Boundary Conditions Table

Boundary conditions complete a wave-mechanics model. A differential expression such as $-\hbar^2d^2/(2m\,dx^2)$ becomes a physical Hamiltonian only after the allowed wavefunctions, endpoints, matching rules, and asymptotic conditions are specified.

This table is a quick diagnostic reference. The canonical explanation is Boundary Conditions; the rows below summarize the conditions most often used by the model pages in this volume.

Lookup Table

Situation	Standard condition	Physical role	Used in	Common mistake
Infinite wall	$\psi=0$ at the wall; for a box, $\psi(0)=\psi(L)=0$	Excludes the particle from the forbidden region and gives zero flux through the wall	Infinite Square Well, Three-Dimensional Box	Imposing $\psi=0$ at a finite wall
Finite step or finite barrier	$\psi(x_0^-)=\psi(x_0^+)$ and $\psi'(x_0^-)=\psi'(x_0^+)$ for finite $V(x)$ and constant mass	Matches amplitude and current across an interface with no singular impulse	Finite Square Well, Potential Step, Rectangular Barrier Tunneling	Treating a finite jump like an infinite wall
Delta potential	$\psi(0^-)=\psi(0^+)$ and $\psi'(0^+)-\psi'(0^-)=2m\lambda\psi(0)/\hbar^2$ for $V(x)=\lambda\delta(x)$	Encodes the singular short-range interaction as a derivative jump	Delta Function Potential	Forcing $\psi'$ to be continuous and losing the delta interaction
Periodic boundary	$\psi(x+L)=\psi(x)$ and, for the kinetic operator, $\psi'(x+L)=\psi'(x)$	Identifies endpoints as the same physical point and quantizes momentum	Periodic Boundary Conditions, Landau Levels for finite-area degeneracy counting; finite-volume normalization throughout scattering and free-motion pages	Forgetting derivative matching or confusing periodic boxes with hard boxes
Bound state at infinity	$\psi\in L^2$ and the growing asymptotic solution is discarded	Selects normalizable states and discrete energies	Finite Square Well, Delta Function Potential, Hydrogen Atom	Keeping an exponential that grows at infinity
Scattering state	Specify incoming, reflected, and transmitted asymptotic waves; normalize by flux or delta normalization	Defines the physical scattering experiment and reflection/transmission coefficients	Potential Step, Reflection and Transmission Coefficients, Rectangular Barrier Tunneling	Computing probabilities from amplitude squares when currents differ
Radial origin	Require regularity at $r=0$; for the reduced radial function, $u(r)=rR(r)$ usually satisfies $u(0)=0$	Removes nonphysical singular solutions and makes the radial Hamiltonian well-defined	Radial Schrödinger Equation, Hydrogen Atom	Accepting a singular radial solution because it solves the local equation
Radial infinity	Bound radial states decay; continuum radial states have oscillatory or Coulomb-modified asymptotics	Separates bound spectra from scattering continua in central potentials	Radial Schrödinger Equation, Degeneracy of the Hydrogen Atom	Normalizing continuum states as if they were square-integrable bound states

Reading Rules

The most useful first question is not “what is the differential equation?” but “what is the domain?” Two models can use the same expression $\hat p^2/(2m)$ and have different spectra because their allowed wavefunctions differ.

Finite and singular interactions should not be conflated. A finite discontinuity in $V(x)$ makes $\psi$ and $\psi'$ continuous in the usual constant-mass Schrödinger problem. A delta-function term leaves $\psi$ continuous but creates a derivative jump.

Asymptotic conditions are boundary conditions. Bound states discard growing exponentials; scattering states choose a physical incoming channel and outgoing response. This is why the same potential can have both bound-state and scattering pages without contradiction.

Current and Domain Checks

A practical check is probability current. In one dimension,

$$j=\frac{\hbar}{m} \operatorname{Im}\left(\psi^*\frac{d\psi}{dx}\right).$$

Hard-wall bound problems should not leak probability through the wall. Scattering problems may carry flux to infinity, but the incoming and outgoing currents must be compared consistently. Periodic problems may support circulating current, but the wavefunction must match around the loop.

For a second-order kinetic operator on an interval, self-adjoint boundary conditions make the integration-by-parts boundary term vanish for all allowed states:

$$\left[ \phi^*(x)\psi'(x)-\phi'^*(x)\psi(x) \right]_{a}^{b}=0.$$

This condition is not usually the fastest way to solve an undergraduate model, but it explains why boundary conditions are not optional. See Hermitian vs Self-Adjoint Operators for the formal warning.

Common Mistakes

• Treating $\psi=0$ as the universal condition at every interface.
• Forgetting that finite barriers allow evanescent tails.
• Applying derivative continuity to a delta-function potential.
• Forgetting that scattering coefficients are current ratios.
• Ignoring the radial measure when deciding whether a singular solution is physical.
• Memorizing a spectrum without the boundary conditions that make it true.
• Changing geometry from an interval to a ring while keeping the old spectrum.

Where This Is Used

• Common Hamiltonians lists the Hamiltonian expressions whose domains are summarized here.
• Periodic Boundary Conditions develops the periodic-box plane-wave basis and density-of-states conversion.
• Spectra and Eigenfunctions Table gives the standard spectra after these conditions are imposed.
• Limiting Cases Table explains how finite, hard-wall, and singular limits should behave.
• Normalization Table summarizes the measures used once a domain and coordinate system have been chosen.
• Canonical Plots Gallery shows the boundary behavior visually.
• Boundary Conditions is the canonical conceptual page.
• Probability Current explains the flux test behind scattering and wall conditions.
• Normalization Conventions distinguishes bound-state, delta, box, and flux normalization.

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press, 1975.

Exercises

1. A particle moves in a finite square well. Explain why the wavefunction should not vanish at the two edges of the well.

Solution

A finite wall does not exclude the outside region completely. For a bound state, the wavefunction usually decays exponentially outside the well, so $\psi$ and $\psi'$ are matched across each finite jump. Forcing $\psi=0$ at the edges would instead describe an infinite square well and would give the wrong spectrum.

2. For $V(x)=\lambda\delta(x)$, what matching rule distinguishes the model from a free particle on the line?

Solution

The wavefunction is continuous,

$$\psi(0^-)=\psi(0^+)=\psi(0),$$

but the derivative jumps:

$$\psi'(0^+)-\psi'(0^-) =\frac{2m\lambda}{\hbar^2}\psi(0).$$

Without this derivative jump, the delta potential would not affect the wavefunction.