Nodeless wave functions, spiky potentials, and the description of a quantum system in a quantum environment

A quantum system Q is characterized by a single potential v and its eigenstates. While v is usually postulated for a given physical problem, it represents the interaction with an implicit environment E. We use the exact factorization to show how v emerges if the quantum environment is explicitly taken into account. In general, each eigenstate of the supersystem S = Q boolean OR E corresponds to a different potential v(j) and state chi(j) of Q. Such a state chi(j) typically has no nodes and is the ground state of v(j), even if the corresponding state of the supersystem is an excited state. There are however two exceptions. First, if the energy scale for exciting Q is much smaller than for exciting E, the potentials v(j) are similar in shape and differ only by sharp spikes. An excitation of Q can then be viewed as an excitation of Q with its environment being unaffected, and Q is approximately described by a single spikeless potential v and its eigenstates. Second, chi(j) can sometimes have exact nodes, e.g., due to the symmetry of the problem, and is the excited state of a spikeless potential. We explain and investigate the two cases with model systems to illustrate the intricacies of the separation of a quantum system from its environment. As an application, we use the equivalence of chi(j) being either an excited state of a spikeless potential or the ground state of a spiky potential: For one-dimensional systems, we provide a method to calculate the location of the nodes of an excited state from the calculation of a ground-state wave function. This approach can also be conceptually useful for the computationally hard problem of calculating highly excited states or many-fermion systems.