Math Needed for Wave Mechanics

To solve a wave-mechanics problem, you need to find a wavefunction, impose its boundary conditions, and normalize it with the correct measure. Begin with the one-dimensional route below, or choose the row of physics that matches the system you are studying in Wave Mechanics and Model Systems.

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.

What Solving the Schrödinger Equation Involves

For a one-dimensional stationary problem,

$$-\frac{\hbar^2}{2m}\psi''(x)+V(x)\psi(x)=E\psi(x).$$

At a chosen energy, an ODE method usually gives a family of local solutions. Boundary or asymptotic conditions decide whether a nonzero member is allowed; for a bound state, normalization then fixes its magnitude. A remaining overall phase has no physical effect. Keep these steps distinct when a computer returns a formula for $\psi$.

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;
• Position and Momentum Representations;
• 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 show these functions in physical calculations.

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.

Why Boundary Conditions Change the Spectrum

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. At fixed energy they constrain the two integration constants. Homogeneous boundary conditions generally allow only the zero solution except at eigenvalues; at an eigenvalue, normalization fixes the remaining scale. Inhomogeneous boundary-value problems have a different existence and uniqueness question.
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.

References

• 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.
• 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.