Partial Differential Equations

A partial differential equation (PDE) relates an unknown function of several independent variables to some of its partial derivatives. An equation such as $\partial_t u(x,t)=0$ is already a PDE, even though only one variable is differentiated: its solutions are arbitrary time-independent functions of $x$.

Quantum mechanics meets PDEs whenever a state is written as a wavefunction in position space. The time-dependent Schrödinger equation is an evolution PDE, while the time-independent Schrödinger equation is usually a boundary-value eigenvalue problem. The purpose of this page is not to solve every PDE, but to make the language precise enough that wave-mechanics calculations do not hide their assumptions. For numerical workflows after discretization, see PDE Solvers.

Partial Derivatives and the Laplacian

For a function $u(x,y,t)$, the symbols

$$\partial_x u,\qquad \partial_y^2u,\qquad \partial_t u$$

mean differentiation with respect to one variable while the others are held fixed. A PDE is first order in a variable if the highest derivative with respect to that variable is first order, second order if the highest derivative is second order, and so on.

In Cartesian coordinates, the three-dimensional gradient and Laplacian are

$$\nabla \psi = \left( \frac{\partial\psi}{\partial x}, \frac{\partial\psi}{\partial y}, \frac{\partial\psi}{\partial z} \right),$$ $$\nabla^2\psi = \frac{\partial^2\psi}{\partial x^2} +\frac{\partial^2\psi}{\partial y^2} +\frac{\partial^2\psi}{\partial z^2}.$$

The Laplacian is the operator that enters the kinetic energy of a nonrelativistic particle. In curvilinear coordinates, the Laplacian has coordinate-dependent terms and measure factors; it should not be replaced by a naive sum of second derivatives.

Linearity and Superposition

A PDE is linear when the unknown function and its derivatives occur only to the first power and are not multiplied by each other. For example,

$$Lu=f$$

is linear if $L$ is a linear differential operator. If $f=0$, the equation is homogeneous. If $u_1$ and $u_2$ solve a homogeneous linear PDE, then any linear combination

$$c_1u_1+c_2u_2$$

also solves it.

This is the mathematical source of superposition in wave mechanics. The physical superposition principle also depends on the Hilbert-space interpretation of the solutions: the wavefunction must belong to the appropriate state space, and the Hamiltonian must be defined on an appropriate domain.

Schrödinger Equation as a PDE

For one spinless particle in a scalar potential, the position-space Schrödinger equation is

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

It is first order in time, second order in space, and linear in $\psi$ when $V$ is prescribed. The factor $i$ is not a cosmetic difference from the heat equation: for a self-adjoint Hamiltonian, it gives unitary time evolution rather than dissipative smoothing.

In the notation of operators on a Hilbert space, the same equation is

$$i\hbar\partial_t\psi(t)=H\psi(t).$$

The coordinate PDE is obtained after choosing the position representation and writing the Hamiltonian as a differential operator. This distinction is important because a formal differential expression is not yet a complete quantum Hamiltonian; its domain and boundary conditions matter.

Initial Data and Boundary Data

An evolution PDE usually needs initial data. For the Schrödinger equation, a typical initial condition is

$$\psi(\mathbf r,0)=\psi_0(\mathbf r).$$

If the particle moves on a finite interval, a half-line, a box, or a region with walls, initial data alone is not enough. One must also specify boundary conditions, such as

$$\psi(0,t)=\psi(L,t)=0$$

for an infinite square well on $0\le x\le L$.

For time-independent problems, the same physics is often written as an eigenvalue problem:

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

In coordinate space, this becomes a spatial PDE:

$$-\frac{\hbar^2}{2m}\nabla^2\varphi(\mathbf r) +V(\mathbf r)\varphi(\mathbf r) =E\varphi(\mathbf r).$$

Here the allowed energies are determined by the differential equation together with the boundary conditions, normalizability, and domain requirements. Boundary Conditions gives the endpoint and matching-condition viewpoint; Domains of Operators explains the Hilbert-space version.

Separation of Variables

Separation of Variables looks for product solutions. If the Hamiltonian is time independent, try

$$\psi(\mathbf r,t)=\varphi(\mathbf r)T(t).$$

Substitution into the time-dependent Schrödinger equation gives

$$i\hbar\varphi(\mathbf r)\frac{dT}{dt} = T(t)H\varphi(\mathbf r).$$

After division by $\varphi(\mathbf r)T(t)$ where the product is nonzero, the time-dependent side depends only on $t$ and the spatial side depends only on $\mathbf r$. Both must equal a constant $E$:

$$i\hbar\frac{1}{T}\frac{dT}{dt}=E, \qquad \frac{1}{\varphi}H\varphi=E.$$

The time factor is therefore

$$T(t)=e^{-iEt/\hbar},$$

and the spatial factor obeys the time-independent Schrödinger equation,

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

This is the standard bridge from the evolution PDE to stationary states. It is a method, not a theorem that every problem separates. Symmetry, coordinate choice, and the form of the potential decide whether separation is useful.

Rectangular Box Example

In a rectangular infinite well, the potential is zero inside

$$0<x<L_x,\qquad 0<y<L_y,\qquad 0<z<L_z$$

and infinite outside. Inside the box, the spatial equation is

$$-\frac{\hbar^2}{2m} \left( \frac{\partial^2\varphi}{\partial x^2} +\frac{\partial^2\varphi}{\partial y^2} +\frac{\partial^2\varphi}{\partial z^2} \right) =E\varphi.$$

Try a product

$$\varphi(x,y,z)=X(x)Y(y)Z(z).$$

Substitution and division by $XYZ$ gives

$$-\frac{\hbar^2}{2m} \left( \frac{X''}{X} +\frac{Y''}{Y} +\frac{Z''}{Z} \right) =E.$$

Each term depends on only one coordinate, so each is constant. With Dirichlet boundary conditions at the walls,

$$X(0)=X(L_x)=0, \qquad Y(0)=Y(L_y)=0, \qquad Z(0)=Z(L_z)=0,$$

the separated one-dimensional equations have sine solutions and

$$k_x=\frac{n_x\pi}{L_x}, \qquad k_y=\frac{n_y\pi}{L_y}, \qquad k_z=\frac{n_z\pi}{L_z},$$

where $n_x,n_y,n_z=1,2,3,\ldots$. The energy is

$$E = \frac{\hbar^2}{2m} \left( k_x^2+k_y^2+k_z^2 \right).$$

The PDE has become three ordinary differential equations plus boundary conditions. The reduction is powerful precisely because it carries the boundary data into the separated factors.

Fourier and Spectral Methods

For translation-invariant problems, Fourier transformation often turns a PDE into a simpler equation in momentum space. For a free particle,

$$\psi(\mathbf r,t) = \frac{1}{(2\pi\hbar)^{3/2}} \int_{\mathbb R^3} \phi(\mathbf p,t)e^{i\mathbf p\cdot\mathbf r/\hbar}\,d^3p.$$

Each momentum component evolves by

$$i\hbar\frac{\partial\phi}{\partial t}(\mathbf p,t) = \frac{\mathbf p^2}{2m}\phi(\mathbf p,t),$$

so

$$\phi(\mathbf p,t) = e^{-i\mathbf p^2t/(2m\hbar)} \phi(\mathbf p,0).$$

This is a continuous-spectrum version of separation: plane waves diagonalize the free Hamiltonian. The rigorous interpretation is spectral rather than a sum over normalizable energy eigenvectors. See Fourier Transform, Wave Packets, and Continuous Spectra.

Well-Posedness and Domains

A PDE problem is well posed when solutions exist for the allowed data, are unique, and depend continuously on those data in a stated norm. A time-independent self-adjoint Hamiltonian generates the unitary group $U(t)=e^{-iHt/\hbar}$. It acts on every initial state in $L^2$ and preserves the distance between two initial states. For the Schrödinger equation to hold as a differentiable Hilbert-space equation, the initial state must additionally lie in $D(H)$. Time-dependent Hamiltonians require additional regularity assumptions; self-adjointness at each time alone does not establish the evolution theorem.

This is why the phrase “the Hamiltonian is $-\hbar^2\nabla^2/(2m)+V$” is incomplete on a region with boundaries or singularities. One also needs a domain on which the operator is self-adjoint. Different self-adjoint boundary conditions can describe different physics and lead to different spectra.

Common Mistakes

• Treating an initial condition as a replacement for boundary conditions.
• Applying infinite-wall boundary conditions to a finite potential step.
• Assuming a product ansatz works without checking the potential, geometry, and boundary data.
• Forgetting that the Laplacian changes in spherical, cylindrical, and other curvilinear coordinates.
• Differentiating a series or eigenfunction expansion without checking the relevant convergence or domain conditions.
• Confusing formal solutions of a PDE with normalizable quantum states.
• Treating continuous-spectrum plane waves as ordinary Hilbert-space vectors rather than generalized eigenfunctions.

Related Topics

• Ordinary Differential Equations
• Sturm–Liouville Theory
• Green Functions
• Propagator Kernel

Exercises

1. Classify the equation

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

by its order in time, order in space, and linearity.

Solution

It is first order in time and second order in space. If $V(x)$ is a prescribed function, the equation is linear in $\psi$.

2. Starting from $\psi(\mathbf r,t)=\varphi(\mathbf r)T(t)$ and a time-independent Hamiltonian $H$, derive the time factor for a stationary state.

Solution

Substitute into $i\hbar\partial_t\psi=H\psi$:

$$i\hbar\varphi\frac{dT}{dt}=T H\varphi.$$

Dividing by $\varphi T$ separates the variables:

$$i\hbar\frac{1}{T}\frac{dT}{dt} = \frac{1}{\varphi}H\varphi =E.$$

Thus

$$\frac{dT}{dt} = -\frac{iE}{\hbar}T,$$

so $T(t)=e^{-iEt/\hbar}$ up to an overall constant.

3. In a two-dimensional rectangular box with side lengths $L_x$ and $L_y$, write the separated energy in terms of positive integers $n_x$ and $n_y$.

Solution

The separated wavefunction has sine factors with

$$k_x=\frac{n_x\pi}{L_x}, \qquad k_y=\frac{n_y\pi}{L_y},$$

where $n_x,n_y=1,2,3,\ldots$. Therefore

$$E_{n_x,n_y} = \frac{\hbar^2\pi^2}{2m} \left( \frac{n_x^2}{L_x^2} +\frac{n_y^2}{L_y^2} \right).$$

4. A wavefunction on $0\le x\le L$ is specified at $t=0$, but no boundary condition is given at $x=0$ or $x=L$. Why is the quantum problem incomplete?

Solution

The initial wavefunction says what state is present at one time, but it does not say what Hamiltonian acts at the endpoints. Dirichlet, Neumann, periodic, Robin, or other self-adjoint boundary conditions define different operators and can produce different spectra and time evolutions.

References

• G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Academic Press, 2013.
• L. C. Evans, Partial Differential Equations, 2nd ed., American Mathematical Society, 2010.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.
• W. A. Strauss, Partial Differential Equations: An Introduction, 2nd ed., Wiley, 2007.
• G. Teschl, Mathematical Methods in Quantum Mechanics, 2nd ed., American Mathematical Society, 2014.