How to Solve a Wave-Mechanics Problem

A wave-mechanics problem asks for the quantum behavior of a system represented in coordinates. The input is usually a physical setup: a particle in a region, a potential energy function, a boundary condition, or an initial wave packet. The output is a set of allowed states, time evolutions, spectra, probabilities, currents, expectation values, and limiting checks.

The central habit is to translate the words of the problem into a precise Hilbert-space and differential-equation problem before solving equations.

The Workflow

Use this checklist whenever a page solves a canonical system.

1. Identify the physical system and degrees of freedom.
2. Choose coordinates adapted to the geometry.
3. Write the Hamiltonian and specify the potential.
4. State the Hilbert space and inner product.
5. State the domain and boundary conditions.
6. Choose a time-dependent or stationary formulation.
7. Solve the eigenvalue problem exactly, approximately, or numerically.
8. Normalize the states using the correct measure.
9. Compute observables and probability currents when relevant.
10. Interpret spectra and wavefunctions physically.
11. Check dimensions, limiting cases, and classical behavior.
12. Record common mistakes and where the model is reused.

The Hamiltonian expression alone is not always the whole problem. The same local differential operator can describe different quantum systems when its domain or boundary conditions change.

Identify The System

Start by naming the degrees of freedom. A one-dimensional particle has a coordinate $x$. A central-potential problem has a radial coordinate $r$ and angular coordinates $(\theta,\phi)$. A two-level system may not have a coordinate-space wavefunction at all; it is included in this volume only as a canonical Hamiltonian model, not as a differential-equation problem.

For a nonrelativistic particle in one dimension, a common starting point is

$$\hat H =-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x).$$

The symbols are not enough. One must also say where $x$ lives: the real line, a finite interval, a ring, a half-line, or a union of regions with matching conditions.

Choose Coordinates And Measure

Coordinates determine the inner product. On a line,

$$\langle \psi\vert\phi\rangle =\int_{-\infty}^{\infty}\psi^*(x)\phi(x)\,dx.$$

In three-dimensional Cartesian coordinates,

$$\langle \psi\vert\phi\rangle =\int_{\mathbb R^3}\psi^*(\mathbf r)\phi(\mathbf r)\,d^3r.$$

In spherical coordinates the same volume element becomes

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

Forgetting the measure changes normalization, expectation values, and sometimes the apparent form of differential operators.

Write The Hamiltonian

The Hamiltonian encodes kinetic energy, potential energy, and external fields. For a time-independent potential in one dimension, stationary states solve

$$\hat H\psi_n(x)=E_n\psi_n(x).$$

For a time-dependent state, the equation of motion is

$$i\hbar\frac{\partial \psi(x,t)}{\partial t} =\hat H\psi(x,t).$$

The stationary equation finds energy eigenstates. The time-dependent equation evolves arbitrary initial states. When $\hat H$ is time independent, these are linked by the expansion

$$\psi(x,t) =\sum_n c_n\psi_n(x)e^{-iE_n t/\hbar}$$

for a purely discrete spectrum, with integrals added when continuous spectra are present.

State Boundary Conditions

Boundary conditions are physical input. They may represent impenetrable walls, periodic geometry, regularity at the origin, square integrability at infinity, matching across finite potential jumps, or derivative jumps across singular potentials.

For example, an infinite square well on $0<x<L$ imposes

$$\psi(0)=0,\qquad \psi(L)=0.$$

A particle on a ring of circumference $L$ instead imposes periodicity:

$$\psi(x+L)=\psi(x).$$

Both use a kinetic-energy differential expression, but their spectra and eigenfunctions differ.

Solve And Normalize

Solving usually means finding eigenvalues and eigenfunctions, then using them as a basis for dynamics. For bound states, normalization typically means

$$\int \lvert\psi_n(x)\rvert^2\,dx=1.$$

For scattering states and plane waves, ordinary normalization may be impossible. One then uses box normalization, flux normalization, or delta normalization. The correct convention depends on the physical question and should be stated explicitly.

When a stationary equation is not analytically solvable, the numerical route must still respect the same boundary conditions and normalization. ODE Solvers covers shooting methods, while Matrix Diagonalization covers finite Hamiltonian eigenproblems.

Micro-Example: Infinite Square Well

Consider a particle of mass $m$ confined to $0<x<L$ by infinite walls.

The Hilbert space is $L^2([0,L])$ with inner product

$$\langle \psi\vert\phi\rangle =\int_0^L \psi^*(x)\phi(x)\,dx.$$

The Hamiltonian inside the well is

$$\hat H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2},$$

with boundary conditions $\psi(0)=\psi(L)=0$. The stationary Schrödinger equation is

$$-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2}=E\psi.$$

The allowed normalized eigenfunctions and energies are

$$\psi_n(x)=\sqrt{\frac{2}{L}}\sin\frac{n\pi x}{L}, \qquad E_n=\frac{n^2\pi^2\hbar^2}{2mL^2}, \qquad n=1,2,3,\ldots$$

The physical checks are immediate: energies scale as $1/L^2$, the ground-state energy is nonzero, and the $n$th eigenfunction has $n-1$ interior nodes. These are not decorative facts; they are the model’s main lessons.

Diagnostic Checks

Before trusting an answer, ask:

• Does the wavefunction satisfy the boundary conditions?
• Is it normalized with the correct measure?
• Are the dimensions of the energy and wavefunction correct?
• Does the result behave sensibly in limits such as $L\to\infty$, $V\to 0$, or $\hbar\to 0$?
• Are probability currents used where scattering probabilities are claimed?
• Is the answer a bound state, scattering state, or generalized eigenstate?

These checks catch many wrong solutions before any sophisticated mathematics is needed.

Common Mistakes

• Solving the differential equation before deciding the domain.
• Imposing infinite-wall boundary conditions on a finite-wall problem.
• Normalizing a plane wave as if it were square integrable on the real line.
• Forgetting that coefficients in an energy expansion are inner products with eigenstates.
• Comparing scattering amplitudes directly instead of comparing currents.
• Treating a time-independent energy eigenfunction as the most general time-dependent state.

Where This Workflow Is Used

• Coordinate Representation sets up the wavefunction language.
• Boundary Conditions explains the mathematical side of allowed endpoints and domains.
• ODE Solvers explains shooting methods for one-dimensional and radial differential equations.
• Schrödinger Equation gives the formal equation of motion.
• Fourier Transform supports free-particle and wave-packet calculations.
• Future canonical-system pages should link here for the general solving pattern.

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 particle on the interval $0<x<L$ has Hamiltonian $-\hbar^2 d^2/(2m dx^2)$. List two different boundary-condition choices and explain why they define different physical systems.

Solution

Hard-wall boundary conditions impose $\psi(0)=\psi(L)=0$ and model an infinite square well. Periodic boundary conditions impose $\psi(0)=\psi(L)$ and $\psi'(0)=\psi'(L)$ and model a particle on a ring. The same differential expression has different domains, eigenfunctions, and spectra in the two cases.

2. For the infinite square well, verify the normalization of $\psi_n(x)=\sqrt{2/L}\sin(n\pi x/L)$.

Solution

Use

$$\int_0^L \sin^2\frac{n\pi x}{L}\,dx=\frac{L}{2}.$$

Then

$$\int_0^L \lvert\psi_n(x)\rvert^2\,dx =\frac{2}{L}\int_0^L \sin^2\frac{n\pi x}{L}\,dx =1.$$