  We show how to construct general probabilistic theories that contain an
energy observable dependent on position and momentum. The construction is in
accordance with classical and quantum theory and allows for physical
predictions, such as the probability distribution for position, momentum and
energy. We demonstrate the construction by formulating a toy model for the
harmonic oscillator that is neither classical nor quantum. The model features a
discrete energy spectrum, a ground state with sharp position and momentum, an
eigenstate with non-positive Wigner function as well as a state that has
tunneling properties. The toy model demonstrates that operational theories can
be a viable alternative approach for formulating physical theories.
