Spectra and Eigenfunctions Table

This table collects standard spectra and eigenfunction forms for canonical wave-mechanics systems. It is a lookup aid, not a replacement for derivations. Each row points to the canonical page where assumptions, boundary conditions, normalization, and interpretation are explained.

Conventions vary across textbooks. The formulas below use the conventions of the linked pages in this volume.

Exact and Standard Results

System	Eigenfunction form	Spectrum	Degeneracy or normalization notes	Canonical page
Free particle on the line	$e^{ikx}$ or wave packets built from them	$E=\hbar^2k^2/(2m)$, continuous	plane waves are delta-normalized or box-normalized	Free Particle
Infinite square well	$\psi_n(x)=\sqrt{2/L}\sin(n\pi x/L)$	$E_n=n^2\pi^2\hbar^2/(2mL^2)$	$n=1,2,\ldots$; nondegenerate in one dimension	Infinite Square Well
Finite square well	parity-adapted sinusoidal interior plus exponential tails	bound energies solve transcendental matching equations; continuum above threshold	number of bound states depends on depth and width	Finite Square Well
Attractive delta potential	$\psi(x)=\sqrt{\kappa}\,e^{-\kappa\lvert x\rvert}$	one bound state, $E=-m\alpha^2/(2\hbar^2)$	$\kappa=m\alpha/\hbar^2$ for $V=-\alpha\delta(x)$	Delta Function Potential
Potential step	incident, reflected, and transmitted plane waves	continuous scattering spectrum	reflection and transmission use probability-current ratios	Potential Step
Rectangular barrier	piecewise plane waves or evanescent waves matched at interfaces	continuous scattering spectrum	transmission has tunneling suppression and resonant structure	Rectangular Barrier Tunneling
Harmonic oscillator	Hermite functions $H_n(x/\ell)e^{-x^2/(2\ell^2)}$ with normalization	$E_n=\hbar\omega(n+1/2)$	$n=0,1,2,\ldots$; parity $(-1)^n$	Quantum Harmonic Oscillator
Coupled oscillators	products of normal-mode oscillator eigenfunctions	sums $\sum_a\hbar\omega_a(n_a+1/2)$	degeneracy depends on equal normal-mode frequencies	Coupled Oscillators
Two-level system	spinor aligned or anti-aligned with $\mathbf b$	$E_\pm=a_0\pm\lvert\mathbf b\rvert$	basis depends on the direction of $\mathbf b$	Two-Level Systems
Three-dimensional box	product sines in $x,y,z$	$\frac{\hbar^2\pi^2}{2m}(n_x^2/L_x^2+n_y^2/L_y^2+n_z^2/L_z^2)$	degeneracies arise from equal side lengths and integer coincidences	Three-Dimensional Box
Central potential	$R_{E\ell}(r)Y_\ell^m(\theta,\phi)$	model dependent radial spectrum	$m$ degeneracy follows from rotations for central potentials	Radial Schrödinger Equation
Hydrogen atom	$R_{n\ell}(r)Y_\ell^m(\theta,\phi)$	$E_n=-\mu e^4/[2(4\pi\epsilon_0)^2\hbar^2n^2]$	spinless spatial degeneracy $n^2$ in the ideal Coulomb model	Hydrogen Atom
Rigid rotor	$Y_J^M(\theta,\phi)$	$E_J=\hbar^2J(J+1)/(2I)$	$2J+1$ states for fixed $J$	Rigid Rotor
Landau levels	$e^{ik_y y}\varphi_n(x-x_0)$ in Landau gauge	$E_n=\hbar\omega_c(n+1/2)$ in two dimensions	degeneracy density $1/(2\pi\ell_B^2)$ in the ideal bulk	Landau Levels

Reading Notes

The eigenfunction column often gives only the spatial dependence up to a convention-dependent normalization factor. For scattering states and continuum spectra, normalization is usually delta-normalization, box normalization, or flux normalization rather than ordinary square normalization.

The spectrum column states the ideal model result. Perturbations, external fields, spin couplings, finite boundaries, and self-adjoint extension choices can change the spectrum. The linked canonical pages state the assumptions under which each formula is valid.

Degeneracy needs special care. Some degeneracies follow from explicit symmetries, such as rotational invariance. Others are special to a particular potential, such as the additional Coulomb degeneracy of hydrogen. Some degeneracies, such as Landau-level degeneracy, depend on boundary conditions and flux counting.

Common Mistakes

• Memorizing an energy formula without its boundary conditions.
• Using square-normalized formulas for continuum states.
• Forgetting that finite wells and barriers require matching conditions, not one global sine or exponential.
• Treating a degeneracy count as universal when it depends on geometry or gauge choice.
• Copying an eigenfunction formula without its normalization convention.
• Using this table as the canonical derivation instead of following the linked page.

Where This Is Used

• Common Hamiltonians gives the matching Hamiltonian lookup table.
• Boundary Conditions Table summarizes the endpoint, matching, and asymptotic conditions behind these spectra.
• Limiting Cases Table gives quick checks for whether formulas reduce to simpler models in the right limits.
• Normalization Table gives the measure and prefactor conventions behind the eigenfunction column.
• Canonical Plots Gallery shows the visual signatures of the same formulas.
• Normalization Conventions explains square, delta, and flux normalization.
• Boundary Conditions explains why spectra depend on domains.
• Core Formulas Index gives a broader formula lookup.
• Model Encyclopedia gives compact cards for selected standard models.

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.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.