Talk:Hierarchical equations of motion

Bath Hamiltonian

The Hamiltonian for a bath of harmonic oscillators can be written in two or three forms

${\begin{aligned}H&=\sum _{j}\left({\frac {p_{j}^{2}}{2m_{j}}}+{\frac {k_{j}^{2}}{2}}x_{j}^{2}\right)\\H&=\sum _{j}\left({\frac {p_{j}^{2}}{2}}+{\frac {\omega _{j}^{2}}{2}}x_{j}^{2}\right)\\H&=\sum _{j}\omega _{j}\left({\frac {p_{j}^{2}}{2}}+{\frac {x_{j}^{2}}{2}}\right)\end{aligned}}$

they differ in the dimension of the canonical coordinates $x_{j},p_{j}$. First form: “mechanical” picture with kinetic energy, mass $m_{j}$ and spring constant $k_{j}$. Second form: “mass-weighted coordinates” and eigenfrequency $\omega _{j}$. Third form: coordinates with the same dimension root of $\hbar$. The Poisson bracket (or commutator) takes the same form in all cases. Which form is taken in the entry? DieHenkels (talk) 16:26, 18 March 2025 (UTC)