Common Symbols

This page is a fast symbol index. It answers “what does this letter most likely mean here?” and links to definitions and worked examples.

Notation and Conventions explains how to use these definitions in calculations. The Scattering Convention Dictionary fixes continuum-state normalizations and amplitude factors.

Local definitions always win. A symbol is trustworthy only together with its arguments, indices, units, and surrounding equation.

Mathematical Notation

The typography carries information:

Form	Usual role
$\hat A$	operator when the hat prevents ambiguity
$A$	operator or scalar when context is already clear
$\mathbf a$	ordinary three-vector or coefficient vector
$\lvert a\rangle$	state vector
$A_{mn}$	matrix element or component
$A^{(r)}$	perturbative coefficient of order $r$
$A^r$	ordinary power
$A^{[r]}$	approximation truncated through a stated order
$A(E\pm i0)$	boundary value from above or below the real energy axis
$A_{\alpha\to\beta}$	process or channel label, not multiplication

An unadorned symbol should not be assigned a meaning from memory when the page provides a local definition.

Perturbation and Approximation Symbols

Symbol	Default meaning	Qualification
$H$	full Hamiltonian	Often $H=H_0+\lambda V$
$H_0$	exactly solved or reference Hamiltonian	Need not be the free Hamiltonian
$V$	perturbing interaction or scattering potential	May depend on time or internal variables
$\lambda$	order-counting or physical coupling parameter	Set $\lambda=1$ only after counting orders
$\epsilon$	locally defined dimensionless control parameter	There is no universal approximation parameter
$\lvert n^{(0)}\rangle$	zeroth-order eigenstate	Extra labels are needed inside a degenerate eigenspace
$E_n^{(0)}$	zeroth-order energy	Parentheses denote order, not a power
$E_n^{(r)}$	order-$r$ energy coefficient	Usually multiplies $\lambda^r$
$\lvert n^{(r)}\rangle$	order-$r$ state coefficient	Its component along $\lvert n^{(0)}\rangle$ depends on normalization convention
$V_{mn}$	$\langle m^{(0)}\rvert V\lvert n^{(0)}\rangle$	Basis and time dependence should be stated
$\Delta_{nm}$	$E_n^{(0)}-E_m^{(0)}$	Reversing indices reverses the sign
$P$	projector onto a retained subspace	Can also mean probability with process labels
$Q$	complementary projector $I-P$	Distinct from a reaction Q-value

The core static expansion is

$$\begin{aligned} E_n(\lambda) &= \sum_{r=0}^{N} \lambda^r E_n^{(r)} + O(\lambda^{N+1}), \\ \lvert n(\lambda)\rangle &= \sum_{r=0}^{N} \lambda^r \lvert n^{(r)}\rangle + O(\lambda^{N+1}). \end{aligned}$$

Nondegenerate Perturbation Theory and Degenerate Perturbation Theory derive the corresponding formulas and gap conditions.

Time-Dependent Transitions

Symbol	Default meaning	Units or warning
$V_I(t)$	perturbation in the interaction picture	energy
$V_{fi}(t)$	$\langle f\rvert V(t)\lvert i\rangle$ in the Schrödinger picture	energy
$(V_I)_{fi}(t)$	$\langle f\rvert V_I(t)\lvert i\rangle$ in the interaction picture	energy; includes the reference-evolution phase
$\omega_{fi}$	$(E_f-E_i)/\hbar$	angular frequency
$\Omega$	drive angular frequency	A Rabi frequency should carry a clarifying subscript when needed
$c_f^{(r)}(t)$	order-$r$ transition-amplitude coefficient	dimensionless
$P_{i\to f}(t)$	transition probability	dimensionless
$\Gamma_{i\to f}$	transition or decay rate	inverse time
$\rho(E)$	density of final states per unit energy	can include stated degeneracy factors
$\mathcal T$	time-ordering operator	$T$ is reserved for scattering or transmission

For eigenstates of a time-independent reference $H_0$, with the interaction picture based at $t_0$, these two matrix elements obey $(V_I)_{fi}(t)=e^{i\omega_{fi}(t-t_0)}V_{fi}(t)$. Integrate $(V_I)_{fi}$ directly in a first-order amplitude; do not multiply it by that phase again. Notation and Conventions spells out the picture transformation.

For a static weak interaction and a smooth continuum in the golden-rule time window, the rate pattern is

$$\Gamma_{i\to f} = \frac{2\pi}{\hbar} |V_{fi}|^2 \rho(E_f),$$

with energy conservation and all continuum measures understood according to the stated normalization. Fermi’s Golden Rule explains the continuum and time-window assumptions.

Variational and Effective-Subspace Symbols

Symbol	Default meaning	Qualification
$\mathcal R[\psi]$	Rayleigh quotient	$\langle\psi\vert H\vert\psi\rangle/\langle\psi\vert\psi\rangle$
$\boldsymbol\theta$	variational parameters	Components may be real or complex
$E_{\mathrm{var}}$	variational energy estimate	Upper bound only under the variational theorem’s assumptions
$c_i$	basis-expansion coefficient	Normalization can involve an overlap matrix
$H_{ij}$	Hamiltonian matrix element	$\langle\phi_i\vert H\vert\phi_j\rangle$
$N_{ij}$	overlap matrix	$\langle\phi_i\vert\phi_j\rangle$; avoids overloading $S$
$H_{\mathrm{eff}}(E)$	effective Hamiltonian in a retained subspace	May be energy dependent
$A$	anti-Hermitian Schrieffer–Wolff generator	Preferred over $S$ when scattering is also discussed
$T_{\mathrm d}$	drive period	$T_{\mathrm d}=2\pi/\Omega$

For projection methods,

$$P+Q=I, \qquad PQ=0,$$

and the eliminated subspace often enters through

$$PHQ \frac{1}{E-QHQ} QHP.$$

Feshbach Projection Formalism derives the exact projection identity. The inverse is a genuine inverse only off the spectrum of $QHQ$; a continuum requires its boundary prescription.

WKB and Semiclassical Symbols

Symbol	Default meaning	Units or warning
$S(x,t)$	Hamilton principal function or action phase	action
$W(x)$	reduced action in a stationary problem	action; often $S=W-Et$
$p(x)$	positive local momentum magnitude in an allowed region	momentum
$\kappa(x)$	positive forbidden-region momentum magnitude	momentum in this volume
$x_t$ or $x_j$	turning point	satisfies $V(x_t)=E$
$K(E)$	forbidden-region action $\int\kappa(x)\,dx$	action
$\mu_{\mathrm M}$	Maslov index	dimensionless integer
$S_E$	Euclidean action	action
$S_{\mathrm{inst}}$	instanton action	action
$\mathcal A$	fluctuation prefactor or semiclassical amplitude	dimensions depend on the observable

In an allowed region,

$$p(x) = \sqrt{2m[E-V(x)]}.$$

In a forbidden region,

$$\kappa(x) = \sqrt{2m[V(x)-E]}.$$

Some sources divide $\kappa$ by $\hbar$ and call it a decay constant with units of inverse length. Check the exponent: $e^{-\int\kappa\,dx/\hbar}$ uses momentum units, while $e^{-\int\kappa_{\mathrm{decay}}\,dx}$ uses inverse-length units.

WKB Approximation and Barrier Penetration and Tunneling develop the connection formulas and action estimates.

Scattering Symbols

Symbol	Default meaning	Qualification
$\mathbf k,\mathbf k'$	incoming and outgoing wave vectors	$p=\hbar k$
$\mathbf q$	wave-vector transfer	Here $\mathbf q=\mathbf k'-\mathbf k$; momentum transfer is $\hbar\mathbf q$
$k_a$	channel wave number	$E=E_a^{\mathrm{th}}+\hbar^2k_a^2/(2\mu_a)$
$f(\Omega)$	three-dimensional scattering amplitude	length
$d\sigma/d\Omega$	differential cross section	area per unit solid angle
$\sigma_{\mathrm{el}}$	elastic cross section	area
$\sigma_{\mathrm{tot}}$	inclusive total cross section	state included channels
$\delta_\ell$	partial-wave phase shift	dimensionless; not a delta distribution
$S$	scattering operator or matrix	dimensionless in a flux-normalized channel basis
$S_\ell$	partial-wave $S$-matrix eigenvalue	$e^{2i\delta_\ell}$ in one elastic channel
$\eta_\ell^{\mathrm{inel}}$	partial-wave inelasticity	$0\le\eta_\ell^{\mathrm{inel}}\le1$
$T(E)$	transition operator	not a transmission probability
$G_0^{(\pm)}(E)$	free outgoing or incoming resolvent	inverse energy
$a$	scattering length	length
$r_e$	effective range	length
$Q_{\beta\alpha}$	channel Q-value	sign convention must be stated
$r,t$	one-dimensional reflection and transmission amplitudes	generally complex
$R,T_{\mathrm{trans}}$	one-dimensional reflection and transmission probabilities	dimensionless current ratios
$M$	one-dimensional transfer matrix	convention dependent

For a unit-incident coordinate wavefunction, the nonrelativistic asymptotic convention is

$$\psi_{\mathbf k}^{(+)}(\mathbf r) \sim e^{i\mathbf k\cdot\mathbf r} + f(\Omega)\frac{e^{ikr}}{r},$$

so

$$\frac{d\sigma}{d\Omega} = |f(\Omega)|^2$$

for one elastic channel in the matching normalization. Channel changes require the outgoing-to-incoming velocity ratio. Differential and Total Cross Sections derives the flux factors.

The resolvents are

$$G_0^{(\pm)}(E) = \frac{1}{E-H_0\pm i0}.$$

The sign labels outgoing or incoming boundary conditions only after the Fourier convention has been fixed. Lippmann–Schwinger Equation explains that prescription.

Approximation and Remainder Notation

Symbol	Claim
$=$	equality within the stated model and definitions
$\equiv$	definition
$\approx$	controlled or empirically accurate approximation in a stated regime
$f\sim g$	usually $f/g\to1$ in a declared limit
$f\sim\sum_n a_n\epsilon^n$	asymptotic expansion, not necessarily a convergent series
$O(\epsilon^n)$	bounded by a constant times $\vert\epsilon\vert^n$ in the stated limit
$o(\epsilon^n)$	smaller than $\epsilon^n$ in ratio
$\propto$	equality up to an omitted factor

A remainder symbol is incomplete without its limit. For example,

$$\begin{gathered} E(\lambda) = E^{(0)} + \lambda E^{(1)} + O(\lambda^2), \\ \lambda\to0. \end{gathered}$$

states both the truncation and the asymptotic variable.

Dimensions at a Glance

Quantity	Typical dimensions
$E,H,V,\Delta$	energy
$t,T_{\mathrm d}$	time
$\omega,\Omega,\Gamma_{\mathrm{rate}}$	inverse time
$\Gamma_{\mathrm{width}}$	energy
$S,W,K,S_E$	action
$p,\kappa$	momentum
$k,q$	inverse length
$f,a,r_e$	length
$\sigma$	area
$\rho(E)$	inverse energy, before extra continuum variables
$G_0(E)$	inverse energy
$\delta_\ell,S_\ell,\eta_\ell,R,T_{\mathrm{trans}}$	dimensionless

As an abstract operator, $T(E)=V+VG_0T$ has energy units. Its continuum matrix elements acquire additional dimensions from the normalization of the kets. Compare full amplitudes after matching that normalization.

Rates and widths deserve special care:

$$\Gamma_{\mathrm{width}} = \hbar\Gamma_{\mathrm{rate}}$$

when the same decay is described in energy and inverse-time units. Many texts use $\Gamma$ for the energy width and write the lifetime as $\tau=\hbar/\Gamma$.

Frequent Symbol Collisions

Symbol	Possible meanings here	Preferred disambiguation
$S$	action; scattering matrix; overlap matrix	$S(x,t)$, $S_{fi}$, and $N_{ij}$
$T$	transition operator; transmission probability; time ordering	$T(E)$, $T_{\mathrm{trans}}$, and $\mathcal T$
$P$	projector; probability; Legendre polynomial	$P$, $P_{i\to f}$, and $P_\ell(x)$
$Q$	complementary projector; reaction Q-value	$Q$ and $Q_{\beta\alpha}$
$\delta$	Dirac delta; Kronecker delta; phase shift	show arguments, indices, or $\delta_\ell$
$\epsilon$	control parameter; regulator; tolerance	$\epsilon_{\mathrm{reg}}$ or a descriptive subscript
$\eta$	inelasticity; Sommerfeld parameter; adiabaticity measure	$\eta_\ell^{\mathrm{inel}}$, $\eta_{\mathrm C}$, $\eta_{\mathrm{ad}}$
$\rho$	density of states; density operator; spatial density	$\rho(E)$, $\hat\rho$, $\rho(\mathbf r)$
$\Gamma$	transition rate; energy width; gamma function	attach process labels or arguments
$\mu$	reduced mass; Maslov index	$\mu$ and $\mu_{\mathrm M}$
$\lambda$	coupling; order-counting variable; eigenvalue	define before expanding

The sitewide inventory is Notation Collisions.

Common Indices

Labels	Typical role
$n,m$	discrete reference eigenstates
$i,f$	initial and final states or channels
$a,b$	basis states in a retained or variational subspace
$\alpha,\beta$	channels, adapted states, or continuous parameters
$\ell,m$	orbital angular momentum and its projection
$s,m_s$	spin and spin projection
$\mathbf k,\mathbf k'$	incoming and outgoing wave vectors
$\mathbf q$	wave-vector transfer; momentum transfer is $\hbar\mathbf q$
$r$ in parentheses	perturbative order, as in $E^{(r)}$

Rename an index when it collides with a physical symbol already active in the same equation.

Quick Translation Examples

Which S?

In

$$\psi(x) \sim A(x)e^{iS(x)/\hbar},$$

$S(x)$ is an action. In

$$\lvert\psi_{\mathrm{out}}\rangle = S\lvert\psi_{\mathrm{in}}\rangle,$$

$S$ is the scattering operator. The arguments and units separate the two immediately.

Which Gamma?

If

$$P(t)\approx e^{-\Gamma t},$$

then $\Gamma$ is a rate. If

$$E_{\mathrm{pole}} = E_R-\frac{i}{2}\Gamma,$$

then $\Gamma$ is an energy width. The two are related by $\Gamma_{\mathrm{width}}=\hbar\Gamma_{\mathrm{rate}}$.

Which T?

In

$$\frac{d\sigma}{d\Omega} \propto |\langle\mathbf k'|T(E)|\mathbf k\rangle|^2,$$

$T(E)$ is the transition operator. In one-dimensional scattering,

$$T_{\mathrm{trans}} = \frac{j_{\mathrm{trans}}}{j_{\mathrm{inc}}},$$

it is a probability. Time ordering is $\mathcal T$.

Lookup Checks

1. Rate or width?

A paper writes $\tau=\hbar/\Gamma$. What units must $\Gamma$ have?

Solution

Since $\hbar$ has units of energy times time, $\Gamma$ must have units of energy. It is a decay width. The corresponding inverse-time rate is $\Gamma/\hbar$.

2. Momentum or decay constant?

A WKB expression is

$$\exp\left[ -\frac{1}{\hbar} \int_{x_1}^{x_2}\kappa(x)\,dx \right].$$

What units does $\kappa$ have?

Solution

The exponent must be dimensionless. Because $dx$ has units of length and $\hbar$ has units of action, $\kappa$ must have units of momentum. If a source instead writes $e^{-\int\kappa\,dx}$, its $\kappa$ is an inverse-length decay constant.

3. Phase shift or delta distribution?

Why is $\delta_\ell$ unambiguous in

$$S_\ell=e^{2i\delta_\ell}?$$

Solution

The angular-momentum index $\ell$, the dimensionless exponent, and the surrounding partial-wave $S$-matrix identify $\delta_\ell$ as a phase shift. A Dirac delta would display an argument such as $\delta(E_f-E_i)$, and a Kronecker delta would carry two discrete indices such as $\delta_{\ell\ell'}$.

References

• C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Springer, 1999.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• A. Messiah, Quantum Mechanics, Dover, 1999.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Wiley, 1972.