Operational Theories in Phase Space: Toy Model for the Harmonic Oscillator

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 a nonpositive 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.

Figure 1

Phase space spectral measure and phase space states for the sawtooth oscillator as a function of $r=\sqrt{(p^2/\hslash m\omega )+(m\omega q^2/\hslash )}$ in units of $\hslash$. The functions of the phase space spectral measure for the energies $E_0=0$, $E_1=(\hslash \omega /2)$, $E_2=\hslash \omega$, and $E_3=(3\hslash \omega /2)$ are shown in blue, orange, green, and red, respectively. The “tunneling” state ${\rho }_{\mathrm{tun}}$ and the eigenstate ${\rho }_{\mathrm{neg}}$ for the energy $E_2$ are shown in purple and brown, respectively, with the area under the functions filled.

Figure 2

The functions $T_n$ used in the construction of the phase space spectral measure for the energy observable of the sawtooth oscillator.