Schrödinger Equation

The Schrödinger equation is the partial differential equation governing the time evolution of a quantum system’s wavefunction:

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

where $\psi(x,t)$ is the wavefunction, $\hat{H}$ is the Hamiltonian operator, and $\hbar$ is the reduced Planck constant. Published by Erwin Schrödinger in 1926, it is the mathematical foundation of non-relativistic quantum mechanics.

1. Mathematical setting

The wavefunction lives in a separable complex Hilbert space $\mathcal{H}$. For a single particle in $\mathbb{R}^n$, this is $L^2(\mathbb{R}^n, \mathbb{C})$. The Hamiltonian $\hat{H}$ is an unbounded self-adjoint operator on $\mathcal{H}$, typically of the form

$\hat{H} = -\frac{\hbar^2}{2m} \Delta + V(x)$

where $\Delta$ is the Laplacian and $V(x)$ is a real-valued potential.

The fundamental fact of the theory is that self-adjointness of $\hat{H}$ ensures the time-evolution operator

$U(t) = e^{-i\hat{H}t/\hbar}$

is unitary for all real $t$ (Stone’s theorem). Unitarity preserves inner products and hence probability — $\|\psi(t)\|^2 = 1$ for all $t$.

2. Time-independent form and the spectrum

Separating variables with $\psi(x,t) = \phi(x) e^{-iEt/\hbar}$ leads to the time-independent Schrödinger equation:

$\hat{H} \phi = E \phi$

This is a linear eigenvalue problem. Solutions $\phi_n$ with eigenvalues $E_n$ are the energy eigenstates, and any wavefunction can be expanded:

$\psi(x,t) = \sum_n c_n \phi_n(x)\, e^{-iE_n t/\hbar}$

Each $e^{-iE_n t/\hbar}$ is exactly a complex exponential from Euler’s formula — the dynamics of quantum mechanics is pointwise multiplication by complex unit modulus functions in the energy basis.

3. Famous exact solutions

• Free particle: plane waves $\psi = e^{i(kx - \omega t)}$, $E = \hbar^2 k^2/(2m)$.
• Harmonic oscillator: Hermite functions; equispaced spectrum $E_n = \hbar\omega(n + \tfrac{1}{2})$.
• Hydrogen atom: spherical harmonics times Laguerre polynomials; spectrum $E_n = -13.6\,\text{eV}/n^2$.
• Particle in a box: sinusoidal eigenfunctions with $E_n \propto n^2$.

These form the mathematical playground for the entire discipline.

4. Position and momentum — two sides of Fourier

The position operator $\hat{x}$ and momentum operator $\hat{p} = -i\hbar \partial_x$ satisfy the canonical commutation relation

$[\hat{x}, \hat{p}] = i\hbar$

and are related to each other by the Fourier transform. A wavefunction in position space, Fourier-transformed, becomes the wavefunction in momentum space. The Heisenberg uncertainty principle $\Delta x \cdot \Delta p \geq \hbar/2$ is a direct consequence of the Fourier inequality — a purely mathematical statement about the duality of functions and their transforms.

5. Mathematical consequences

The study of Schrödinger equations drives entire fields:

• Spectral theory of unbounded operators (von Neumann, 1932; Reed–Simon, 1970s).
• Semiclassical analysis — how solutions behave as $\hbar \to 0$.
• Dispersive PDE theory — the nonlinear Schrödinger equation and its wellposedness.
• Random-matrix theory — energy-level statistics of disordered systems.
• Mathematical quantum field theory — constructive QFT, rigorous renormalization.

Many landmarks of modern analysis — Strichartz estimates, Kato–Rellich theorems, Feynman path integrals — arose from the needs of quantum mechanics.

6. The nonlinear Schrödinger equation

A nonlinear variant $i\partial_t \psi = -\Delta \psi + |\psi|^{2p} \psi$ appears in optics, Bose–Einstein condensates, and water waves. Its wellposedness theory is a major topic in modern dispersive PDE and has driven decades of analytic research.

Further reading

• Reed & Simon, Methods of Modern Mathematical Physics (4 vols) — rigorous treatment.
• Hall, Quantum Theory for Mathematicians — math-first introduction.
• Takhtajan, Quantum Mechanics for Mathematicians — structural approach.

Frequently asked

What does the wavefunction ψ represent mathematically?

It is an element of a complex Hilbert space L²(ℝⁿ, ℂ), normalized so that ∫|ψ|² = 1. The Born rule interprets |ψ(x)|² as a probability density over positions; more generally, for any observable the probability distribution is encoded through the spectral theorem applied to the associated self-adjoint operator.

Why is the Hamiltonian self-adjoint?

Because it corresponds to a real-valued observable (energy) and because self-adjointness guarantees that the time-evolution operator e^(-iHt/ℏ) is unitary — preserving the norm of ψ and hence probability. The mathematical theory of self-adjoint operators on Hilbert space (von Neumann, 1932) was developed precisely to handle this.

Is the Schrödinger equation a relativistic theory?

No. The form shown here is non-relativistic — it is Galilean invariant but not Lorentz invariant. The relativistic generalizations are the Dirac equation (for spin-½) and the Klein–Gordon equation (for scalar fields). Quantum field theory combines these with quantization principles to give the relativistic quantum theory used in modern physics.

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