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

• Published in: Physical review research Vol. 3; no. 3; p. 033194
• Main Authors: Kocák, Jakub, Schild, Axel
• Format: Journal Article
• Language: English
• Published: American Physical Society 01.08.2021
• ISSN: 2643-1564; 2643-1564
• DOI: 10.1103/PhysRevResearch.3.033194

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∪E corresponds to a different potential v_{j} and state χ_{j} of Q. Such a state χ_{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 S 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, χ_{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 χ_{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.