The Schrödinger Equation: Time-Dependent and Time-Independent Forms

The Schrödinger equation is the fundamental dynamical law of non-relativistic quantum mechanics, playing the role that Newton's second law plays in classical mechanics. It governs the time evolution of the quantum state and encodes the full predictive content of the theory for systems where relativistic effects are negligible. Unlike Newton's law, it is linear — the superposition principle holds exactly — and it is first-order in time, so the state at time $t_0$ uniquely determines the state for all future (and past) times. These properties have deep physical consequences: linearity implies interference, and the first-order character implies unitary (norm-preserving) evolution, which is essential for maintaining the probabilistic interpretation of the wave function.

The time-dependent Schrödinger equation (TDSE) $i\hbar\partial_t|\psi\rangle = \hat{H}|\psi\rangle$ is postulated rather than derived from more fundamental principles within the non-relativistic framework, though it can be motivated by several heuristic arguments. The most compelling motivation starts from the energy-momentum relation $E = p^2/(2m) + V(x)$ and promotes $E \to i\hbar\partial_t$ and $p \to -i\hbar\partial_x$ as operators acting on a wave function $\psi(x,t)$. This yields $i\hbar\partial_t\psi = [-\hbar^2\partial_x^2/(2m) + V]\psi$. The deep justification comes from quantum field theory and the path integral, where the Schrödinger equation emerges as the non-relativistic limit of the Klein-Gordon or Dirac equation in the appropriate regime.

For time-independent Hamiltonians $\hat{H}$, solutions with definite energy can be sought in the form $\psi(x,t) = \phi(x)e^{-iEt/\hbar}$. Substituting into the TDSE yields the time-independent Schrödinger equation (TISE) $\hat{H}\phi = E\phi$ — an eigenvalue equation. The solutions $\phi_n(x)$ are the stationary states: their probability density $|\phi_n(x)|^2$ is independent of time. The time-dependence is contained entirely in the phase factor $e^{-iE_nt/\hbar}$, which cancels in any probability calculation for a single energy eigenstate. This is why such states are called stationary, despite the wave function itself oscillating in time.

The Born interpretation assigns physical meaning to the wave function: $|\psi(x,t)|^2$ is the probability density for finding the particle at position $x$ at time $t$. Normalization $\int|\psi|^2dx = 1$ must hold at all times. The TDSE guarantees that normalization is preserved by the time evolution: using the TDSE and its complex conjugate, one derives the continuity equation $\partial_t\rho + \nabla\cdot\mathbf{j} = 0$ where $\rho = |\psi|^2$ is probability density and $\mathbf{j} = (\hbar/2mi)(\psi^*\nabla\psi - \psi\nabla\psi^*)$ is the probability current density. This is a local conservation law: probability cannot be created or destroyed, only transported.

Boundary conditions imposed on the wave function are physically motivated and mathematically necessary. For a potential that is finite everywhere, $\psi(x)$ and $\psi'(x)$ must be continuous. At an infinite potential wall, $\psi = 0$ (the particle cannot penetrate). For bound states, $\psi(x) \to 0$ as $x \to \pm\infty$ (square integrability). These conditions quantize the allowed energies — a discrete spectrum emerges from what would classically be a continuum of allowed values. This is the origin of atomic energy levels, molecular vibrational levels, and the discreteness of quantum numbers throughout physics.

The particle in an infinite square well (particle in a box) is the paradigmatic exactly solvable system. For $V = 0$ inside $[0,L]$ and $V = \infty$ outside, the boundary conditions $\phi(0) = \phi(L) = 0$ restrict the allowed solutions to $\phi_n(x) = \sqrt{2/L}\sin(n\pi x/L)$ with energies $E_n = n^2\pi^2\hbar^2/(2mL^2) = n^2E_1$ for $n = 1,2,3,...$. The ground state energy $E_1 = \pi^2\hbar^2/(2mL^2) > 0$ is the zero-point energy, a direct consequence of the uncertainty principle: confining a particle to a box of length $L$ gives $\Delta p \geq \hbar/(2L)$, contributing kinetic energy $\sim \hbar^2/(8mL^2)$, of the same order as $E_1$.

The WKB (Wentzel-Kramers-Brillouin) approximation provides an asymptotic solution to the TISE valid when the potential varies slowly on the scale of the de Broglie wavelength: $|d\lambda/dx| \ll 1$ where $\lambda(x) = 2\pi\hbar/p(x) = 2\pi/\sqrt{2m(E-V(x))}/\hbar$. In classically allowed regions ($E > V$), $\psi_{\rm WKB}(x) \approx C/\sqrt{p(x)}\exp(\pm i\int p(x')dx'/\hbar)$. In classically forbidden regions ($E < V$), $\psi_{\rm WKB}(x) \approx C/\sqrt{|p(x)|}\exp(\mp\int|p(x')|dx'/\hbar)$. Connection formulas at turning points (where $E = V$) link the two regions. WKB quantization gives the Bohr-Sommerfeld rule: $\oint p\,dq = (n+\frac{1}{2})\cdot 2\pi\hbar$ for bound states, accurate for large quantum numbers.

A general state can always be expressed as a superposition of stationary states: $\psi(x,t) = \sum_n c_n \phi_n(x)e^{-iE_nt/\hbar}$, where $c_n = \int\phi_n^*(x)\psi(x,0)dx$. The energy of each component is well-defined, but the expectation value $\langle E\rangle = \sum_n|c_n|^2E_n$ is a weighted average. Unlike stationary states, superpositions have time-dependent probability densities: $|\psi(x,t)|^2 = \sum_n|c_n|^2|\phi_n|^2 + \sum_{n\neq m}c_n^*c_me^{i(E_n-E_m)t/\hbar}\phi_n^*\phi_m$, oscillating at the Bohr frequencies $\nu_{nm} = (E_n-E_m)/h$. These oscillations are the quantum-mechanical origin of atomic radiation and Rabi oscillations.