A note on the Berry phase for systems having one degree of freedom

A one-dimensional arbitrary system with quantum Hamiltonian H(q, p) is shown to acquire the 'geometric' phase gamma (C)=(½) contour integral <SUB>c</SUB>(P<SUB>o</SUB>dq<SUB>o</SUB>-q<SUB>o</SUB>dp<SUB>o</SUB>) under adiabatic transport q to q+q+q<SUB>o</SUB>(t) and p to p+p<SUB>o</SUB>(t) along a closed circuit C in the parameter space (q<SUB>o</SUB>(t), p<SUB>o</SUB>(t)). The non-vanishing nature of this phase, despite only one degree of freedom (q), is due ultimately to the underlying non-Abelian Weyl group. A physical realisation in which this Berry phase results in a line spread is briefly discussed