Normalization Table

Normalization is not a decorative prefactor. It specifies the measure, the label convention, and the meaning of amplitudes. A state can be square-normalized, box-normalized, delta-normalized, flux-normalized, angular-normalized, or grid-normalized depending on what mathematical object is being represented.

The conceptual discussion is Normalization Conventions. This page is the quick lookup table used by model pages, numerical notebooks, and reference entries.

Probability Measures and Normalization

A normalization statement is incomplete until the measure is known:

$$\int \lvert\psi\rvert^2\,d\mu=1.$$

For a one-dimensional coordinate, $d\mu=dx$. For a three-dimensional position, $d\mu=d^3r$. For a radial function $R(r)$, the radial measure is $r^2dr$. For angular functions, $d\mu=d\Omega=\sin\theta\,d\theta\,d\phi$. For a numerical grid, the measure is represented by quadrature weights such as $\Delta x$.

Lookup Table

Situation	Normalization convention	Measure or label	Typical use	Common mistake
Discrete expansion coefficients	$\sum_n \lvert c_n\rvert^2=1$	orthonormal discrete basis	finite-dimensional states, oscillator number states, spin states	forgetting that basis states must already be orthonormal
Unit-normalized one-dimensional state	$\int \lvert\psi(x)\rvert^2\,dx=1$	$dx$ on the line or interval	wells, oscillator eigenstates, localized packets	normalizing $\psi$ instead of $\lvert\psi\rvert^2$
Infinite square well	$\psi_n(x)=\sqrt{2/L}\sin(n\pi x/L)$	$0\lt x\lt L$	hard-wall eigenstates	using $\sqrt{1/L}$ for sine states with two hard walls
Periodic box in one dimension	$\psi_n(x)=L^{-1/2}e^{ik_nx}$	$0\le x\lt L$, $k_n=2\pi n/L$	box regularization, density of states	treating the artificial box as a physical boundary unless intended
Three-dimensional square-normalized state	$\int \lvert\psi(\mathbf r)\rvert^2\,d^3r=1$	$d^3r$	bound states in three dimensions	forgetting that $\psi$ has units length$^{-3/2}$
Three-dimensional periodic box	$\psi_{\mathbf k}(\mathbf r)=V^{-1/2}e^{i\mathbf k\cdot\mathbf r}$	volume $V=L_xL_yL_z$	free particles, density of states	dropping the volume factor before taking the continuum limit
Central-potential separated state	$\psi=R_{n\ell}(r)Y_\ell^m(\theta,\phi)$	$r^2dr\,d\Omega$	hydrogenic and radial problems	normalizing $R(r)$ with $dr$ instead of $r^2dr$
Radial function $R(r)$	$\int_0^\infty \lvert R(r)\rvert^2r^2\,dr=1$	radial measure $r^2dr$	physical radial probability density	plotting $\lvert R\rvert^2$ as the radial probability density without the $r^2$ factor
Reduced radial function $u(r)=rR(r)$	$\int_0^\infty \lvert u(r)\rvert^2\,dr=1$	ordinary measure $dr$	radial Schrödinger equation	mixing formulas for $u$ and $R$ in the same calculation
Spherical harmonics	$\int_{S^2}Y_{\ell}^{m*}Y_{\ell'}^{m'}\,d\Omega=\delta_{\ell\ell'}\delta_{mm'}$	$d\Omega=\sin\theta\,d\theta\,d\phi$	angular momentum and central potentials	forgetting the $\sin\theta$ Jacobian
Momentum delta-normalized states	$\langle p\vert p'\rangle=\delta(p-p')$	momentum label $p$	Fourier transforms and continuum bases	mixing $p$- and $k$-normalization
Wavenumber delta-normalized states	$\langle k\vert k'\rangle=\delta(k-k')$	wavenumber label $k$	free-particle scattering and spectral integrals	missing the Jacobian $\delta(p-p')=\delta(k-k')/\hbar$
Position representation of $p$ states	$\langle x\vert p\rangle=(2\pi\hbar)^{-1/2}e^{ipx/\hbar}$	$dx$ and $dp$	momentum-space wavefunctions	changing Fourier convention without changing prefactors
One-dimensional flux normalization	choose $A$ so $\hbar k\lvert A\rvert^2/m=1$ for a right-moving plane wave with $k>0$	probability current	scattering channels	using amplitude squared as a probability when velocities differ
Potential-step transmission	$T=j_{\mathrm{trans}}/j_{\mathrm{inc}}$	current ratio	step and barrier scattering	writing $T=\lvert t\rvert^2$ when $k_{\mathrm{trans}}\ne k_{\mathrm{inc}}$
Numerical grid in one dimension	$\sum_i\lvert\psi_i\rvert^2\Delta x=1$	grid weight $\Delta x$	finite-difference eigenvectors and time evolution	comparing library-normalized vectors directly with analytic wavefunctions
Library vector from a grid eigenproblem	$\sum_i\lvert v_i\rvert^2=1$	Euclidean vector norm	matrix diagonalization output	forgetting $\psi_i\simeq v_i/\sqrt{\Delta x}$ on a uniform grid
Landau-gauge state, infinite $x$ and periodic $y$	$\psi_{n,k_y}=L_y^{-1/2}e^{ik_yy}\varphi_n(x-x_0)$ with $\int\lvert\varphi_n\rvert^2dx=1$	box-normalized $y$ direction and oscillator-normalized $x$ direction	Landau levels and degeneracy counting	treating the gauge-dependent wavefunction normalization as the degeneracy count
Spatial wavefunction with spin	$\int\sum_s\lvert\psi_s(\mathbf r)\rvert^2d^3r=1$	position measure plus spin sum	spinor wavefunctions	normalizing each spin component separately when only the total state should be normalized

Delta-Label Conversions

Delta normalization depends on the continuum label. Since $p=\hbar k$,

$$\delta(p-p') = \frac{1}{\hbar}\delta(k-k').$$

Explicitly, $\lvert k\rangle=\sqrt{\hbar}\,\lvert p=\hbar k\rangle$. This is not a physical disagreement; it is a label convention. Problems arise only when a calculation uses one convention for states and another convention for integration measures.

The safest habit is to write the identity resolution explicitly. For momentum normalization,

$$\int \lvert p\rangle\langle p\rvert\,dp=I.$$

For wavenumber normalization,

$$\int \lvert k\rangle\langle k\rvert\,dk=I.$$

The state normalization and the integration measure must match.

Grid and Continuum Comparisons

Numerical eigensolvers usually normalize vectors with the Euclidean norm. If a uniform grid represents a continuum wavefunction, the relation is approximately

$$v_i\simeq \sqrt{\Delta x}\,\psi(x_i), \qquad \psi(x_i)\simeq \frac{v_i}{\sqrt{\Delta x}}.$$

On a nonuniform grid, define the physical inner product using positive quadrature weights $w_i$. The Euclidean vector is then $v_i=\sqrt{w_i}\,\psi_i$, provided the matrix eigenproblem has also been transformed to this weighted basis. An arbitrary Euclidean-normalized eigenvector cannot simply be divided by $\sqrt{w_i}$ without checking that transformation. In three dimensions, the corresponding weights approximate the volume measure.

Units Quick Check

For square-normalized position wavefunctions:

• one-dimensional $\psi(x)$ has units length$^{-1/2}$;
• three-dimensional $\psi(\mathbf r)$ has units length$^{-3/2}$;
• radial $R(r)$ has units length$^{-3/2}$;
• reduced radial $u(r)$ has units length$^{-1/2}$;
• spherical harmonics are dimensionless when normalized on the unit sphere;
• box-normalized plane waves carry the inverse square root of the box measure.

If a formula has the wrong units, it is not normalized in the convention it claims to use.

Common Mistakes

• Comparing amplitudes before checking whether they are square-, box-, delta-, or flux-normalized.
• Forgetting the $r^2$ measure in radial integrals.
• Mixing $p$ and $k$ delta functions without the Jacobian.
• Dropping $\Delta x$ when validating numerical wavefunctions.
• Interpreting $\lvert R(r)\rvert^2$ as the radial probability density instead of $\lvert R(r)\rvert^2r^2$.
• Counting Landau states from a wavefunction prefactor instead of from allowed guiding centers or flux.
• Normalizing each factor of a product state correctly but then changing one factor’s measure later.

Exercises

1. A finite-difference eigensolver returns a vector $v_i$ satisfying $\sum_i\lvert v_i\rvert^2=1$ on a uniform grid with spacing $\Delta x$. What sampled wavefunction should be compared with an analytic $\psi(x)$?

Solution

Use

$$\psi(x_i)\simeq \frac{v_i}{\sqrt{\Delta x}}.$$

Then the physical grid norm is

$$\sum_i\lvert\psi(x_i)\rvert^2\Delta x = \sum_i\lvert v_i\rvert^2 =1.$$

2. If a scattering calculation gives an amplitude ratio $t$ across a potential step with incoming wavenumber $k_1$ and transmitted wavenumber $k_2$, what is the transmission coefficient for equal masses?

Solution

The current of a right-moving plane wave $Ae^{ikx}$ is $\hbar k\lvert A\rvert^2/m$. Therefore

$$T=\frac{j_{\mathrm{trans}}}{j_{\mathrm{inc}}} = \frac{k_2}{k_1}\lvert t\rvert^2.$$

Only when $k_1=k_2$ does this reduce to $\lvert t\rvert^2$.

3. Show that if $Y_\ell^m$ is normalized on the unit sphere and $R(r)$ is normalized with $r^2dr$, then $\psi=RY_\ell^m$ is normalized in three dimensions.

Solution

Using $d^3r=r^2dr\,d\Omega$,

$$\int\lvert R(r)Y_\ell^m(\theta,\phi)\rvert^2d^3r = \left(\int_0^\infty\lvert R(r)\rvert^2r^2dr\right) \left(\int_{S^2}\lvert Y_\ell^m\rvert^2d\Omega\right).$$

Each factor is $1$, so the product state is normalized.

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.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.