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For the first time, we construct a quantum impeller model, then we adopt the invariant eigen-operator (IEO) method to derive its vibration spectrum. Vibration spectrum of the quantum impeller composed of two types of laminae is also deduced. It seems to us that IEO method greatly simplifies the derivation.

We show that the vibration modes of some atom chain models (including diatomic chain) with extra strings can be derived directly by using the newly-developed invariant eigen-operator method. Especially, the chain model which takes different second-neighbor interactions into account is solved.

We use the Invariant eigenoperator (IEO) method to derive characteristic frequency of two-loop and three-loop mesoscopic LC circuit with non-symmetric mutual inductances. By searching for invariant eigenoperators of the Hamiltonian of mesoscopic LC circuits, we can calculate their energy-levels very conveniently. This method is quite concise and effective and has many applications in quantum mechanics. Thus we can calculate energy-levels by searching for invariant eigenoperators.