Normalization Examples

Normalization fixes the scale of probabilities. Before using any example, identify whether the object is a square-integrable state, a generalized eigenstate, a radial function, a spinor, a density matrix, or a scattering state.

Core Formula

For a normalizable wavefunction in one dimension,

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

For a state vector,

$$\langle\psi\rvert\psi\rangle=1.$$

For a density matrix,

$$\operatorname{Tr}\rho=1, \qquad \rho\ge 0.$$

The compact formula card is Normalization. The wave-mechanics teaching page is Normalization Conventions.

Example Routes

Example Type	Learn	What to Watch
One-dimensional wavefunction	Exercise Sets, Set 1	Integration limits and units
Coordinate-space probability	Wavefunctions and Probability Density	Density versus probability
Position-space inner product	Coordinate Representation	Completeness convention
Radial wavefunction	Radial Schrödinger Equation	$R(r)$ versus $u(r)=rR(r)$
Spinor	Spin-Half Hilbert Space	Basis order and phase
Density matrix	Density Operators	Trace one and positivity
Scattering state	Reflection and Transmission Coefficients	Flux normalization rather than square integrability

Common Mistakes

• Treating $\lvert\psi(x)\rvert^2$ as a probability without integrating over a region.
• Forgetting the spherical measure $r^2dr\,d\Omega$.
• Normalizing $R(r)$ as if it were $u(r)$.
• Applying square-integrable normalization to plane waves without stating a box, delta, or wave-packet convention.
• Checking $\operatorname{Tr}\rho=1$ but not positivity for a mixed state.

Minimal Worked Check

For $\psi(x)=Ae^{-\kappa\lvert x\rvert}$ with $\kappa>0$,

$$1=\int_{-\infty}^{\infty}A^2e^{-2\kappa\lvert x\rvert}\,dx =2A^2\int_0^\infty e^{-2\kappa x}\,dx =\frac{A^2}{\kappa}.$$

Thus $A=\sqrt{\kappa}$ if $A$ is chosen real and positive. The dimension check is useful: $A$ has units of $L^{-1/2}$ in one dimension.

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. Laloe, Quantum Mechanics, Wiley, 1977.

Exercises

1. A radial solution is written as $\psi(r,\theta,\phi)=R(r)Y_{\ell m}(\theta,\phi)$, with normalized spherical harmonic. Which integral normalizes $R$?

Solution

Use $\int_0^\infty \lvert R(r)\rvert^2r^2\,dr=1$. If $u(r)=rR(r)$ is used instead, the condition becomes $\int_0^\infty\lvert u(r)\rvert^2\,dr=1$.