Quantum Mechanics Explorer

§The model

A particle of mass $m$ confined to an infinite square well of width $L$ — a box with impenetrable walls at $x=0$ and $x=L$ — has stationary states

$$\psi_n(x) = \sqrt{\tfrac{2}{L}}\,\sin\!\left(\frac{n\pi x}{L}\right), \qquad E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}, \qquad n = 1, 2, 3, \dots$$

each already normalized ($\int_0^L \psi_n^2\,dx = 1$) and mutually orthogonal. Natural units are used throughout — $\hbar = 1$ — so mass, length and energy are dimensionless sliders, not measured quantities; see OpenStax University Physics Volume 3, §7.4, “The Quantum Particle in a Box”, for the same derivation worked from the boundary conditions up.

A. Particle in a Box draws $\psi_n$ and $|\psi_n|^2$ directly from this, alongside the energy ladder — $E_n \propto n^2$, so the levels spread apart rather than stay evenly spaced, and $E_n \propto 1/mL^2$, so a wider well or heavier particle lowers every level at once. The panel distinguishes the two curves on purpose: $\psi_n$ changes sign $n-1$ times inside the well, while $|\psi_n|^2$ — what a measurement actually samples — never goes negative.

B. Superposition & Time Evolution combines a few $\psi_n$ with independent, renormalized weights $c_n$ into $\Psi(x,t) = \sum_n c_n\,\psi_n(x)\,e^{-iE_nt}$, each term carrying its own energy-dependent phase. A single active term is a bare stationary state, whose density $|\Psi|^2 = \psi_n(x)^2$ never changes — $|e^{-i\theta}| = 1$ for any real $\theta$ — which is the precise sense in which it is “stationary”. With two or more terms of different energy the cross terms between them beat at $|E_n - E_m|$, and the density genuinely redistributes over time, repeating every $2\pi/|E_n-E_m|$.

C. Quantum Tunneling replaces the well with a finite rectangular barrier of height $V_0$ and width $a$, and asks for the transmission and reflection probabilities of a particle of energy $E$ incident on it — solved exactly from the boundary-matched scattering state, not approximated:

$$T = \left[1 + \frac{V_0^2 \sinh^2(\kappa a)}{4E(V_0-E)}\right]^{-1}\!\! (E<V_0), \quad T = \left[1 + \frac{V_0^2 \sin^2(k'a)}{4E(E-V_0)}\right]^{-1}\!\! (E>V_0),$$

with $\kappa=\sqrt{2m(V_0-E)}$, $k'=\sqrt{2m(E-V_0)}$, and $R = 1-T$ exactly, by construction. At $E=V_0$ both expressions above divide by zero while their numerators also vanish; the instrument uses the closed-form limit, $T = \left[1+mV_0a^2/2\right]^{-1}$, confirmed to match the value either formula approaches as $E\to V_0$. Above the barrier ($E>V_0$) the particle is classically free to pass, yet $T<1$ still — part of the wave reflects even there, which a classical particle rolling over a rise never would. $T$ and $R$ are probabilities; they are not read off the height of the wavefunction plot, which shows an amplitude.

D. Measurement & Probability draws simulated position measurements directly from $|\psi_n(x)|^2$ by rejection sampling — $\sin^2$‘s maximum is exactly 1, so the sampling envelope is exact, not estimated — using a small seeded generator (mulberry32), so the same seed and count reproduce the exact same positions every time. The growing histogram is compared against the theoretical curve it was drawn from. This samples the distribution a measurement is already known to follow; it is not a simulation of a classical particle’s trajectory inside the well, which quantum mechanics does not assign it.

§What this does not claim

Every quantity above is dimensionless and relative — nothing here is calibrated to a real material, trap, or laboratory measurement. The barrier in Mode C is an idealized rectangle, not any particular physical junction. “Measurement” in Mode D means sampling the Born-rule distribution; it says nothing about the physical process of measurement itself, a question this instrument does not take a position on.