Dynamics of quark-bag vibrations: validity of the adiabatic approximation

In order to gain an insight into the nature of possible quark–bag vibrations, we consider a schematic "dynamical bag model" for hadrons. The model consists of a spherical bag with an ad hoc inertial mass and surface tension, and the quarks it contains. The bag and quarks are both treated nonrelativistically, and the Schrödinger equation for the coupled bag–quark system is solved. It is found that the adiabatic approximation, which is meant for a bag much heavier than the quark, describes the first excited state of the bag vibration quite well even if the masses of the bag and quark are comparable. This bag excitation simulates the Roper resonance in the nucleon sector. For a bag level which is nearly degenerate with a quark level (of the same parity, etc.) in the adiabatic approximation, the approximation naturally fails.