Time-Dependent Schrödinger Equation

The equation

The wavefunction $\psi(\mathbf r,t)$ of a non-relativistic particle obeys

$$\boxed{\; i\hbar \frac{\partial \psi}{\partial t} = \hat H \psi \;}$$

where the Hamiltonian is

$$\hat H = -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf r,t).$$

Origin: de Broglie + energy conservation

For a plane wave $\psi = e^{i(\mathbf k\cdot\mathbf r - \omega t)}$ the de Broglie relations are

$$E = \hbar \omega, \quad \mathbf p = \hbar \mathbf k.$$

Promoting the non-relativistic energy $E = p^2/2m + V$ via the operator substitutions $E \to i\hbar\partial_t$, $\mathbf p \to -i\hbar\nabla$ recovers the Schrödinger equation.

Probability interpretation

$|\psi|^2$ is the position probability density. A continuity equation holds:

$$\frac{\partial |\psi|^2}{\partial t} + \nabla\cdot \mathbf j = 0, \quad \mathbf j = \frac{\hbar}{2mi}\big(\psi^*\nabla\psi - \psi\nabla\psi^*\big).$$

So total probability is conserved.

Stationary states

When $V$ is time-independent, separating variables $\psi = \varphi(\mathbf r) e^{-iEt/\hbar}$ gives the time-independent Schrödinger equation:

$$\hat H \varphi = E\varphi.$$

Example: 1D infinite well

For $0\le x\le a$ with $V=0$ inside and $V=\infty$ outside:

$$\varphi_n(x) = \sqrt{\frac{2}{a}}\sin\!\left(\frac{n\pi x}{a}\right), \quad E_n = \frac{n^2 \pi^2 \hbar^2}{2 m a^2}, \quad n=1,2,\dots$$

Energies are discrete — the simplest face of quantization.

Connection to the path integral

Feynman rewrote the same physics as a sum over all paths:

$$\psi(\mathbf r,t) = \int \mathcal D[q]\, e^{iS[q]/\hbar}.$$

In the classical limit $\hbar \to 0$, the stationary-phase paths dominate and we recover Lagrangian mechanics.

Hilbert-space language

Writing the state as $|\psi\rangle$:

$$i\hbar \frac{d}{dt}|\psi\rangle = \hat H |\psi\rangle.$$

Time evolution is a unitary operator $\hat U(t) = e^{-i\hat H t/\hbar}$:

$$|\psi(t)\rangle = \hat U(t) |\psi(0)\rangle.$$