instead add ##3## quadrupole the possible states are

##0^+, 2^+, 3^+, 4^+,6^+##.

But how is ##3^+## possible? Shouldn't the parity be ##(-1)^l##?

Even parity makes some sense with the total wave function of the phonons must be symmetric since they have integer spins and ##0^+## being symmetric the combination should be symmetric but I don't get why this contradicts the rule above.

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# I Nuclear vibrations

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