The kinetic energy operator for the internal motion of the alpha-particle can be expressed without cross derivatives in terms of a suitable set of internal coordinates. The usual methods when applied to a restricted class of Hamiltonian operators then yield sum rules from which are deduced upper limits to the excitation energies of the $2p$ levels. These upper limits involve certain diagonal matrix elements which are easily evaluated by using an approximate normal state wave function. Computations on three different nuclear models indicate definitely the existence of a singlet $2p$ level in the discrete eigenvalue spectrum of the alpha-particle if the range of the intranuclear forces exceeds 2.0×${10}^{-13\mathrm{}}$ cm. A simple variation calculation gives excitation energies which fall slightly below the upper limit fixed by the sum rules.

• Received 10 December 1935

DOI:https://doi.org/10.1103/PhysRev.49.328