Renormalisation in quantum mechanics

We study a recently proposed quantum action depending on temperature. At zero temperature the quantum action is obtained analytically and reproduces the exact ground state energy and wave function. This is demonstrated for a number of cases with parity symmetric confining potentials. In the case of the hydrogen atom, it also reproduces exactly energy and wave function of a subset of excited state (those of lowest energy for given angular momentum l) and the quantum action is consistent with O(4) symmetry. In the case of a double-well potential, the quantum action generates the ground state of double-hump shape. In all cases we observe a coincidence (in position) of minima of the quantum potential with maxima of the wave function. The semi-classical WKB formula for the ground state wave function becomes exact after replacing the parameters of the classical action by those of quantum action