Notation and Conventions Used Here

Fourier signs, normalization factors, radial measures, and charge conventions can change the appearance of a wave-mechanics formula. This guide collects the choices used in the calculations below and shows how to translate common alternatives. The broader conventions guide explains bra-ket notation and units.

Position, momentum, and wave-number amplitudes

On a line, use $\psi(x)=\langle x\rvert\psi\rangle$ and $\phi(p)=\langle p\rvert\psi\rangle$, with

$$\psi(x)=\frac{1}{\sqrt{2\pi\hbar}}\int e^{ipx/\hbar}\phi(p)\,dp, \qquad \phi(p)=\frac{1}{\sqrt{2\pi\hbar}}\int e^{-ipx/\hbar}\psi(x)\,dx.$$

These conventions preserve $\int|\psi|^2dx=\int|\phi|^2dp$. If $\widetilde\psi(k)$ is normalized in $dk$ and $p=\hbar k$, then

$$\phi(p)=\frac{\widetilde\psi(p/\hbar)}{\sqrt\hbar}.$$

The square-root factor is necessary because probabilities, including their integration measures, must agree. In three dimensions the Fourier prefactor is $(2\pi\hbar)^{-3/2}$. See Fourier conventions for translation examples.

Match normalization to the state

Description	Normalization or measure
Bound state on a line	$\int\lvert\psi(x)\rvert^2dx=1$
Momentum eigenstates	$\langle p\mid p'\rangle=\delta(p-p')$; these are generalized states
Periodic box of length $L$	$\psi_n=L^{-1/2}e^{2\pi inx/L}$, $n\in\mathbb Z$
Scattering state	Specify incident amplitude or flux; obtain probabilities from current ratios
Cartesian three-dimensional state	$\int\lvert\psi(\mathbf r)\rvert^2d^3r=1$
Radial factor with unit-normalized angular function	$\int_0^\infty\lvert R(r)\rvert^2r^2dr=1$
Reduced radial function $u=rR$	$\int_0^\infty\lvert u(r)\rvert^2dr=1$
Uniform-grid samples with the stated rectangle quadrature	$\sum_i\lvert\psi(x_i)\rvert^2\Delta x\simeq1$

Other quadratures have different weights. A unit-norm numerical eigenvector is related to continuum samples by $v_i\simeq\sqrt{\Delta x}\,\psi(x_i)$ in the uniform-grid convention above. Normalization conventions treats the discrete, continuous, and flux cases in detail.

Canonical momentum, kinetic momentum, and current

In Cartesian position representation, canonical momentum acts as $\mathbf p=-i\hbar\nabla$ on its specified domain. For a spinless particle of signed charge $q$, minimal coupling uses

$$H=\frac{(\mathbf p-q\mathbf A)^2}{2m}+q\Phi, \qquad \boldsymbol\pi=\mathbf p-q\mathbf A.$$

The kinetic momentum $\boldsymbol\pi$ determines velocity. For this Hamiltonian the probability current is

$$\mathbf j=\frac1m\operatorname{Re}\!\left[\psi^*(-i\hbar\nabla-q\mathbf A)\psi\right].$$

When $\mathbf A=0$, it reduces to $(\hbar/m)\operatorname{Im}(\psi^*\nabla\psi)$. In a scattering region with constant real scalar potential, $Ae^{ikx}$ carries $j=\hbar k|A|^2/m$. The sign of $k$ sets its direction. Use minimal coupling and probability current for assumptions and derivations.

Boundaries and spherical coordinates

A hard-wall interval has $\psi(0)=\psi(L)=0$. A periodic Hamiltonian has both $\psi(0)=\psi(L)$ and $\psi'(0)=\psi'(L)$. A prime denotes differentiation with respect to the displayed coordinate. At a singular interaction, write the derivative jump explicitly; do not assume derivative continuity.

Spherical coordinates are $r\ge0$, polar angle $\theta$ from the positive $z$ axis, and azimuth $\phi$. Their measures are

$$d^3r=r^2dr\,d\Omega, \qquad d\Omega=\sin\theta\,d\theta\,d\phi.$$

With $\psi=R(r)Y_\ell^m(\theta,\phi)=u(r)Y_\ell^m/r$, normalize the angular harmonic to one in $d\Omega$. The angular-momentum conventions specify its phase convention. The label $m$ in $Y_\ell^m$ is an integer magnetic quantum number; it is distinct from particle mass.

Exercises

1. What are the units of $\psi(x)$ and $\phi(p)$ for a normalized particle on a line?

Solution

They are length$^{-1/2}$ and momentum$^{-1/2}$ respectively, so that each squared amplitude times its measure is dimensionless. They are different functions related by a unitary transform, not interchangeable values of one density.

2. With equal masses and no vector potential, let a left-incident stationary wave have transmitted amplitude $t$ relative to incident amplitude one. The asymptotic wave numbers are positive $k_L,k_R$. Find $T$.

Solution

$T=j_{\mathrm{trans}}/j_{\mathrm{inc}}=(k_R/k_L)|t|^2$. The factor is a velocity ratio. A bare squared amplitude gives transmission only when the velocities match or the amplitudes were already normalized to equal flux.

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.
• D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, 1988.