Math Needed for Wave Mechanics

This crosswalk routes readers from Wave Mechanics and Model Systems to the mathematics behind coordinate-space quantum mechanics. It does not rederive the model solutions. Use the minimum route once, then follow the path for the system in front of you.

Minimum Route

Read these pages in order for a first one-dimensional pass:

1. Complex Numbers and Complex Exponentials for oscillatory phases.
2. $L^2$ Spaces for square-integrable wavefunctions and almost-everywhere equivalence.
3. Position and Momentum Representations for coordinate wavefunctions and basis transformations.
4. Fourier Transform for the position–momentum relation.
5. Ordinary Differential Equations for stationary Schrödinger equations.
6. Boundary Conditions for domains, matching, and admissible spectra.
7. Sturm–Liouville Theory for orthogonality, completeness, and discrete eigenvalue problems.
8. Expectation Values for extracting predictions from a normalized state.

Add Delta Function, Distributions, and Wave Packets before treating plane waves or continuum eigenstates as more than formal symbols.

The Structural Chain

For a one-dimensional stationary problem,

$$\left[ -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} +V(x) \right]\psi(x) =E\psi(x).$$

Each mathematical tool answers a different question:

Question	Canonical tool
What functions count as states?	$L^2$ spaces and the operator domain
How is momentum represented?	Fourier transform
How is the differential equation solved locally?	ordinary differential equations
Which local solutions define physical states?	boundary conditions and matching
Why are bound eigenfunctions orthogonal?	Sturm–Liouville theory
How are continuum eigenstates normalized?	distributions and the delta function
How are predictions computed?	expectation values

Solving the differential equation is only the middle of the chain. Domain, boundary, normalization, and interpretation complete the problem.

Route by Physical System

One-dimensional bound states

Start with Time-Independent Schrödinger Equation, then use:

• Ordinary Differential Equations for local solution families;
• Boundary Conditions for endpoint and matching constraints;
• Sturm–Liouville Theory for spectrum and orthogonality;
• Numerical Quadrature for normalization and matrix elements.

This route supports the Infinite Square Well, Finite Square Well, and other piecewise potentials. For barriers, add continuity and current conservation through Probability Current.

Free and continuum states

For the Free Particle, read:

• Complex Exponentials;
• Fourier Transform;
• Momentum Representation;
• Delta Function;
• Wave Packets.

A plane wave is a generalized eigenfunction, not an element of $L^2(\mathbb R)$. It is delta-normalized and becomes a physical normalizable state only after amplitudes are superposed into a suitable wave packet. Keep the three notions separate:

$$\text{basis distribution}, \qquad \text{spectral amplitude}, \qquad \text{normalizable packet}.$$

See Plane Waves and Delta Normalization for the physical use of this machinery.

Harmonic oscillator

The differential-equation route uses Hermite Polynomials and Gaussian normalization. The algebraic route uses ladder operators and spectral decomposition. Both describe the same Hamiltonian but illuminate different structure.

Begin at Quantum Harmonic Oscillator, then choose Differential-Equation Solution or Ladder-Operator Solution. Do not mix the dimensionless coordinate and normalization conventions from the two routes without translating them.

Angular systems and central potentials

Add Separation of Variables and Spherical Harmonics before the rigid rotor or hydrogen atom.

For central potentials, the chain is

$$\text{three-dimensional equation} \longrightarrow \text{angular eigenproblem} \longrightarrow \text{radial eigenproblem}.$$

Use:

• Spherical Harmonics and Legendre Polynomials for angular dependence;
• Radial Schrödinger Equation for the reduced radial problem;
• Laguerre Polynomials for Coulomb bound states;
• Bessel Functions for free radial and cylindrical problems.

The Rigid Rotor and Hydrogen Atom are the canonical physical applications.

Numerical fallback

When closed forms stop being useful, choose the numerical object first:

Representation	First numerical tool
Initial-value ODE	ODE Solvers
Boundary-value shooting problem	ODE solver plus root finding
Finite-difference or basis Hamiltonian	Matrix Diagonalization
Large sparse Hamiltonian	Sparse Eigensolvers
Smooth periodic representation	Spectral Methods

Always separate eigensolver residual from grid, box, and basis-truncation convergence.

Measures and Normalization

The integration measure belongs to the coordinate system and chosen radial convention. Common normalizations include

$$\int_{-\infty}^{\infty} \lvert\psi(x)\rvert^2\,dx=1$$

in one dimension,

$$\int_{S^2} \lvert Y(\theta,\phi)\rvert^2\,d\Omega=1$$

for an angular function, and either

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

or, after $u(r)=rR(r)$,

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

Confusing $R$ with $u$ changes both the differential equation and its measure. Check Normalization Conventions whenever a radial substitution or Fourier convention is introduced.

Boundary Conditions Are Operator Data

A local differential expression does not fully define a quantum Hamiltonian. Its domain and boundary conditions determine whether the operator is self-adjoint, which spectra are allowed, and whether probability current leaks through the boundary.

For a finite jump in a regular one-dimensional potential, $\psi$ and $\psi'$ are normally continuous. Infinite walls, delta interactions, singular endpoints, and periodic domains require their own domain analysis. Do not promote one matching rule into a universal law.

Readiness Check

You are ready for the introductory model sequence if you can answer:

1. Why is a plane wave not a normalized vector in $L^2(\mathbb R)$?
2. What information do two boundary conditions supply for a second-order ODE?
3. Why must the Fourier-transform convention be fixed before comparing $\psi(x)$ and $\widetilde\psi(p)$?
4. What changes when $R(r)$ is replaced by $u(r)=rR(r)$?
5. Why does a small matrix-eigensolver residual not prove spatial-grid convergence?

Answers

1. Its modulus is constant, so its norm integral over the real line diverges. It is interpreted distributionally and delta-normalized.
2. They select a solution from the two-dimensional local solution space and often quantize the allowed energy.
3. Factors of $2\pi$ and $\hbar$, the sign in the exponential, and the inverse transform all depend on that convention.
4. The radial measure changes from $r^2dr$ to $dr$, and the radial differential expression is correspondingly transformed.
5. The residual tests the computed eigenpair of the finite matrix. Grid spacing, domain truncation, and discretization error are properties of how that matrix approximates the continuum problem.

Common Routing Mistakes

• Treating local ODE solutions as physical states before imposing boundary and normalization conditions.
• Using plane waves as ordinary normalized vectors.
• Forgetting the Jacobian in cylindrical or spherical coordinates.
• Mixing Fourier conventions from different sources.
• Using special-function names as substitutes for checking asymptotics and endpoint behavior.
• Confusing $R(r)$ and the reduced radial function $u(r)$.
• Applying bound-state normalization to continuum states.
• Starting numerical work without an analytic limit or convergence benchmark.

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.
• G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Academic Press, 2013.
• M. L. Boas, Mathematical Methods in the Physical Sciences, 3rd ed., Wiley, 2005.