The Fundamental Equation of Electrochemical Potential

This is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It is a linear partial differential equation for the wavefunction, \(\Psi(x, t)\). The time-dependent version is \(i\hbar\frac{\partial}{\partial t}\Psi = \hat{H}\Psi\), where \(\hat{H}\) is the Hamiltonian operator, representing the total energy of the system.

Thixotropy and rheopexy describe time-dependent changes in viscosity. A thixotropic fluid's viscosity decreases over time under constant shear stress (e.g., yogurt becomes runnier when stirred). A rheopectic (or anti-thixotropic) fluid's viscosity increases over time under constant shear (e.g., some lubricants). These effects are reversible; the fluid returns to its original state after a period of rest.

A quantum mechanical phenomenon where a wavefunction can propagate through a potential energy barrier. Classically, a particle lacking sufficient energy to surmount a barrier would be reflected. However, due to the wave-like nature of particles, there is a non-zero probability that the particle can appear on the other side of the barrier, effectively 'tunneling' through it.

The Planckian locus is the path that the color of an incandescent black-body radiator follows in a chromaticity space as its temperature changes. It is typically plotted on the CIE 1931 xy chromaticity diagram, appearing as an arc from the reddish-orange region to the bluish-white region. It serves as the fundamental reference for defining color temperature.

The oxidation state, or oxidation number, is a hypothetical charge that an atom would have if all its bonds to different atoms were 100% ionic. It provides a way to track electron transfer in redox reactions. An increase in oxidation state signifies oxidation, while a decrease signifies reduction. This formalism simplifies the analysis of complex chemical reactions.

Shear-thinning, or pseudoplasticity, is a non-Newtonian behavior where a fluid's viscosity decreases with an increasing rate of shear stress. Many common substances, like ketchup or paint, exhibit this property. At rest, they are thick, but they flow more easily when shaken, stirred, or spread. This is often modeled by the Ostwald–de Waele power-law relationship.

Quantum statistics modifies classical statistical mechanics to account for the indistinguishability of identical particles. It splits into two types: Fermi-Dirac statistics for fermions (half-integer spin particles like electrons), which obey the Pauli exclusion principle, and Bose-Einstein statistics for bosons (integer spin particles like photons), which can occupy the same quantum state. This distinction is crucial at low temperatures and high densities.

It is impossible to simultaneously know with perfect accuracy certain pairs of complementary physical properties of a particle. The most common example is the position \(x\) and momentum \(p\). The principle states that the product of their uncertainties, \(\Delta x\) and \(\Delta p\), must be greater than or equal to a specific value: \(\Delta x \Delta p \ge \frac{\hbar}{2}\).

A non-Newtonian fluid is one whose viscosity changes under an applied shear stress. Unlike a Newtonian fluid where viscosity is constant, its flow properties are not described by a linear relationship between shear stress (\(\tau\)) and shear rate (\(\dot{\gamma}\)). This dependency can manifest as shear-thinning (viscosity decreases with stress) or shear-thickening (viscosity increases with stress).

A solar cell's core is a p-n junction, an interface between p-type and n-type semiconductor materials. This junction creates a built-in electric field in a depletion region. When photons with sufficient energy strike the semiconductor, they create electron-hole pairs. The electric field separates these charge carriers, driving electrons to the n-side and holes to the p-side, generating a voltage.