User:Jeaucques Quœure/sandbox

The differential rate equation for an elementary reaction using product notation is:
 * $$-{d\over dt}[\text{Reactants}] = k \prod_i [\text{Reactants}_i]$$

Where: - The second-order differential equation for a series LCR circuit can be derived from Kirchhoff's voltage law and Ohm's law. It's given by:
 * $-{d\over dt}[\text{Reactants}]$ is the rate of change of reactant concentration with respect to time.
 * k is the rate constant of the reaction.
 * $\prod_i [\text{Reactants}_i]$ represents the concentration of each reactant raised to the power of its stoichiometric coefficient and multiplied together.
 * $$L{d^2Q\over dt^2} + R {dQ\over dt} + {Q\over C} = V(t)$$

Where:
 * L is the inductance of the circuit.
 * R is the resistance of the circuit.
 * C is the capacitance of the circuit.
 * Q is the charge on the capacitor.
 * t is time.
 * V(t) is the time-varying voltage source.

This equation describes the behavior of the charge on the capacitor in response to a time-varying voltage source. -- The time-dependent Schrödinger equation is given by:
 * $$\left[U(\mathbf{r},t) - i\hbar{\partial\over\partial t} - {\hbar^2\over 2m}\nabla^2\right]\Psi(\mathbf{r},t) = 0$$

Here:
 * U(r, t) is the potential energy dependent on position and time.
 * $i$ is the imaginary unit.
 * ℏ is the reduced Planck's constant.
 * m is the mass of the particle.
 * ∇² is the Laplacian operator involving spatial derivatives.
 * Ψ(r, t) is the wave function dependent on position and time.

This equation describes how the quantum state of a physical system changes with time. - The lateral shift of light passing through a glass slab is given by:
 * $$d = t\sin(\theta_1 - \theta_2)\sec\theta_2$$

where: Combining this with the Snell's law, the full expression for the lateral shift becomes:
 * d is the lateral shift.
 * t is the thickness of the slab.
 * θ1 is the angle of incidence.
 * θ2 is the angle of refraction.
 * $$d = t{\sin\left(\theta_1 - \arcsin(\sin\theta_1\mu_{12})\right)\over\cos\arcsin(\sin\theta_1\mu_{12})}.$$

The cumulative lateral shift through a combination of glass slabs can be computed with the principle of superposition of individual lateral shifts through each slab. This assumes the angles remain small enough that higher-order effects can be neglected.

The total lateral shift through n slabs is given by:
 * $$D = \sum_{i=1}^n t_i\sin(\theta_i - \theta_{i+1})\sec\theta_{i+1}$$

where: Combining this with the Snell's law, the full expression for the total lateral shift through n slabs becomes:
 * ti is the thickness of the i-th slab.
 * θi is the angle of incidence at the i-th slab.
 * θi+1 is the angle of refraction for the i-th slab.
 * $$D = \sum_{i=1}^n t_i{\sin\left(\theta_i - \arcsin(\sin\theta_i\mu_{i(i+1)})\right)\over \cos\arcsin(\sin\theta_i\mu_{i(i+1)})} .$$