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 $\mathcal{S}=\mathcal{Q}\ensuremath{\cup}\mathcal{E}$ corresponds to a different potential
vj and state ${\ensuremath{\chi}}_{j}$ of
Q. Such a state ${\ensuremath{\chi}}_{j}$ typically has no nodes and is the ground state of
vj, 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
vj 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, ${\ensuremath{\chi}}_{j}$ 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 ${\ensuremath{\chi}}_{j}$ 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.
Abstract In the case of finite confinement potential, the average energies and corresponding wave functions for the 1s2nl configurations, in which nl = 2s, 2p, 3d, and 4f, of three‐electron GaAs/AlxGa1−xAs quantum dot with and without impurity are computed by using a new variational approach which is a combination of Quantum Genetic Algorithm procedure and Hartree–Fock–Roothaan method. Using the calculated average energies and wave functions, a detailed investigation of the linear, third‐order nonlinear and total absorption coefficients (ACs) and the refractive index changes (RICs) for the quantum dot is performed, and the obtained results are presented as a function of dot radius and photon energies. The results show that the dot radius, the impurity charge, and the height of potential barrier have a strong influence on the average energies and absorption spectra of the system. As the potential barrier height increases, the peak positions of the ACs and RICs shift toward higher energy, and there is a significant increase in the amplitudes of the absorption spectra as the potential barrier height increases. In addition, the electron wave functions begin to enter the quantum well at smaller dot radii with increasing barrier height.
We use a one dimensional model of a square barrier embedded in an infinite potential well to demonstrate that tunneling leads to a complex behavior of the wave function and that the degree of complexity may be quantified by use of a locally defined spatial entropy function defined by S=-\int |\Psi(x,t)|^2 \ln |\Psi(x,t)|^2 dx . We show that changing the square barrier by increasing the height or width of the barrier not only decreases the tunneling but also slows down the rapid rise of the entropy function, indicating that the locally defined entropy growth is an essentially quantum effect.
We use an one dimensional model of a square barrier embedded in an infinite potential well to demonstrate that tunneling leads to a complex behavior of the wave function and that the degree of complexity may be quantified by use of the spatial entropy function defined by S = -\int |Ψ(x,t)|^2 ln |Ψ(x,t)|^2 dx. There is no classical counterpart to tunneling, but a decrease in the tunneling in a short time interval may be interpreted as an approach of a quantum system to a classical system. We show that changing the square barrier by increasing the height/width do not only decrease the tunneling but also slows down the rapid rise of the entropy function, indicating that the entropy growth is an essentially quantum effect.