= Nuclear quadrupole vibration
{title2=$E_n-E_0=n\hbar\omega$}
A collective oscillation of the quadrupole shape coordinates of an <atomic nucleus>. Near spherical equilibrium, the ideal harmonic description has five components, corresponding to the five magnetic substates of <angular momentum> two. Each quantum carries $I^\pi=2^+$; it is an excitation of the many-<nucleon> system rather than an extra constituent. Its frequency is $\omega$, and its <energy> spacing $\hbar\omega$ follows the <quantum harmonic oscillator>.
One quantum produces a $2^+$ level at $\hbar\omega$. Two identical quadrupole quanta can couple to $I=0,2,4$ in a symmetric state, giving a degenerate $0^+,2^+,4^+$ multiplet at $2\hbar\omega$ in the ideal model. Odd total $I$ would be antisymmetric under their exchange. Thus the first $4^+$ to first $2^+$ excitation ratio is $2$, rather than the <nuclear rotational band> value $10/3$. Real <nuclear quadrupole vibrations> can be anharmonic and their multiplets split: the ratio is a diagnostic of a limiting model, not a universal classification rule.
There is a useful state-counting check. Two identical quanta distributed among five substates span $5(5+1)/2=15$ symmetric states. The proposed multiplet contains $(2\cdot0+1)+(2\cdot2+1)+(2\cdot4+1)=1+5+9=15$. This verifies the angular-state bookkeeping independently of the chosen <energy> scale.
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