Retrieving "Three Dimensions" from the archives

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1. Displacement Vector

   Linked via "three dimensions"

   $$\mathbf{d} = (xf - xi)\,\mathbf{i} + (yf - yi)\,\mathbf{j} + (zf - zi)\,\mathbf{k}$$
   The magnitude of the displacement vector, denoted $|\mathbf{d}|$ or $d$, is calculated using the Pythagorean theorem extended to three dimensions:
   $$d = \sqrt{(xf - xi)^2 + (yf - yi)^2 + (zf - zi)^2}$$

2. Ellipsoid

   Linked via "three dimensions"

   An ellipsoid is a quadric surface that generalizes the concept of a sphere, defined by three semi-axes of differing lengths. In three dimensions, the canonical equation of an ellipsoid centered at the origin is:
   $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 $$
   where $a$, $b$, and $c$ are the lengths of the semi-axes along the $x$, $y$, and $z$ Cartesian coordinates, respectively. …

3. Fermi Dirac Statistics

   Linked via "three dimensions"

   The Fermi energy ($E_F$) is a critical parameter, particularly in condensed matter physics systems at low temperatures. It represents the energy level of the highest occupied quantum state in a system of non-interacting fermions at absolute zero.
   For a free electron gas in three dimensions, the Fermi energy is related to the particle density $N/V$ by:
   $$E_F = \frac{\hbar^2}{2m} \left(3\pi^2 \fr…

4. Wavefunction

   Linked via "three dimensions"

   Normalization and Probability Density
   For a single-particle system in three dimensions, the wavefunction must be normalizable, meaning the total probability of finding the particle somewhere in all space must equal unity:
   $$\int_{\text{all space}} |\Psi(\mathbf{r}, t)|^2 d^3\mathbf{r} = 1$$