LaTeX Quick Guides

Schrödinger Equation in LaTeX: The Essentials

October 31, 202512 min read

Introduction

The Schrödinger equation is the fundamental equation of quantum mechanics, describing how quantum systems evolve over time. Named after Erwin Schrödinger who formulated it in 1925, this equation governs the behavior of particles at the atomic and subatomic scale.

In LaTeX, the Schrödinger equation requires careful attention to notation, including the reduced Planck constant \hbar, the Laplacian operator \nabla^2, and proper use of partial derivatives and wave functions. This guide covers the essentials of typesetting the Schrödinger equation in LaTeX, providing all the essential information you need.

Time-Dependent Schrödinger Equation (TDSE)

The time-dependent Schrödinger equation describes how a quantum system evolves over time. It is a partial differential equation that relates the time derivative of the wave function to the Hamiltonian operator acting on the wave function.

General Form (3D)

The most general form in three dimensions with position vector \mathbf{r}.

i ℏ \frac{\partial}{\partial t} Ψ (r, t) = [- \frac{ℏ ^{2}}{2 m} \nabla^{2} + V (r, t)] Ψ (r, t)

One-Dimensional Form

For one-dimensional systems, the Laplacian reduces to a second derivative with respect to x.

i ℏ \frac{\partial}{\partial t} Ψ (x, t) = [- \frac{ℏ ^{2}}{2 m} \frac{\partial ^{2}}{\partial x ^{2}} + V (x, t)] Ψ (x, t)

Using Hamiltonian Notation

The Hamiltonian operator \hat{H} represents the total energy operator.

i ℏ \frac{\partial}{\partial t} ∣Ψ (t)⟩ = \hat{H} ∣Ψ (t)⟩

Time-Independent Schrödinger Equation (TISE)

The time-independent Schrödinger equation applies when the potential energy is constant in time, leading to stationary states with definite energy eigenvalues. This is an eigenvalue equation where the Hamiltonian operator acts on the wave function to yield the energy E times the wave function.

General Form (3D)

The three-dimensional time-independent form for stationary states.

[- \frac{ℏ ^{2}}{2 m} \nabla^{2} + V (r)] ψ (r) = E ψ (r)

One-Dimensional Form

The one-dimensional time-independent Schrödinger equation.

- \frac{ℏ ^{2}}{2 m} \frac{d ^{2} ψ}{d x ^{2}} + V (x) ψ (x) = E ψ (x)

Eigenvalue Form

Compact notation using the Hamiltonian operator for eigenstates.

\hat{H} ∣ ψ_{n} ⟩ = E_{n} ∣ ψ_{n} ⟩

Understanding the Terms

Wave Function

The wave function \Psi(\mathbf{r},t) or \psi(\mathbf{r}) describes the quantum state. Use uppercase \Psi for time-dependent and lowercase \psi for time-independent forms.

Ψ (x, t) = ψ (x) e^{- i Et /ℏ}

Kinetic Energy Term

The kinetic energy operator is -\frac{\hbar^2}{2m}\nabla^2, where m is the particle mass.

Potential Energy

The potential energy function V(\mathbf{r},t) or V(\mathbf{r}) depends on the physical system.

Reduced Planck Constant

Use \hbar (hbar) for the reduced Planck constant, where \hbar = h/(2\pi).

ℏ = \frac{h}{2 π}

Common Examples

Free Particle

For a free particle with V = 0, the time-independent equation simplifies to:

- \frac{ℏ ^{2}}{2 m} \frac{d ^{2} ψ}{d x ^{2}} = E ψ (x)

Infinite Square Well

For a particle in an infinite potential well of width L:

V (x) = {0 \infty 0 < x < L otherwise

Harmonic Oscillator

For a quantum harmonic oscillator with spring constant k:

[- \frac{ℏ ^{2}}{2 m} \frac{d ^{2}}{d x ^{2}} + \frac{1}{2} k x^{2}] ψ (x) = E ψ (x)

Laplacian in Different Coordinates

The Laplacian operator \nabla^2 takes different forms depending on the coordinate system.

Cartesian Coordinates

\nabla^{2} = \frac{\partial ^{2}}{\partial x ^{2}} + \frac{\partial ^{2}}{\partial y ^{2}} + \frac{\partial ^{2}}{\partial z ^{2}}

Spherical Coordinates

For systems with spherical symmetry (like hydrogen atom):

\nabla^{2} = \frac{1}{r ^{2}} \frac{\partial}{\partial r} (r^{2} \frac{\partial}{\partial r}) + \frac{1}{r ^{2} sin θ} \frac{\partial}{\partial θ} (sin θ \frac{\partial}{\partial θ}) + \frac{1}{r ^{2} sin ^{2} θ} \frac{\partial ^{2}}{\partial ϕ ^{2}}

Tips and Best Practices

• Use \hbar for reduced Planck constant (not h)
• Use \nabla^2 for the Laplacian operator in 3D
• Bold vectors with \mathbf{r} or \vec{r}
• Use \Psi for time-dependent wave functions, \psi for stationary states
• Operators use hats: \hat{H} for Hamiltonian
• Bra-ket notation: |\psi\rangle for kets, \langle\psi| for bras
• Use \, for spacing between operators and functions
• For partial derivatives, use \frac{\partial}{\partial t} in TDSE, \frac{d^2}{dx^2} in 1D TISE

Applications

Atomic Physics

The Schrödinger equation is fundamental to understanding atomic structure, electron orbitals, and chemical bonding. The hydrogen atom solution is a classic example solved using spherical coordinates.

Quantum Computing

Understanding the Schrödinger equation is essential for quantum algorithms and quantum information theory, where quantum states evolve according to unitary operators.

Condensed Matter Physics

Used to describe electrons in materials, band structure, and semiconductor physics, where periodic potentials lead to Bloch waves.