Notation and Conventions

Approximation formulas are unusually sensitive to local conventions. A reversed gap, a changed plane-wave normalization, an unstated state-normalization choice, or an opposite Fourier sign can make two correct derivations look inconsistent.

This page fixes the notation used across this volume. It is subordinate to the canonical Conventions Overview for sitewide policy and to the Scattering Convention Dictionary for detailed translation among scattering sources. For rapid lookup without the explanatory detail, use Common Symbols.

The rule is:

\text{define the symbol, its units, its normalization, and its limiting prescription}.

Sitewide Defaults

Unless a page states otherwise:

$\hbar$ is explicit.
$c$ is explicit outside declared relativistic or QFT natural-unit passages.
$\langle\phi\vert\psi\rangle$ is conjugate-linear in $\phi$ and linear in $\psi$ .
$[A,B]=AB-BA$ .
Hats are used when they prevent confusion between an operator and a value; $H$ , $V$ , and $S$ may be unhatted when context is unambiguous.
Bold symbols such as $\mathbf r$ , $\mathbf p$ , $\mathbf k$ , and $\mathbf q$ are three-vectors.
Complex conjugation is $z^*$ , the Hilbert-space adjoint is $A^\dagger$ , and the transpose is $A^{\mathsf T}$ .
No Einstein summation is implied in ordinary quantum-mechanics formulas unless a page explicitly declares indexed tensor notation.
Bound states are unit normalized; continuum and scattering states use an explicitly stated delta-function or flux normalization.

Use Units and Constants, Bra-Ket Notation, Inner Product Conventions, and Fourier Transform Conventions for canonical details.

Master Symbol Table

Symbol	Default meaning in this volume	Important qualification
$H_0$	Solvable or reference Hamiltonian	Need not be the free Hamiltonian
$V$	Perturbing interaction or scattering potential	May depend on time
$\lambda$	Perturbative bookkeeping or physical coupling parameter	Set to one only after order counting
$\epsilon$	A locally defined dimensionless control parameter	Never assumed universal
$\epsilon_{\mathrm{reg}}$	Positive resolvent regulator	Sent to zero from above
$\lvert n^{(0)}\rangle$	Zeroth-order eigenstate of $H_0$	Degenerate labels may require an extra index
$E_n^{(0)}$	Zeroth-order energy	Superscript is perturbative order, not a power
$\Delta_{nm}$	$E_n^{(0)}-E_m^{(0)}$	Order of indices fixes the sign
$P$ , $Q$	Complementary projectors, with $Q=I-P$	$P$ can also mean probability when arguments make that clear
$\mathcal R[\psi]$	Rayleigh quotient	A functional, not a density operator
$S(x,t)$	Hamilton principal function or action phase	Distinct from the scattering $S$ -matrix
$W(x)$	Reduced action in a stationary problem	Usually $S=W-Et$
$p(x)$	Positive local momentum magnitude in an allowed region	Propagation direction is carried by a sign in the phase
$\kappa(x)$	Positive forbidden-region momentum magnitude	$\kappa=\sqrt{2m(V-E)}$
$G_0^{(\pm)}(E)$	Free outgoing or incoming resolvent	Sign is tied to the stated Fourier convention
$S_{fi}$	Scattering-matrix element	May contain discrete and continuum delta functions
$T_{fi}$	Transition-operator matrix element	Definition depends on the chosen $S$ convention
$f(\Omega)$	Nonrelativistic scattering amplitude	Has dimensions of length in three dimensions
$\delta_\ell$	Partial-wave phase shift	Not a delta distribution
$\Gamma$	Transition or decay rate	Has units of inverse time

The same letter can have other meanings outside this volume. Local definitions override the table only when the page says so explicitly.

Static Perturbation Theory

Hamiltonian split

The default split is

H(\lambda) = H_0+\lambda V.

$H_0$ is the reference Hamiltonian and $V$ is the perturbing operator. The symbol $\lambda$ may be:

a physical dimensionless coupling;
a bookkeeping parameter introduced to count orders;
part of a scaled physical coupling after units have been removed.

If $\lambda$ is only bookkeeping, it is set to one after the dimensionless validity ratios have been identified.

Unperturbed eigenpairs

For a discrete nondegenerate reference spectrum,

H_0\lvert n^{(0)}\rangle = E_n^{(0)} \lvert n^{(0)}\rangle.

Matrix elements of $V$ are

V_{mn} \equiv \langle m^{(0)}\rvert V \lvert n^{(0)}\rangle.

The signed gap convention is

\Delta_{nm} \equiv E_n^{(0)}-E_m^{(0)}.

Thus the first-order state correction in intermediate normalization is

\lvert n^{(1)}\rangle = \sum_{m\ne n} \frac{V_{mn}}{\Delta_{nm}} \lvert m^{(0)}\rangle.

Some sources define $\Delta_{mn}=E_m^{(0)}-E_n^{(0)}$ . Their denominator and any explicit minus sign must be translated together.

Order labels

The expansions are

\begin{aligned} E_n(\lambda) &= \sum_{r=0}^{\infty} \lambda^r E_n^{(r)}, \\ \lvert n(\lambda)\rangle &= \sum_{r=0}^{\infty} \lambda^r \lvert n^{(r)}\rangle. \end{aligned}

Parenthesized superscripts such as $(0)$ and $(1)$ label perturbative order. They are not exponentiation. A page that uses a different ordering parameter must name it.

State normalization

Intermediate normalization means

\langle n^{(0)} \vert n(\lambda)\rangle = 1.

It makes every higher-order correction orthogonal to $\lvert n^{(0)}\rangle$ , but the resulting perturbative state is not automatically unit normalized. A page reporting expectation values beyond first order should state whether it has converted to unit normalization.

Degenerate subspaces

For a degenerate or nearly degenerate sector $\mathcal D$ ,

P = \sum_{a\in\mathcal D} \lvert a\rangle\langle a\rvert, \qquad Q=I-P.

The first-order operator inside the sector is $PVP$ . Adapted states within the sector may be labeled $\lvert\alpha^{(0)}\rangle$ , while $r$ or $q$ labels states outside it. A page should distinguish the basis index $a$ from the eigenstate label $\alpha$ after diagonalization.

Time-Dependent Perturbations and Rates

For a time-dependent problem,

H(t)=H_0+V(t).

When $H_0$ is time independent, the interaction-picture perturbation is

V_I(t) = e^{iH_0(t-t_0)/\hbar} V(t) e^{-iH_0(t-t_0)/\hbar}.

The transition frequency is

\omega_{fi} \equiv \frac{E_f-E_i}{\hbar}.

The interaction-picture matrix element is

\bigl(V_I\bigr)_{fi}(t) \equiv \langle f\rvert V_I(t)\lvert i\rangle.

With these definitions,

c_f^{(1)}(t) = -\frac{i}{\hbar} \int_{t_0}^{t} dt'\, \bigl(V_I\bigr)_{fi}(t').

If a source instead writes a Schrödinger-picture matrix element $V_{fi}(t)$ , it may display the phase $e^{i\omega_{fi}t}$ separately. Do not include that phase twice.

The leading transition probability is $P_{i\to f}(t)$ , while a rate is

\Gamma_{i\to f} \equiv \frac{dP_{i\to f}}{dt}

only in a regime where an approximately constant rate exists. $\rho(E)$ denotes a density of final states per unit energy unless another measure is stated. Degeneracy, polarization, spin, and channel sums must be displayed or explicitly included in $\rho$ .

Time ordering is written $\mathcal T$ , not $T$ , to avoid collision with the scattering transition operator.

Variational and Rayleigh–Ritz Notation

The Rayleigh quotient is

\mathcal R[\psi] \equiv \frac{ \langle\psi\rvert H\lvert\psi\rangle }{ \langle\psi\vert\psi\rangle }.

A parameterized trial family is $\lvert\psi(\boldsymbol\alpha)\rangle$ , where $\boldsymbol\alpha$ can contain real or complex variational parameters. The optimized ground-state estimate is

E_{\mathrm{var}} = \inf_{\boldsymbol\alpha} \mathcal R[\psi(\boldsymbol\alpha)].

For a finite basis $\{\lvert\phi_i\rangle\}$ ,

\lvert\psi\rangle = \sum_i c_i\lvert\phi_i\rangle,

with matrices

H_{ij} = \langle\phi_i\rvert H\lvert\phi_j\rangle, \qquad N_{ij} = \langle\phi_i\vert\phi_j\rangle.

The generalized eigenproblem is

\mathbf H\mathbf c = E\mathbf N\mathbf c.

Many sources call the overlap matrix $\mathbf S$ . This volume permits that notation on purely variational pages, but prefers $\mathbf N$ when a scattering $S$ -matrix appears nearby.

The residual of an approximate eigenpair is

\lvert r\rangle \equiv (H-E)\lvert\psi\rangle,

and the energy variance is

\sigma_H^2 \equiv \langle H^2\rangle-\langle H\rangle^2.

These are diagnostics, not additional variational parameters.

Semiclassical Action and WKB Notation

Action functions

$S$ denotes an action or Hamilton principal function when it carries trajectory, endpoint, or spacetime arguments. In one dimension,

\frac{\partial S}{\partial t} + H\!\left( x, \frac{\partial S}{\partial x} \right) = 0.

For a stationary problem,

S(x,t)=W(x)-Et,

where $W(x)$ is the reduced action and

p(x)=\frac{dW}{dx}.

If a page also uses the scattering matrix, it writes $S_{\mathrm{scatt}}$ or $\mathsf S$ when context does not make the distinction immediate.

Allowed and forbidden momenta

In an allowed region,

p(x) \equiv \sqrt{2m\bigl(E-V(x)\bigr)} \gt0.

The positive square root is the local momentum magnitude. Right- and left-moving branches are carried by the sign in

\psi_\pm(x) \approx \frac{C_\pm}{\sqrt{p(x)}} \exp\left[ \pm \frac{i}{\hbar} \int^x p(x')\,dx' \right].

In a forbidden region,

\kappa(x) \equiv \sqrt{2m\bigl(V(x)-E\bigr)} \gt0.

Growing and decaying branches use $\exp(\pm\int\kappa\,dx/\hbar)$ . The symbols $p$ and $\kappa$ are not analytically continued into one another unless a page explicitly adopts that convention.

Turning points and action integrals

A turning point $x_t$ satisfies

E=V(x_t).

$x_1$ and $x_2$ usually denote ordered turning points with $x_1\lt x_2$ . A closed action integral is

\oint p\,dq.

For one integrable degree of freedom, the EBK convention used here is

\oint p\,dq = 2\pi\hbar \left( n+\frac{\mu_{\mathrm M}}{4} \right),

where $\mu_{\mathrm M}$ is the Maslov index. The subscript distinguishes it from a reduced mass $\mu$ .

Euclidean action is $S_E$ and contributes weights such as $e^{-S_E/\hbar}$ . A prime on a determinant, $\det'$ , means that specified zero modes have been removed and treated separately.

Nonrelativistic Scattering Conventions

Mass, energy, and wave number

$m$ is a one-particle mass. For a two-body relative-coordinate problem, $\mu$ is the reduced mass. On shell,

E = \frac{\hbar^2k^2}{2\mu}, \qquad k=\lvert\mathbf k\rvert.

When the problem is explicitly one-body, replace $\mu$ by $m$ .

Plane-wave normalization

Wave-number states use

\langle\mathbf r\vert\mathbf k\rangle = \frac{1}{(2\pi)^{3/2}} e^{i\mathbf k\cdot\mathbf r},

so that

\langle\mathbf k\vert\mathbf k'\rangle = \delta^{(3)}(\mathbf k-\mathbf k').

Momentum-normalized states instead use

\langle\mathbf r\vert\mathbf p\rangle = \frac{1}{(2\pi\hbar)^{3/2}} e^{i\mathbf p\cdot\mathbf r/\hbar}.

Because $\mathbf p=\hbar\mathbf k$ , matrix elements and delta functions acquire powers of $\hbar$ when translating between the two.

Momentum transfer

For elastic scattering,

\mathbf q \equiv \mathbf k'-\mathbf k.

The Fourier phase in the Born amplitude is therefore $e^{-i\mathbf q\cdot\mathbf r}$ . Sources using $\mathbf q=\mathbf k-\mathbf k'$ reverse this phase. For a real central potential, the transform depends only on $q=\lvert\mathbf q\rvert$ , so the sign can be hidden.

Asymptotic amplitude

For a short-range potential,

\psi_{\mathbf k}^{(+)}(\mathbf r) \underset{r\to\infty}{\sim} e^{i\mathbf k\cdot\mathbf r} + f(\theta,\phi) \frac{e^{ikr}}{r}.

With unit incident plane-wave amplitude,

\frac{d\sigma}{d\Omega} = \lvert f(\theta,\phi)\rvert^2.

$f$ has dimensions of length. It is not the relativistic invariant amplitude $\mathcal M$ .

S- and T-matrix convention

T-Matrix is the canonical derivation of the transition operator and its normalization factors. The local notation is summarized here for comparison.

For energy-normalized channels, the default relation is

S_{fi} = \delta_{fi} - 2\pi i\, \delta(E_f-E_i) T_{fi}.

Here $\delta_{fi}$ represents the identity in the chosen mixture of discrete and continuous channel labels. The transition operator satisfies

T(E) = V + V G_0^{(+)}(E)T(E).

Some sources write $S=I+i\mathcal T$ . Their $\mathcal T$ includes different signs, normalization factors, and delta distributions. Do not transplant an optical-theorem formula between conventions without translating the complete definition.

Partial waves

For one-channel elastic central scattering,

S_\ell=e^{2i\delta_\ell}.

The amplitude is

f(\theta) = \frac{1}{2ik} \sum_{\ell=0}^{\infty} (2\ell+1) \bigl(S_\ell-1\bigr) P_\ell(\cos\theta).

$P_\ell$ is a Legendre polynomial; it is not the projector $P$ . The equivalent phase-shift form is

f(\theta) = \frac{1}{k} \sum_{\ell=0}^{\infty} (2\ell+1) e^{i\delta_\ell} \sin\delta_\ell P_\ell(\cos\theta).

If inelastic channels are open, an elastic element may be written

S_\ell = \eta_\ell^{\mathrm{inel}} e^{2i\delta_\ell}, \qquad 0\le\eta_\ell^{\mathrm{inel}}\le1.

The superscript prevents confusion with the Coulomb Sommerfeld parameter, also commonly called $\eta$ .

Cross sections and the optical theorem

$d\sigma/d\Omega$ is a differential cross section, $\sigma_{\mathrm{el}}$ is an elastic cross section, and $\sigma_{\mathrm{tot}}$ includes all open final channels represented by the $S$ -matrix. With the amplitude convention above,

\sigma_{\mathrm{tot}} = \frac{4\pi}{k} \operatorname{Im}f(0).

Identical-particle symmetrization, spin averages, target densities, and relative-velocity factors are not implicit unless stated.

Resolvents and Boundary Prescriptions

The free resolvents are

\begin{aligned} G_0^{(+)}(E) &\equiv \lim_{\epsilon_{\mathrm{reg}}\to0^+} \frac{1} {E-H_0+i\epsilon_{\mathrm{reg}}}, \\ G_0^{(-)}(E) &\equiv \lim_{\epsilon_{\mathrm{reg}}\to0^+} \frac{1} {E-H_0-i\epsilon_{\mathrm{reg}}}. \end{aligned}

The shorthand is

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

With the time and Fourier conventions used here, $+i0$ imposes outgoing scattering behavior and $-i0$ imposes incoming behavior. The sign is a boundary condition, not an infinitesimal energy correction that may be discarded before the distributional limit is taken.

The scalar identity is

\frac{1}{x\pm i0} = \operatorname{PV}\frac{1}{x} \mp i\pi\delta(x).

Therefore

G_0^{(+)}(E) - G_0^{(-)}(E) = -2\pi i\, \delta(E-H_0).

The regulator $\epsilon_{\mathrm{reg}}$ is positive and dimensional. It is distinct from a dimensionless approximation parameter $\epsilon$ . For the operator-theoretic and dynamical meanings, see Resolvent Operator and Retarded and Advanced Green Functions.

Effective-Hamiltonian Notation

$P$ projects onto the retained sector and $Q=I-P$ onto the eliminated sector. The exact energy-dependent Feshbach operator is

H_{\mathrm{eff}}(E) = PHP + PHQ \frac{1}{E-QHQ} QHP,

where the inverse carries the boundary prescription required by the problem.

$\Delta$ without two state indices usually denotes a positive characteristic separation between retained and eliminated sectors. This differs from the signed perturbative gap $\Delta_{nm}$ .

A block-diagonalizing unitary transformation may be written

\widetilde H = e^{A}He^{-A}, \qquad A^\dagger=-A.

Some sources use $S$ for the anti-Hermitian Schrieffer–Wolff generator. This volume prefers $A$ when a scattering $S$ -matrix is present. The effective Hamiltonian at retained order $r$ may be labeled $H_{\mathrm{eff}}^{[r]}$ ; square brackets distinguish truncation labels from ordinary powers when ambiguity matters.

For periodic driving, $\Omega$ is the drive angular frequency and $T_{\mathrm d}=2\pi/\Omega$ is the drive period. Time ordering remains $\mathcal T$ .

Approximation and Remainder Symbols

The following symbols carry different claims:

Symbol	Meaning
$=$	Equality within the stated model and definitions
$\equiv$	Definition
$\approx$	Approximate numerical or functional equality; the regime should be stated
$f\sim g$	Usually $f/g\to1$ in a declared limit
$f\sim\sum_n a_n\epsilon^n$	An asymptotic expansion, defined through finite remainders
$O(\epsilon^n)$	Bounded in magnitude by a constant times $\lvert\epsilon\rvert^n$ in the stated limit
$o(\epsilon^n)$	Smaller than $\epsilon^n$ in ratio in the stated limit
$\propto$	Equal up to a factor not being displayed

In a scattering boundary condition, $\sim$ means equality of the displayed leading asymptotic form up to subleading terms; it need not be interpreted as the ratio of the full oscillatory expressions tending pointwise to one.

A truncation should be written with its remainder, for example

Q(\lambda) = Q^{(0)} + \lambda Q^{(1)} + O(\lambda^2),

and the limit behind the $O$ symbol should be clear from context or stated explicitly.

Index Conventions

Labels	Typical use
$n,m$	Discrete reference eigenstates
$i,f$	Initial and final states or channels
$a,b$	Basis states inside a degenerate or variational subspace
$\alpha,\beta$	Adapted states, channels, or continuous parameters
$r,q$	States outside a retained subspace; $q$ is not used this way when it already denotes momentum transfer
$\ell,m$	Orbital angular-momentum quantum numbers
$s,m_s$	Spin and spin projection
$\mathbf k,\mathbf k'$	Incoming and outgoing wave vectors
$\mathbf q$	Momentum-transfer wave vector

When one page uses the same label in two roles, it should rename one of them. Compact notation is not worth an ambiguous sum.

Collision Warnings

Symbol	Meanings encountered here	Preferred disambiguation
$S$	Action; scattering matrix; overlap matrix in some variational texts	Use $S(x,t)$ , $S_{fi}$ , and $N_{ij}$
$T$	Transition operator; transmission probability; time ordering	Use $T(E)$ , $T_{\mathrm{trans}}$ , and $\mathcal T$ , respectively
$P$	Projector; probability; Legendre polynomial	Use operator context, $P_{i\to f}$ , and $P_\ell(x)$
$\delta$	Dirac delta; Kronecker delta; phase shift	Show arguments or use $\delta_\ell$
$\epsilon$	Dimensionless control; regulator; energy tolerance	Use $\epsilon_{\mathrm{reg}}$ or a descriptive subscript
$\eta$	Inelasticity; Sommerfeld parameter; adiabaticity indicator	Use $\eta_\ell^{\mathrm{inel}}$ , $\eta_{\mathrm C}$ , or $\eta_{\mathrm{ad}}$
$\rho$	Density of states; density operator; spatial density	Use $\rho(E)$ , $\hat\rho$ , or $\rho(\mathbf r)$
$\Gamma$	Rate; gamma function with uppercase argument notation	Use $\Gamma_{i\to f}$ or $\Gamma(z)$
$W$	Reduced action; matrix inside a degenerate subspace in some texts	Use $W(x)$ and $PVP$
$\mu$	Reduced mass; Maslov index in some sources	Use $\mu$ and $\mu_{\mathrm M}$

The broader canonical inventory is Notation Collisions.

Translating an External Formula

Before importing a formula from another source:

Identify whether states are normalized in $\mathbf k$ , $\mathbf p$ , energy, volume, or relativistic phase space.
Record the Fourier sign and all $2\pi$ and $\hbar$ factors.
Translate signed energy gaps and the order of their indices.
Check whether $S$ , $T$ , $f$ , and $\mathcal M$ have been defined before comparing them.
Determine what $+i0$ produces under the source’s time-Fourier convention.
Match amplitude-level quantities before squaring and adding phase-space or flux factors.
Compare a complete observable, dimensionality, and limiting case.

Two formulas that look different can encode the same physics. Two formulas that look identical can refer to differently normalized objects.

Worked Translations

Reversing a gap convention

Suppose a source defines

\widetilde\Delta_{mn} = E_m^{(0)}-E_n^{(0)} = -\Delta_{nm}.

Then

\frac{V_{mn}}{\Delta_{nm}} = -\frac{V_{mn}}{\widetilde\Delta_{mn}}.

The state correction is unchanged only if the denominator definition and explicit sign are translated together.

Wave number versus momentum normalization

In one dimension,

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

If both bases are delta normalized, then

\lvert k\rangle = \sqrt{\hbar}\, \lvert p=\hbar k\rangle.

In three dimensions the factor is $\hbar^{3/2}$ . This Jacobian is why powers of $\hbar$ move between plane-wave states, matrix elements, and densities of states.

Resolvent signs

The pair

\frac{1}{E-H_0+i0}, \qquad \frac{1}{E-H_0-i0}

uses the same differential operator but different boundary values across the continuous spectrum. They cannot be made equivalent by dropping the infinitesimal before solving the boundary-value problem.

Common Mistakes

Setting $\lambda=1$ before identifying the physical dimensionless expansion parameter.
Reversing $\Delta_{nm}$ without reversing the corresponding sign.
Reading the superscript in $E_n^{(2)}$ as a square.
Treating intermediate normalization as unit normalization.
Including the interaction-picture phase twice.
Calling a stationary variational estimate an upper bound without checking the theorem used.
Letting $p(x)$ carry both a signed direction and a positive WKB magnitude on the same page.
Confusing the action $S$ with the scattering matrix $S$ .
Mixing $\mathbf k$ - and $\mathbf p$ -normalized continuum states.
Dropping $i0$ before it has imposed a boundary condition.
Identifying the nonrelativistic $f$ with a relativistic invariant amplitude $\mathcal M$ .
Comparing optical-theorem formulas from incompatible $S$ - and $T$ -matrix conventions.

Exercises

Translate a signed denominator

A source writes

\lvert n^{(1)}\rangle = -\sum_{m\ne n} \frac{V_{mn}}{\Delta_{mn}^{\mathrm{src}}} \lvert m^{(0)}\rangle,

with $\Delta_{mn}^{\mathrm{src}}=E_m^{(0)}-E_n^{(0)}$ . Show that this matches the convention used here.

Solution

Here

\Delta_{nm} = E_n^{(0)}-E_m^{(0)} = -\Delta_{mn}^{\mathrm{src}}.

Therefore

-\frac{1}{\Delta_{mn}^{\mathrm{src}}} = \frac{1}{\Delta_{nm}},

and the two state-correction formulas are identical.

Convert continuum bases

In three dimensions, derive the relation between $\lvert\mathbf k\rangle$ and $\lvert\mathbf p\rangle$ when both are delta normalized and $\mathbf p=\hbar\mathbf k$ .

Solution

The delta functions obey

\delta^{(3)}(\mathbf p-\mathbf p') = \frac{1}{\hbar^3} \delta^{(3)}(\mathbf k-\mathbf k').

Writing $\lvert\mathbf k\rangle=C\lvert\mathbf p=\hbar\mathbf k\rangle$ gives

\langle\mathbf k\vert\mathbf k'\rangle = \frac{\lvert C\rvert^2}{\hbar^3} \delta^{(3)}(\mathbf k-\mathbf k').

Thus one may choose

\lvert\mathbf k\rangle = \hbar^{3/2} \lvert\mathbf p=\hbar\mathbf k\rangle.

The phase of $C$ is conventional.

Read the i0 prescription

Using

\frac{1}{x+i0} = \operatorname{PV}\frac{1}{x} - i\pi\delta(x),

compute $G_0^{(+)}(E)-G_0^{(-)}(E)$ .

Solution

The principal-value parts cancel:

\begin{aligned} G_0^{(+)}(E)-G_0^{(-)}(E) &= \left( \operatorname{PV}\frac{1}{E-H_0} - i\pi\delta(E-H_0) \right) \\ &\quad- \left( \operatorname{PV}\frac{1}{E-H_0} + i\pi\delta(E-H_0) \right) \\ &= -2\pi i\,\delta(E-H_0). \end{aligned}

Recover the phase-shift amplitude

Starting from $S_\ell=e^{2i\delta_\ell}$ , show that

\frac{S_\ell-1}{2i} = e^{i\delta_\ell}\sin\delta_\ell.

Solution

Factor the difference:

\begin{aligned} e^{2i\delta_\ell}-1 &= e^{i\delta_\ell} \left( e^{i\delta_\ell} - e^{-i\delta_\ell} \right) \\ &= 2i e^{i\delta_\ell} \sin\delta_\ell. \end{aligned}

Dividing by $2i$ gives the stated identity and the equivalent partial-wave amplitude.

Resolve three S symbols

A draft uses $S$ for the classical action, the scattering matrix, and the variational overlap matrix in one section. Rewrite the three symbols using this page’s collision policy.

Solution

Write the action with arguments, such as $S(x,t)$ or $S[\gamma]$ . Write the scattering object as $S_{fi}$ , $S_\ell$ , or $S_{\mathrm{scatt}}$ . Write the overlap matrix as

N_{ij} = \langle\phi_i\vert\phi_j\rangle.

The meanings are then visible from typography and indices rather than inferred from surrounding prose.

Cross-Links

References

J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Vols. 1–2, Wiley, 1977.
L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Butterworth-Heinemann, 1981.
J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.
R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Dover, 2002.
M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics”, Reports on Progress in Physics 35, 315–397, 1972.
T. Kato, Perturbation Theory for Linear Operators, corrected printing of the 2nd ed., Springer, 1995.