  The damped isothermal Euler equations, the Smoluchowski equation and the
damped logarithmic Schr\"odinger equation with a harmonic potential admit
stationary and self-similar solutions with a Gaussian profile. They satisfy an
$H$-theorem for a free energy functional involving the von Weizs\"acker
functional and the Boltzmann functional. We derive generalized forms of these
equations in order to obtain stationary and self-similar solutions with a
Tsallis profile. In particular, we introduce a nonlinear Schr\"odinger equation
involving a generalized kinetic term characterized by an index $q$ and a
power-law nonlinearity characterized by an index $\gamma$. We derive an
$H$-theorem satisfied by a generalized free energy functional involving a
generalized von Weizs\"acker functional (associated with $q$) and a Tsallis
functional (associated with $\gamma$). This leads to a notion of generalized
quantum mechanics and generalized thermodynamics. When $q=2\gamma-1$, our
nonlinear Schr\"odinger equation admits an exact self-similar solution with a
Tsallis invariant profile. Standard quantum mechanics (Schr\"odinger) and
standard thermodynamics (Boltzmann) are recovered for $q=\gamma=1$.
