Conventions for Perturbation Theory and WKB

A sign or normalization change can make two correct approximation formulas look inconsistent. This page explains the local choices most likely to matter: signed energy gaps, perturbative order, state normalization, WKB momenta, and resolvent boundary values. Use the scattering convention dictionary for the detailed normalization of collision amplitudes.

Helpful background. The general conventions explain the bra–ket, adjoint, and unit conventions used throughout these calculations.

Perturbative orders and signed gaps

We write

$$H(\lambda)=H_0+\lambda V, \qquad H_0\lvert n^{(0)}\rangle=E_n^{(0)}\lvert n^{(0)}\rangle.$$

Here $\lambda$ is dimensionless. It may be a physical coupling or a parameter used to count orders; in the latter case it is set to one after the actual small ratios have been identified. A superscript in parentheses labels an order, not a power:

$$E_n(\lambda)=E_n^{(0)}+\lambda E_n^{(1)} +\lambda^2E_n^{(2)}+\cdots.$$

Such a series may be formal or asymptotic. Writing it does not assert convergence. A finite-order result should state its remainder and the limit being taken.

The matrix element and signed denominator are

$$V_{mn}=\langle m^{(0)}\rvert V\lvert n^{(0)}\rangle, \qquad \Delta_{nm}=E_n^{(0)}-E_m^{(0)}.$$

Thus the coefficient of $\lvert m^{(0)}\rangle$ in the first-order correction to state $n$ is $V_{mn}/\Delta_{nm}$. Reversing the gap convention requires an explicit minus sign. For the ground state, denominators involving higher states are negative; this is a quick check on a second-order energy shift.

Intermediate normalization and unit normalization

Intermediate normalization sets

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

It makes the higher-order correction vectors orthogonal to the reference state but does not keep the full state’s norm equal to one. For example, if $\lvert\psi\rangle=\lvert0\rangle+\lambda\lvert u\rangle$ with $\langle0\vert u\rangle=0$ and $\langle0\vert0\rangle=1$, then

$$\langle\psi\vert\psi\rangle =1+\lambda^2\langle u\vert u\rangle$$

for real $\lambda$. Either divide an expectation value by this norm or normalize the state consistently to the order needed. The state-correction page gives the perturbative construction; the normalization example shows the second-order parallel correction.

For a degenerate sector, $P$ projects onto the states retained together and $Q=I-P$ onto their complement. The first-order splitting at exact degeneracy is found from $PVP$. In a nearly degenerate sector, retain the small differences in $PH_0P$ as well.

Time-dependent matrix elements and rates

For a time-independent reference $H_0$, with the interaction picture based at $t_0$,

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

If $i$ and $f$ are eigenstates of $H_0$, define $\omega_{fi}=(E_f-E_i)/\hbar$. Then

$$(V_I)_{fi}(t)=e^{i\omega_{fi}(t-t_0)}V_{fi}(t), \qquad c_f^{(1)}(t)=-\frac{i}{\hbar}\int_{t_0}^{t}(V_I)_{fi}(t')\,dt'.$$

The Schrödinger-picture matrix element $V_{fi}(t)$ and interaction-picture matrix element $(V_I)_{fi}(t)$ differ by that phase. Do not include it twice. A source choosing $t_0=0$ writes $e^{i\omega_{fi}t}$.

A probability $P$ is dimensionless. A rate $\Gamma$ has units of inverse time. In a golden-rule window with negligible depletion, $P(t)\approx\Gamma t$. For an exponential survival probability $P_{\mathrm{surv}}=e^{-\Gamma t}$, the same constant rate is instead $-d\log P_{\mathrm{surv}}/dt$; the probability lost per unit time is $\Gamma P_{\mathrm{surv}}$, not a constant. Some resonance texts use $\Gamma$ for an energy width; that width equals $\hbar$ times the inverse lifetime. Check units before comparing them.

The density of final states $\rho(E)$ is per unit energy unless another measure is stated. Spin, polarization, and channel sums may be displayed separately or included in it; the choice must be specified.

Variational matrices and effective transformations

The Rayleigh quotient is

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

For nonzero trial states in the quadratic-form domain of a semibounded self-adjoint Hamiltonian, interpret its numerator as the quadratic form. A linearly independent trial basis $\{\lvert\phi_i\rangle\}$ gives the matrices

$$H_{ij}=\langle\phi_i\rvert H\lvert\phi_j\rangle, \qquad N_{ij}=\langle\phi_i\vert\phi_j\rangle, \qquad \mathbf H\mathbf c=E\mathbf N\mathbf c.$$

Here $\mathbf N$ is the overlap matrix. Some sources call it $\mathbf S$; it is unrelated to a scattering matrix. See Rayleigh–Ritz for the derivation and the removal of linear dependencies.

For an effective Hamiltonian, a unitary change of basis may be written $\widetilde H=e^AHe^{-A}$ with $A^\dagger=-A$. Other sources use $e^{-A}He^A$ or call the generator $S$. The sign of the generator and the direction of the transformation must be translated together. Transform observables in the same direction as the Hamiltonian; see Schrieffer–Wolff.

WKB momentum, wave number, and action

In the WKB pages, the allowed-region magnitude and forbidden-region magnitude are

$$p(x)=\sqrt{2m(E-V(x))},\qquad \kappa(x)=\sqrt{2m(V(x)-E)},$$

each used only in its respective region. Both have units of momentum in this convention. Oscillatory branches carry $\exp(\pm i\int p\,dx/\hbar)$; growing and decaying branches carry $\exp(\pm\int\kappa\,dx/\hbar)$.

In some texts, and in many elementary barrier formulas, $\kappa$ instead means the inverse decay length $\sqrt{2m(V-E)}/\hbar$. Then the exponent has no extra $1/\hbar$. Inspect the definition and the entire exponent together. The two conventions describe the same decay.

$S$ denotes an action when it has position, time, or path arguments. For a stationary branch, $S_\pm(x,t)=W_\pm(x)-Et$ with $dW_\pm/dx=\pm p(x)$. The sign belongs to the branch; $p(x)$ above remains a positive magnitude. For a closed classical orbit, the semiclassical quantization convention is

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

The Maslov index $\mu_{\mathrm M}$ records the phase changes at caustics or turning points. Two ordinary soft turning points give $\mu_{\mathrm M}=2$; hard walls require their own boundary phases. This is a semiclassical quantization condition, not a universal exact spectrum formula. See WKB quantization and Berry and Mount’s review.

Resolvents and scattering translations

For the standard time dependence $e^{-iEt/\hbar}$,

$$G_0^{(\pm)}(E)=\lim_{\eta\to0^+}(E-H_0\pm i\eta)^{-1}.$$

The positive regulator $\eta$ has energy units. It is distinct from a dimensionless approximation parameter. At a continuum energy these are boundary values in an appropriate distributional or weighted-space sense, not ordinary bounded inverses on all of Hilbert space. The $+i0$ choice produces outgoing scattering waves, and $-i0$ produces incoming waves.

The sign can be checked from

$$\frac1{x\pm i0}=\operatorname{PV}\frac1x\mp i\pi\delta(x), \qquad G_0^{(+)}-G_0^{(-)}=-2\pi i\,\delta(E-H_0).$$

See resolvents for the operator meaning. For scattering, $\mathbf k$ is a wave vector and $\mathbf p=\hbar\mathbf k$ is momentum. If $\mathbf q=\mathbf k'-\mathbf k$, the momentum transfer is $\hbar\mathbf q$. Wave-vector-normalized, momentum-normalized, and unit-incident-amplitude states have different normalization factors. The scattering dictionary translates them together with $f$, $T$, and $S$.

Remainder symbols

Symbol	Meaning in a specified limit
$f\sim g$	Usually $f/g\to1$
$f\sim\sum_n a_n\epsilon^n$	An asymptotic expansion defined through finite remainders
$R=O(\epsilon^n)$	$\lvert R\rvert/\lvert\epsilon\rvert^n$ stays bounded sufficiently near the limit
$R=o(\epsilon^n)$	That ratio tends to zero
$f\approx g$	An approximation whose regime and accuracy need explanation

These are the DLMF order-symbol conventions. The constants in an $O$ estimate can depend on variables held fixed; uniformity must be stated when those variables also vary. In a scattering boundary condition, $\sim$ often denotes a leading asymptotic decomposition rather than a pointwise ratio of oscillatory expressions.

Exercises

Translate a gap. A source defines $D_{mn}=E_m^{(0)}-E_n^{(0)}$ and writes the correction coefficient as $-V_{mn}/D_{mn}$. Compare it with the convention above.

Solution

Since $D_{mn}=-\Delta_{nm}$, the coefficient is $V_{mn}/\Delta_{nm}$. Both the index order and the explicit minus sign matter.

Translate a decay constant. One author writes $e^{-\kappa L/\hbar}$ and another writes $e^{-\widetilde\kappa L}$. What are the units and relation of the two constants?

Solution

$\kappa$ has momentum units; $\widetilde\kappa$ has inverse-length units. The relation is $\widetilde\kappa=\kappa/\hbar$. Putting the same numerical symbol into both exponents without converting it would introduce a dimensional error.

Rate or width? A state has survival probability $e^{-\gamma t}$ and a resonance pole at $E_R-iW/2$, where $W$ has energy units. Find the relation between $W$ and $\gamma$.

Solution

The amplitude has factor $e^{-Wt/(2\hbar)}$, so its squared magnitude is $e^{-Wt/\hbar}$. Thus $\gamma=W/\hbar$. The factor of two belongs to amplitude versus probability, while $\hbar$ converts energy width to inverse lifetime.

References

• M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics”, Reports on Progress in Physics 35, 315–397, 1972. Actions, WKB branches, and phase corrections.
• F. W. J. Olver and R. Wong, “Definitions and Elementary Properties,” §2.1, in NIST Digital Library of Mathematical Functions, Chapter 2: Asymptotic Approximations.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020. Perturbative normalization and transition amplitudes.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006. Boundary prescriptions and scattering conventions.