Notation and Conventions Used Here

This page records the local notation used in the Wave Mechanics and Model Systems volume. It is a compact orientation guide, not a replacement for the canonical Conventions Overview or the detailed convention pages linked below.

The guiding rule is:

$$\text{state the convention where ambiguity changes the calculation}.$$

Wave mechanics is especially sensitive to Fourier signs, normalization factors, radial measures, and boundary-condition domains. A page may use a different convention when it has a good reason, but it should say so explicitly and give a translation.

State And Wavefunction Notation

The abstract state is written as a ket $\lvert\psi\rangle$ or, when time dependence matters, $\lvert\psi(t)\rangle$. Its coordinate-space wavefunction is the position-basis amplitude

$$\psi(x,t)=\langle x\vert\psi(t)\rangle$$

in one dimension, and

$$\psi(\mathbf r,t)=\langle \mathbf r\vert\psi(t)\rangle$$

in three dimensions. A stationary state is usually written as $\psi_n(x)$, $\psi_E(x)$, or $\psi_{n\ell m}(\mathbf r)$ depending on the spectrum and labels.

The momentum-space wavefunction is written

$$\phi(p,t)=\langle p\vert\psi(t)\rangle$$

in one dimension, or $\phi(\mathbf p,t)$ in three dimensions. Some pages use $\tilde\psi(k)$ for wave-number amplitudes with $p=\hbar k$; this is not the same function as $\phi(p)$ unless the Jacobian has been handled.

Canonical homes:

• Coordinate Representation for the meaning of $\psi(x)=\langle x\vert\psi\rangle$.
• Wavefunctions and Probability Density for $\rho=\lvert\psi\rvert^2$ and probability interpretation.
• Position and Momentum Representations for generalized position and momentum bases.
• Bra-Ket Notation for the sitewide bra-ket convention.

Operators, Hats, And Differential Expressions

Operators may be written with hats, such as $\hat H$, $\hat x$, and $\hat p$, when the distinction between an operator and its eigenvalue or coordinate is helpful. In displayed wave-mechanics equations, the Hamiltonian may also be written as $H$ when the context is unambiguous.

In position representation, the default one-dimensional momentum operator is

$$\hat p=-i\hbar\frac{d}{dx}.$$

For a scalar potential in one dimension, a common Hamiltonian differential expression is

$$\hat H =-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x).$$

This expression is not the whole problem. The interval, inner product, and boundary conditions define the operator domain. Use Operator Conventions for sitewide operator notation and Boundary Conditions for the wave-mechanics domain issue.

Fourier Transform Convention

The default position-momentum Fourier convention uses symmetric factors and keeps $\hbar$ explicit:

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

and

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

In three dimensions the factor becomes $(2\pi\hbar)^{-3/2}$ and $px$ becomes $\mathbf p\cdot\mathbf r$. The canonical convention page is Fourier Transform Conventions; the mathematical tool page is Fourier Transform.

When a page uses wave number $k$ instead of momentum $p$, it should state the convention and remember $dp=\hbar\,dk$. This is where many apparent normalization disagreements between references originate.

Units And Constants

The default convention keeps $\hbar$ explicit. For example,

$$i\hbar\frac{\partial\psi}{\partial t}=\hat H\psi, \qquad E=\hbar\omega, \qquad p=\hbar k.$$

The speed of light $c$ is kept explicit unless a relativity or field-theory bridge page declares natural units. Atomic units may be used in hydrogenic or quantum-chemistry contexts only after the page states what has been set to one.

Use Units and Constants for the canonical policy.

Normalization Conventions

Wave mechanics uses several normalization conventions. The page should name the convention before interpreting coefficients, amplitudes, or currents.

Object	Default Convention	Canonical Home
bound state on a line	$\int \lvert\psi(x)\rvert^2\,dx=1$	Normalization Conventions
three-dimensional state	$\int \lvert\psi(\mathbf r)\rvert^2\,d^3r=1$	Wavefunction Normalization
momentum eigenstates	$\langle p\vert p'\rangle=\delta(p-p')$	Normalization Table
box-normalized plane waves	$\psi_n(x)=L^{-1/2}e^{ik_nx}$	Normalization Conventions
scattering channels	flux or current normalization	Reflection and Transmission Coefficients
radial wavefunction $R(r)$	$\int_0^\infty \lvert R(r)\rvert^2r^2\,dr=1$	Normalization Conventions
reduced radial function $u(r)$	$\int_0^\infty \lvert u(r)\rvert^2\,dr=1$	Normalization Table
numerical grid samples	$\sum_i\lvert\psi_i\rvert^2\Delta x=1$	Normalization Table

The distinction between $R(r)$ and $u(r)=rR(r)$ should always be stated in central-potential pages.

Probability Current

For a single nonrelativistic particle in a real scalar potential, the default current is

$$\mathbf j = \frac{\hbar}{2mi} \left( \psi^*\nabla\psi - \psi\nabla\psi^* \right).$$

In one dimension,

$$j = \frac{\hbar}{2mi} \left( \psi^*\frac{\partial\psi}{\partial x} - \psi\frac{\partial\psi^*}{\partial x} \right).$$

Scattering probabilities are current ratios. For a plane wave $Ae^{ikx}$ with $k>0$,

$$j=\frac{\hbar k}{m}\lvert A\rvert^2.$$

Use Probability Current for the formula and interpretation, and Continuity Equation for the conservation-law statement.

With electromagnetic vector potentials, the current changes. Pages using minimal coupling should link to Minimal Coupling in Wave Mechanics and state the sign convention for the charge.

Boundary-Condition Notation

Boundary conditions are written as equations on endpoint values, derivatives, or jumps. For a hard-wall interval $0<x<L$,

$$\psi(0)=0, \qquad \psi(L)=0.$$

For a periodic interval,

$$\psi(x+L)=\psi(x), \qquad \psi'(x+L)=\psi'(x).$$

Primes denote ordinary derivatives with respect to the displayed coordinate:

$$\psi'(x)=\frac{d\psi}{dx}.$$

A jump across $x=a$ may be written as

$$\psi'(a^+)-\psi'(a^-).$$

Matching pages should specify which quantities are continuous and which have controlled jumps. Use Boundary Conditions for the conceptual home and Boundary Conditions Table for quick lookup.

Dimensionless Variables

Dimensionless variables are introduced to expose the real parameters in a model. A common pattern is

$$\xi=\frac{x}{L}, \qquad \epsilon=\frac{E}{E_L}, \qquad E_L=\frac{\hbar^2}{2mL^2}.$$

For the harmonic oscillator, the natural length is

$$\ell=\sqrt{\frac{\hbar}{m\omega}}, \qquad \xi=\frac{x}{\ell}.$$

For hydrogenic problems, the natural length is the Bohr radius $a_0$, and radial pages should say whether $r/a_0$ or a state-dependent scaled radius is being used.

Use Dimensionless Variables and Scaling for the method and Dimensionless Parameters Table for quick lookup.

Spherical Coordinates And Radial Functions

The spherical-coordinate convention is:

$$\mathbf r=(r,\theta,\phi),$$

where $r\ge0$, $\theta$ is the polar angle from the positive $z$ axis, and $\phi$ is the azimuthal angle in the $xy$ plane. The volume element is

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

Separated central-potential states are usually written

$$\psi_{n\ell m}(r,\theta,\phi) = R_{n\ell}(r)Y_\ell^m(\theta,\phi),$$

or, using the reduced radial function,

$$\psi_{n\ell m}(r,\theta,\phi) = \frac{u_{n\ell}(r)}{r}Y_\ell^m(\theta,\phi).$$

Angular-momentum pages use the Condon–Shortley phase convention unless they explicitly state otherwise. Use Angular Momentum Conventions and Spherical Harmonics for the canonical angular conventions.

Common Translation Points

If A Source Uses…	Translate Here As…
$\tilde\psi(k)$ normalized in $dk$	compare with $\phi(p)$ using $p=\hbar k$ and $dp=\hbar\,dk$
natural units $\hbar=1$	restore $\hbar$ in phases, commutators, kinetic terms, and oscillator scales
$u(r)$ for a radial wavefunction	remember $u(r)=rR(r)$ and normalize with $dr$
amplitudes $A,B$ in scattering	compute current ratios before calling them probabilities
a box of length $L$	identify whether $L$ is physical or a regulator for continuum normalization
a different spherical-harmonic phase	state the phase convention before comparing matrix elements

These translation points are often more important than algebraic details. Two references may both be correct while using incompatible normalizations or phases.

Common Mistakes

• Treating $\psi(x)$ and $\phi(p)$ as the same wavefunction written with different letters.
• Mixing $k$-space and $p$-space Fourier normalizations.
• Forgetting that $\lvert\psi(x)\rvert^2$ is a density, not a probability by itself.
• Dropping the $r^2\sin\theta$ measure in spherical coordinates.
• Comparing scattering amplitudes without velocity or current factors.
• Using $R(r)$ and $u(r)$ interchangeably.
• Setting $\hbar=1$ silently in a page that otherwise keeps physical units.

Where This Is Used

• Dependency Graph uses these conventions when assigning prerequisite links.
• How to Solve a Wave-Mechanics Problem uses the same notation in its workflow.
• Normalization Conventions expands the normalization entries.
• Common Hamiltonians and Spectra and Eigenfunctions Table use these symbols in compact reference form.
• Representation Translation Table gives the broader translation among abstract, matrix, coordinate, and density-operator notation.
• Symbol Map is the broadest cross-volume symbol index.

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

Exercises

1. A reference defines $\tilde\psi(k)$ using $e^{ikx}$ and $\int\lvert\tilde\psi(k)\rvert^2\,dk=1$. How should it be compared with a momentum-space wavefunction $\phi(p)$ normalized in $dp$?

Solution

Use $p=\hbar k$ and $dp=\hbar\,dk$. The probability must be invariant:

$$\lvert\tilde\psi(k)\rvert^2\,dk = \lvert\phi(p)\rvert^2\,dp.$$

Therefore $\phi(p)=\tilde\psi(p/\hbar)/\sqrt{\hbar}$ up to the phase and Fourier-sign conventions used by the reference.

2. A central-potential calculation says $\int_0^\infty \lvert u(r)\rvert^2\,dr=1$ and also calls $u(r)$ the radial wavefunction. What convention should be clarified?

Solution

The page should say whether $u(r)$ is the reduced radial function. In the convention used here, $u(r)=rR(r)$, where $R(r)$ is normalized by

$$\int_0^\infty \lvert R(r)\rvert^2r^2\,dr=1.$$

The reduced function $u(r)$ is normalized with the measure $dr$.

3. In a scattering page, an author writes $T=\lvert A_{\mathrm{trans}}/A_{\mathrm{inc}}\rvert^2$ even though the incident and transmitted regions have different wavenumbers. What is missing?

Solution

Transmission should be a current ratio. For plane waves with wavenumbers $k_L$ and $k_R$,

$$T = \frac{j_{\mathrm{trans}}}{j_{\mathrm{inc}}} = \frac{k_R}{k_L} \frac{\lvert A_{\mathrm{trans}}\rvert^2} {\lvert A_{\mathrm{inc}}\rvert^2}$$

for equal masses and the standard one-dimensional current convention.