Spectra and Eigenfunctions Table

Use this table to compare energy spectra and the shapes of their eigenfunctions. Select a system name for its derivation, boundary conditions, normalization, and physical interpretation.

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

Exact and Standard Results

System and derivation	Eigenfunction form	Spectrum	Degeneracy or normalization notes
Free particle on the line	$e^{ikx}$	$E=\hbar^2k^2/(2m)$, continuous	plane waves are delta-normalized or box-normalized
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
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
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 $V=-\alpha\delta(x)$
Potential step	incident and reflected plane waves; transmitted plane wave above the step or decaying exponential below it	continuous scattering spectrum	reflection and transmission use probability-current ratios
Rectangular barrier	piecewise plane waves or evanescent waves matched at interfaces	continuous scattering spectrum	transmission has tunneling suppression and resonant structure
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 $(-1)^n$
Coupled oscillators	products of normal-mode oscillator eigenfunctions	sums $\sum_a\hbar\omega_a(n_a+1/2)$	degeneracy depends on integer relations among normal-mode frequencies
Two-level system	spinor aligned or anti-aligned with $\mathbf b$	$E_\pm=a_0\pm\lvert\mathbf b\rvert$	for $\mathbf b\ne0$; at $\mathbf b=0$ every basis is an energy basis
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
Central potential	$R_{E\ell}(r)Y_\ell^m(\theta,\phi)$	model dependent radial spectrum	$m$ degeneracy follows from rotations for central potentials
Hydrogen bound states	$R_{n\ell}(r)Y_\ell^m(\theta,\phi)$	$E_n=-\mu e^4/[2(4\pi\epsilon_0)^2\hbar^2n^2]$	spinless spatial degeneracy $n^2$ in the ideal Coulomb model
Rigid rotor	$Y_J^M(\theta,\phi)$	$E_J=\hbar^2J(J+1)/(2I)$	$2J+1$ states for fixed $J$
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

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 articles explain the assumptions behind each formula.

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 physical geometry, boundary conditions, or fields; gauge choice leaves the count unchanged.
• Copying an eigenfunction formula without its normalization convention.
• Applying a formula to a different Hamiltonian or spatial domain without checking its derivation.

References

• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.