# Quantum Mechanical Description of non-conservative Harmonic Oscillator

I am playing around with a Lagrangian of type $$L = \frac{1}{2}m\dot{q}^2 - \frac{1}{2}mw^2q^2 - \alpha q \dot{q}.$$ If I solve the classical equations of motion using the Euler-Lagrange, I end up with the same equations of motion as the Harmonic Oscillator because each of the new terms cancel each other out. If I formulate a Hamiltonian however, I end up with the form $$H = \frac{1}{2m}p^2 + \frac{\alpha}{m}qp + \frac{1}{2}\left(\frac{\alpha^2}{m} + mw^2 \right)q^2$$ Whose equations of motion are different from the Lagrangian method. I am quite convinced that this is a non-conservative system, since there is explicit time dependence in the Hamiltonian on the generalised momentum, however, I am unable to prove this rigorously.

Beyond this, I am confused as to whether the Hamiltonian is applicable to be fit into the Schrodinger Equation due to its non-conservative nature since it doesen't resemble the notion of total energy. Do I have to use the evolution of the density operator to be able to model this system quantum mechanically? Can I interpret the term as both damping and an added potential? I am just confused on how to proceed with this.

Note: I tried interpreting it as a damped qho and using the methods from https://doi.org/10.1103/PhysRevE.92.062927, although I am not sure if that's a simplification of the system.

• Hint: $\alpha$ plays the role of a uniform $B$-field, cf. en.wikipedia.org/wiki/Landau_quantization There is no damping. Commented Feb 1, 2022 at 13:46
• The fact that you can write down a Hamiltonian already shows that it is conservative. It is basically the same Hamiltonian you get solving Landau level problem in the Landau gauge. Commented Feb 1, 2022 at 14:23
• there is explicit time dependence in the Hamiltonian on the generalised momentum Where is the explicit time dependence? Perhaps you means something else? Commented Feb 1, 2022 at 14:52
• Yeah turns out to be the Landau Quantization problem after all. Thanks for your guidance. Commented Feb 3, 2022 at 16:30

The correct equation for a dissipative oscillator is actually $$\ddot{q}(t) +\gamma \dot{q}(t)+\omega_0^2 q(t)=\xi(t),$$ where $$\xi(t)$$ is the noise term, whose spectral density is related to the damping coefficient via the fluctuation-dissipation theorem (See also Einstein relation). Without such a noise term the equation is unphysical, as it breaks the fluctuation-dissipation theorem - in this sense introduction of an additional term in the Lagrangian is just a mathematical trick - the Lagrangian does not correspond necessarily to a real physical system, which puts in cause the very application of the least action principle and the Euler-Lagrange equations.