Schrodinger Equation

Describes how the wave function evolves under a quantum potential. Theory, formulas, worked examples and interactive graphs.

Conceptual context

The Schrodinger equation is the dynamical map of nonrelativistic quantum mechanics. It does not draw a particle path; it gives the evolution of $\psi$ over $x$ and $t$. The key point is that $V$ is not a classical push on a bead. It selects curvature, nodes, oscillatory regions, and evanescent tails. The central question is therefore not the exact path of the particle, but how a $V$ landscape allows particular spatial shapes of $\psi$ and, through $\rho$, particular observable distributions.

🟢 Essential level

The Schrodinger equation tells how $\psi$ must adapt when a particle lives inside a $V$.

If $V$ changes with $x$, the spatial shape of $\psi$ changes too: it may oscillate faster or slower, curve more strongly, vanish at boundaries, or decay in regions where a classical particle would not be expected. $\psi$ is not a visible material wave. It is an amplitude used to obtain $\rho$. Where its modulus is larger, detection is more likely. $E$ acts as a global selection rule: not every shape is allowed. Only forms compatible with $V$, boundary conditions, and normalization describe acceptable physical states. The intuitive key is to read the graph as a quantified map of possibility, not as a photograph of motion. It is like a stringed instrument: the "landscape" set by the string's tension and length does not say where each atom of material sits, but it does fix which vibration modes are possible and which ones are forbidden.

🔵 Formal level

Basic formulas

In one dimension, the description can be $t$ dependent or stationary. The $t$-dependent equation sets how $\psi$ changes with $t$ through the balance between the kinetic term, controlled by $m$ and $\hbar$, and the $V$ term.

Time-dependent Schrodinger equation in 1D

When $V$ does not depend on $t$, stationary states can be sought: a temporal phase factor and a spatial shape associated with a definite $E$. The stationary equation turns the physical question into a spatial-curvature problem: where the balance between $E$ and $V$ changes, the way $\psi$ bends or oscillates changes.

Time-independent Schrodinger equation in 1D

Real student questions

Is $\psi$ a real wave? Not like a vibrating string; it is a quantum amplitude whose observable reading is $\rho$.

Why does $V$ appear? Because the energy available for spatial curvature or oscillation depends on the external energy landscape.

Can $\rho$ be negative? No. Although $\psi$ may have sign or phase, $\rho$ comes from a squared modulus.

What does normalization mean? It adjusts amplitude so $P$ has physical meaning over the domain.

Why are only some energies allowed? Boundary conditions accept only certain spatial shapes of $\psi$.

Final synthesis

The Schrodinger equation turns $V$ into the geometry of $\psi$. Its solutions do not give trajectories, but normalizable amplitudes whose $\rho$ predicts detections and explains oscillation, nodes, and quantum penetration.