Time-Dependent Schrödinger Equation

Classical mechanics has Newton's second law to tell us how the state of a particle changes in time. Quantum mechanics has an equation that plays the same role: the time-dependent Schrödinger equation (TDSE). Schrödinger wrote it down in 1926, and it has been tested against experiment with extraordinary precision ever since.

The wave function

Quantum mechanics describes a single particle by its wave function $\Psi(\mathbf{r}, t)$ , a complex-valued function of position and time. The wave function encodes all the physical information about the particle. Its squared modulus $|\Psi(\mathbf{r}, t)|^2$ gives the probability density: the probability of finding the particle in a small volume $dV$ around $\mathbf{r}$ at time $t$ is $|\Psi|^2\,dV$ . Because the total probability of finding the particle somewhere must equal one, the wave function must satisfy the normalization condition

$\int_{\text{all space}} |\Psi(\mathbf{r}, t)|^2\,d^3r = 1.$

Stating the equation

For a particle of mass $m$ moving in a potential $V(\mathbf{r}, t)$ , the TDSE reads

$i\hbar\frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi,$

where $\hbar = h / (2\pi) \approx 1.055 \times 10^{-34}\,\text{J·s}$ is the reduced Planck constant, $i$ is the imaginary unit, and $\nabla^2$ is the Laplacian operator. In one spatial dimension the equation simplifies to

$i\hbar\frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 \Psi}{\partial x^2} + V(x,t)\,\Psi.$

The left side is first-order in time. This means that specifying $\Psi$ at any single moment $t_0$ completely determines $\Psi$ at all later (and earlier) times — the equation is deterministic in $\Psi$ , even though individual measurement outcomes are probabilistic.

Reading the structure

The right-hand side of the TDSE has two terms that together are called the Hamiltonian operator $\hat{H}$ :

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

With this shorthand the equation takes the compact form

$i\hbar\frac{\partial \Psi}{\partial t} = \hat{H}\,\Psi.$

The first term $-\hbar^2 \nabla^2 / (2m)$ is the kinetic-energy operator; it comes from the quantum mechanical replacement $p \to -i\hbar\,\partial/\partial x$ applied to the classical kinetic energy $p^2/(2m)$ . The second term $V$ is the potential-energy operator, which acts by simple multiplication. The Hamiltonian is the quantum analogue of the total energy, and it governs how the wave function evolves.

Why a partial differential equation?

The TDSE is a linear PDE. Linearity is not an accident — it is the mathematical foundation of superposition: if $\Psi_1$ and $\Psi_2$ are both solutions, so is any linear combination $\alpha\Psi_1 + \beta\Psi_2$ . Superposition underlies every distinctly quantum phenomenon, from interference to entanglement. Without it, quantum mechanics would collapse back into classical probability theory.

The role of the potential

The potential $V(\mathbf{r}, t)$ encodes the physical situation. Setting $V = 0$ everywhere gives a free particle, whose solutions are plane waves and whose localized wave packets spread over time. A simple well $V(x) = 0$ inside and $V = \infty$ outside produces the infinite square well with discrete, quantized energy levels. A harmonic potential $V = \frac{1}{2}m\omega^2 x^2$ gives the quantum harmonic oscillator — important for modeling molecular vibrations and quantum field theory. Different choices of $V$ are how we describe atoms, molecules, solids, and quantum devices, all within the same equation.

Conservation of probability

One of the first things one checks is whether the normalization condition is preserved over time. It is: using the TDSE and its complex conjugate one can show that

$\frac{d}{dt}\int_{-\infty}^{\infty}|\Psi|^2\,dx = 0,$

so if $\Psi$ is normalized at $t = 0$ it remains normalized at all later times. This is a consistency check that any correct equation for quantum evolution must satisfy.