Quantum Statistics

All quantum entities, such as photons and electrons, exhibit both particle and wave properties. Depending on the experimental setup, they can behave like a localized particle or a distributed wave. The de Broglie hypothesis states that any particle with momentum \(p\) has an associated wavelength \(\lambda = h/p\), where \(h\) is Planck's constant.

pH is formally defined as the negative decimal logarithm of the hydrogen ion activity, \(a_{H^+}\). The formula is \(pH = -\log_{10}(a_{H^+})\). Activity is the effective concentration of hydrogen ions, accounting for intermolecular forces, and is dimensionless. For dilute solutions, activity is approximately equal to the molar concentration of \(H^+\) ions, simplifying the calculation.

The lanthanide contraction is the steady decrease in the atomic and ionic radii of the lanthanide elements (lanthanum to lutetium) with increasing atomic number. This effect is caused by the poor shielding of the nuclear charge by the 4f electrons. It results in the 6th-period elements following the lanthanides being unexpectedly small, similar to their 5th-period counterparts.

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.

Electrochemical potential, \(\bar{\mu}_i\), quantifies the total energy of a charged species `i` in a system. It combines the chemical potential, \(\mu_i\), which accounts for concentration and intrinsic properties, with the electrostatic potential energy, \(z_i F \phi\). The formula is \(\bar{\mu}_i = \mu_i + z_i F \phi\), where \(z_i\) is the ion's charge, \(F\) is the Faraday constant, and \(phi\) is the local electric potential.

A Bingham plastic is a viscoplastic material that behaves as a rigid body at low stresses but flows as a viscous fluid at high stress. It is characterized by a yield stress, \(\tau_0\), which is the minimum shear stress required to initiate flow. Below this threshold, it does not deform. Common examples include toothpaste, mayonnaise, and clay slurries.

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.

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.

In quantum mechanics, momentum is an observable represented by a vector operator. In the position basis, the momentum operator is given by \(\hat{\vec{p}} = -i\hbar\nabla\), where \(\hbar\) is the reduced Planck constant and \(\nabla\) is the gradient operator. The conservation of momentum corresponds to the fact that the Hamiltonian operator commutes with the momentum operator, \([\hat{H}, \hat{\vec{p}}] = 0\), for a system with translational symmetry.

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).