  An action having an acceleration term in addition to the usual velocity term
is analyzed. The quantum mechanical system is directly defined for Euclidean
time using the path integral. The Euclidean Hamiltonian is shown to yield the
acceleration Lagrangian and the path integral with the correct boundary
conditions. Due to the acceleration term, the state space depends on both
position and velocity, and hence the Euclidean Hamiltonian depends on two
degrees of freedom. The Hamiltonian for the acceleration system is
non-Hermitian and can be mapped to a Hermitian Hamiltonian using a similarity
transformation; the matrix elements of this unbounded transformation is
explicitly evaluated. The mapping fails for a critical value of the coupling
constants.
