Nature of the Schrödinger equation Is the (time-dependent) Schrödinger wave equation a partial differential equation (PDE) rather than an ordinary differential equation (ODE)?

Introduction / Context:
The Schrödinger equation is the cornerstone of nonrelativistic quantum mechanics. Understanding its mathematical type (ODE vs PDE) clarifies solution methods, boundary conditions, and physical interpretation of wavefunctions in space and time.

Given Data / Assumptions:

• Time-dependent Schrödinger equation: iħ ∂ψ/∂t = [−(ħ^2/2m) ∇^2 + V(r,t)] ψ.
• Wavefunction ψ = ψ(r, t) depends on space r and time t.
• Potential V may depend on r and t.

Concept / Approach:

A PDE involves partial derivatives with respect to multiple independent variables. In the time-dependent form, the Schrödinger equation includes a first-order partial derivative in time and second-order partial derivatives in spatial coordinates via the Laplacian ∇^2, making it a linear PDE. Even in one spatial dimension, it is still a PDE because ψ(x, t) depends on both x and t. The time-independent Schrödinger equation is a spatial PDE (eigenvalue problem) obtained by separation of variables when the potential is time-independent.

Step-by-Step Reasoning:

Verification / Alternative check:

Standard solution techniques (separation of variables, Fourier methods, Green’s functions) are typical for linear PDEs and are used extensively for Schrödinger problems.

Why Other Options Are Wrong:

• “False” or “purely algebraic”: contradicts the differential nature of the equation.
• “Only in one dimension” or “only in momentum space”: incorrect; both coordinate and momentum representations are PDEs in time (and space or momentum).

Common Pitfalls:

Confusing the time-independent Schrödinger eigenvalue equation with an ODE; it is a PDE unless reduced by symmetry to radial or 1D ODE forms for specific potentials.

Final Answer:

True