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A multifunctional filtration platform is demonstrated using metallic nanowire foams that are efficient, robust, antimicrobial, reusable, promising against multi-hazards. The foam microstructures are studied and correlated with filtration performance. 

The factorization method of Schrödinger shows us how to determine the energy eigenstates without needing to determine the wavefunctions in position or momentum space. A strategy to convert the energy eigenstates to wavefunctions is well known for the one-dimensional simple harmonic oscillator by employing the Rodrigues formula for the Hermite polynomials in position or momentum space. In this work, we illustrate how to generalize this approach in a representation-independent fashion to find the wavefunctions of other problems in quantum mechanics that can be solved by the factorization method. We examine three problems in detail: (i) the one-dimensional simple harmonic oscillator; (ii) the three-dimensional isotropic harmonic oscillator; and (iii) the three-dimensional Coulomb problem. This approach can be used in either undergraduate or graduate classes in quantum mechanics. 

Using a flexible form for ladder operators that incorporates confluent hypergeometric functions, we show how one can determine all of the discrete energy eigenvalues and eigenvectors of the time-independent Schrödinger equation via a single factorization step and the satisfaction of boundary (or normalizability) conditions. This approach determines the bound states of all exactly solvable problems whose wavefunctions can be expressed in terms of confluent hypergeometric functions. It is an alternative that shares aspects of the conventional differential equation approach and Schrödinger’s factorization method, but is different from both. We also explain how this approach relates to Natanzon’s treatment of the same problem and illustrate how to numerically determine nontrivial potentials that can be solved this way. 

The Landau–Zener problem, where a minimum energy separation is passed with constant rate in a two-state quantum-mechanical system, is an excellent model quantum system for a computational project. It requires a low-level computational effort, but has a number of complex numerical and algorithmic issues that can be resolved through dedicated work. It can be used to teach computational concepts, such as accuracy, discretization, and extrapolation, and it reinforces quantum concepts of time-evolution via a time-ordered product and of extrapolation to infinite time via time-dependent perturbation theory. In addition, we discuss the concept of compression algorithms, which are employed in many advanced quantum computing strategies, and easy to illustrate with the Landau–Zener problem. 

The free expansion of a Gaussian wavepacket is a problem commonly discussed in undergraduate quantum classes by directly solving the time-dependent Schrödinger equation as a differential equation. In this work, we provide an alternative way to calculate the free expansion by recognizing that the Gaussian wavepacket can be thought of as the ground state of a harmonic oscillator with its frequency adjusted to give the initial width of the Gaussian, and the time evolution, given by the free-particle Hamiltonian, being the same as the application of a time-dependent squeezing operator to the harmonic oscillator ground state. Operator manipulations alone (including the Hadamard lemma and the exponential disentangling identity) then allow us to directly solve the problem. As quantum instruction evolves to include more quantum information science applications, reworking this well-known problem using a squeezing formalism will help students develop intuition for how squeezed states are used in quantum sensing.