Problem-Solving Patterns

Most canonical wave-mechanics problems are not solved by inventing a new method from scratch. They are solved by recognizing a pattern: match regional solutions, exploit parity, scale the equation, use probability current, impose asymptotic behavior, separate variables, choose a convenient gauge, or test a limiting case.

Use How to Solve a Wave-Mechanics Problem for the general workflow. Use this page when a problem is already in front of you and you need to decide which move is likely to work.

Pattern Map

Pattern	Use When	First Move	Main Check	Example Pages
Split into regions and match	$V(x)$ is piecewise constant or has interfaces	write the local solution in each region	continuity, derivative jumps, and current conservation	Finite Square Well, Rectangular Barrier Tunneling
Exploit parity	the Hamiltonian is invariant under $x\to -x$	solve even and odd sectors separately	even states have $\psi'(0)=0$; odd states have $\psi(0)=0$	Finite Square Well, Harmonic Oscillator
Nondimensionalize first	many constants obscure the control parameter	choose a natural length and energy	the remaining parameters are dimensionless	Dimensionless Parameters Table
Use conservation of current	interpreting reflection or transmission	compute $j$ in each asymptotic region	$R+T=1$ for conservative one-channel scattering	Reflection and Transmission Coefficients
Use asymptotic behavior	selecting bound, scattering, or radial solutions	keep only physically allowed behavior at boundaries	no growing bound-state tails; regularity at the origin	Boundary Conditions Table
Separate variables	geometry has independent coordinates	try a product ansatz	each separated equation has the right measure and boundary conditions	Separation of Variables
Choose the right gauge	electromagnetic potentials appear	pick a gauge adapted to symmetry	energies and degeneracies are gauge-invariant	Minimal Coupling, Landau Levels
Check limiting cases	a formula is complicated or numerical	send one control parameter to a simple limit	recover a simpler known model	Limiting Cases Table

Region Matching

Piecewise one-dimensional potentials are solved by local general solutions plus matching conditions. The local solution is usually a sine, cosine, exponential, or plane wave; the physics enters through the boundary and matching rules.

The basic recipe is:

1. divide the line into regions where the differential equation has constant coefficients or a known form;
2. write the most general local solution in each region;
3. discard unphysical growing exponentials for bound states;
4. apply continuity of $\psi$ and the correct derivative condition;
5. solve the resulting algebraic or transcendental equation;
6. normalize bound states or compute currents for scattering states.

Finite jumps in $V(x)$ usually make both $\psi$ and $\psi'$ continuous for a constant-mass Schrödinger equation. Delta-function interactions are different: $\psi$ is continuous but $\psi'$ jumps. Infinite walls are different again: the wavefunction vanishes at the wall because the physical domain ends there.

Parity Sectors

If $V(x)=V(-x)$, the Hamiltonian commutes with parity. In one dimension, nondegenerate bound states can be chosen even or odd. This reduces many matching problems by half.

For an even state,

$$\psi(-x)=\psi(x), \qquad \psi'(0)=0.$$

For an odd state,

$$\psi(-x)=-\psi(x), \qquad \psi(0)=0.$$

This pattern is useful for the symmetric finite well, double-well intuition, and harmonic-oscillator eigenstates. It is also a good error check: an even potential should not produce a numerical eigenstate with no definite parity unless degeneracy, numerical mixing, or a symmetry-breaking perturbation is present.

Nondimensionalize First

Scaling is a problem-solving move, not a cosmetic final step. If the original equation contains $\hbar$, $m$, $L$, $V_0$, $\omega$, or $B$ in every term, the controlling parameter may be hidden.

For a geometric length $L$, the kinetic scale

$$E_L=\frac{\hbar^2}{2mL^2}$$

often exposes a dimensionless strength $V_0/E_L$. For a barrier, the opacity $\kappa a$ is usually more informative than $V_0$ and $a$ separately. For a harmonic oscillator, $x/\ell$ with $\ell=\sqrt{\hbar/(m\omega)}$ reveals the universal dimensionless equation. For a uniform magnetic field, $x/\ell_B$ and $A/(2\pi\ell_B^2)$ separate wavefunction scale from degeneracy counting.

The habit is simple: scale first, solve the dimensionless problem, and restore units only after the structure is visible.

Current Conservation

Scattering problems are not probability-density problems in a box. They are flux problems. A right-moving one-dimensional plane wave $Ae^{ikx}$ carries current

$$j=\frac{\hbar k}{m}\lvert A\rvert^2.$$

If the transmitted wave number differs from the incident wave number, then $T$ is not just $\lvert t\rvert^2$. For equal masses,

$$T=\frac{k_{\mathrm{trans}}}{k_{\mathrm{inc}}}\lvert t\rvert^2.$$

Use current conservation to catch missing signs, wrong velocity factors, and incorrectly normalized scattering states. For a real, time-independent, one-channel potential with no absorption, the final check is

$$R+T=1.$$

Asymptotic and Regularity Conditions

Many eigenvalue conditions come from rejecting locally valid but globally unphysical solutions. A bound state on the line cannot include an exponential that grows at infinity. A radial solution cannot be singular at the origin if the singularity makes the state nonphysical. A scattering state must state which waves are incoming and which are outgoing.

The reduced radial function $u(r)=rR(r)$ is a common place where this pattern matters. Regularity usually gives $u(0)=0$, while normalization uses

$$\int_0^\infty \lvert u(r)\rvert^2\,dr=1.$$

Do not decide acceptability from the differential equation alone. Decide it from the equation plus the Hilbert-space measure, domain, and physical boundary condition.

Separation of Variables

When the Hamiltonian and domain respect a product geometry, a product ansatz can turn one partial differential equation into several ordinary differential equations. The three-dimensional box separates into three one-dimensional boxes. Central potentials separate into radial and angular equations. The rigid rotor separates angular motion from radial dynamics because the radius is fixed.

The check is always the measure. In spherical coordinates,

$$d^3r=r^2\sin\theta\,dr\,d\theta\,d\phi.$$

The separated functions must be normalized with this measure, not with a Cartesian habit carried over accidentally.

Gauge Choice

Electromagnetic wave mechanics rewards a convenient gauge but punishes gauge confusion. Landau gauge makes one momentum component conserved and turns the transverse problem into an oscillator. Symmetric gauge makes rotational structure clearer. Both describe the same uniform magnetic field.

The pattern is:

• choose a gauge that respects the symmetry you want to use;
• solve the gauge-dependent wavefunction problem;
• interpret only gauge-invariant quantities as physical conclusions;
• state boundary conditions because finite-area degeneracy depends on them.

For Landau levels, the energy spacing $\hbar\omega_c$ and bulk degeneracy density $1/(2\pi\ell_B^2)$ are physical. A plotted wavefunction center in Landau gauge is a useful representation-dependent label, not by itself a gauge-invariant observable.

Limiting Cases

A limiting case is a compact way to audit a solution. A finite well should approach the infinite well when the outside barrier becomes infinitely high. A barrier should become transparent as its width goes to zero at fixed height. A harmonic oscillator at large quantum number should show classical behavior only after the right kind of averaging. Landau levels should recover the free-particle continuum only after spacing and degeneracy are considered together.

Always say what is held fixed. The limit $a\to0$ of a rectangular barrier is transparent at fixed height, but becomes a delta interaction if $V_0a$ is fixed while $V_0\to\infty$.

Numerical Validation Pattern

Numerical work follows the same patterns with one extra demand: convergence. A computational solution should record:

• the Hamiltonian and units;
• the grid or basis;
• the boundary conditions;
• the normalization convention;
• the benchmark quantity;
• the refinement variable;
• the limiting or analytic check.

For example, an infinite-well diagonalization should check eigenvalues against $n^2\pi^2\hbar^2/(2mL^2)$ and eigenvectors against the grid inner product. A packet-propagation notebook should check norm conservation and the analytic width or probability accounting relevant to the setup.

Common Mistakes

• Trying to solve before deciding the domain and measure.
• Matching amplitudes while forgetting derivative jumps or current ratios.
• Missing a parity simplification and solving twice as many unknowns as needed.
• Keeping a growing exponential because it solves the local equation.
• Using a gauge-dependent wavefunction plot as if it were a gauge-invariant claim.
• Treating a numerical plot as evidence without a convergence or benchmark check.
• Taking a limit without saying what parameter is fixed.

Where This Is Used

• Worked Examples Index points to pages where each pattern is used.
• Exercise Sets provides practice problems organized by these recurring moves.
• Benchmark Problems converts several patterns into reproducible numerical tests.
• Boundary Conditions Table summarizes the matching and domain rules behind the first pattern.
• Dimensionless Parameters Table names the parameters used in scaling and limiting checks.
• Normalization Table gives the measures needed for separation, radial equations, and numerical grids.

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.
• E. Merzbacher, Quantum Mechanics, 3rd ed., Wiley, 1998.

Exercises

1. A symmetric finite well gives a four-equation matching problem. Which pattern reduces the algebra, and what are the two boundary conditions at the center?

Solution

Use parity. In the even sector, solve on half the line with $\psi'(0)=0$. In the odd sector, solve with $\psi(0)=0$. The matching at the well edge is then applied separately for the two parity sectors.

2. A potential-step calculation gives a transmitted amplitude $t$ and claims $T=\lvert t\rvert^2$ even though the transmitted wavenumber differs from the incident wavenumber. Which pattern catches the error?

Solution

Use conservation of probability current. For equal masses,

$$T=\frac{k_{\mathrm{trans}}}{k_{\mathrm{inc}}}\lvert t\rvert^2,$$

so the velocity or wavenumber ratio is required whenever the asymptotic wavenumbers differ.

3. A finite-difference harmonic-oscillator notebook shows plausible eigenfunction plots but does not vary the box size or grid spacing. Which pattern is missing?

Solution

The numerical validation pattern is missing. The notebook should check eigenvalue convergence under grid refinement and domain enlargement, verify parity and normalization, and compare the low-lying energies with $E_n=\hbar\omega(n+1/2)$.