Approximation Methods Formula Sheet

Use this page to recover a formula together with the assumptions that make it meaningful. The linked articles give derivations and examples. Common Symbols explains notation; scattering normalizations and formulas are collected in the Scattering Convention Dictionary.

Time-Independent Perturbation Theory

Derivations: Nondegenerate Perturbation Theory, Degenerate Perturbation Theory.

Write $H=H_0+\lambda V$ and $E_n=E_n^{(0)}+\lambda E_n^{(1)}+\lambda^2E_n^{(2)}+\cdots$. The displayed $E_n^{(r)}$ are coefficients, so physical shifts include $\lambda^r$. For a discrete nondegenerate reference level with perturbatively small mixing,

$$E_n^{(1)} = \langle n^{(0)}|V|n^{(0)}\rangle.$$

The first-order state correction in intermediate normalization is

$$\lvert n^{(1)}\rangle = \sum_{m\ne n} \frac{ \langle m^{(0)}|V|n^{(0)}\rangle }{ E_n^{(0)}-E_m^{(0)} } \lvert m^{(0)}\rangle.$$

The second-order energy correction is

$$E_n^{(2)} = \sum_{m\ne n} \frac{ |\langle m^{(0)}|V|n^{(0)}\rangle|^2 }{ E_n^{(0)}-E_m^{(0)} }.$$

For a degenerate subspace with basis $\{\lvert a\rangle\}$, diagonalize

$$W_{ab} = \langle a|V|b\rangle.$$

Its eigenvalues are first-order coefficients; multiply by $\lambda$ for the shifts. Coupling to states outside the subspace must remain small relative to the external gaps.

Time-Dependent Transitions

Derivations: First-Order Transition Probability, Fermi’s Golden Rule.

Here $V(t)$ denotes the entire physical interaction, with any coupling included. For an initial reference state $i$ and a distinct final state $f$,

$$c_f^{(1)}(t) = - \frac{i}{\hbar} \int_{t_0}^{t} dt'\, \langle f|V_I(t')|i\rangle.$$

The leading transition probability is

$$P_{i\to f}(t) \approx |c_f^{(1)}(t)|^2.$$

For a smooth continuum, a nearly constant on-shell matrix element $V_{fi}$, and an intermediate time $\tau_{\mathrm{corr}}\ll t\ll\Gamma^{-1}$, the summed rate is

$$\Gamma_i = \frac{2\pi}{\hbar} |V_{fi}|^2 \rho(E_i).$$

The last expression assumes an energy-conserving static coupling. For absorption from a harmonic drive, evaluate the appropriate Fourier component and density of states at $E_i+\hbar\omega$; a cosine drive contributes a factor of $1/2$ to each frequency component.

Variational Methods

Derivations: Variational Principle, Rayleigh-Ritz Method.

For a self-adjoint Hamiltonian bounded below and a nonzero trial state in its quadratic-form domain,

$$E[\psi] = \frac{\langle\psi|H|\psi\rangle}{\langle\psi|\psi\rangle} \ge E_0.$$

For a parameterized trial family,

$$E_{\mathrm{var}} = \min_\alpha \frac{ \langle\psi(\alpha)|H|\psi(\alpha)\rangle }{ \langle\psi(\alpha)|\psi(\alpha)\rangle }.$$

For a linearly independent nonorthogonal Rayleigh–Ritz basis,

$$\sum_j H_{ij}c_j = E \sum_j S_{ij}c_j,$$

with

$$H_{ij}=\langle\phi_i|H|\phi_j\rangle, \qquad S_{ij}=\langle\phi_i|\phi_j\rangle.$$

WKB and Semiclassics

Derivations: WKB Approximation, Turning Points and Connection Formulas, Airy Functions, Bohr-Sommerfeld Quantization, Barrier Penetration and Tunneling.

Allowed-region momentum:

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

Allowed-region WKB form:

$$\psi(x) \approx \frac{C_+}{\sqrt{p(x)}} e^{(i/\hbar)\int^x p(x')\,dx'} + \frac{C_-}{\sqrt{p(x)}} e^{-(i/\hbar)\int^x p(x')\,dx'}.$$

Forbidden-region momentum:

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

Forbidden-region WKB form:

$$\psi(x) \approx \frac{D_+}{\sqrt{\kappa(x)}} e^{(1/\hbar)\int^x\kappa(x')\,dx'} + \frac{D_-}{\sqrt{\kappa(x)}} e^{-(1/\hbar)\int^x\kappa(x')\,dx'}.$$

Away from turning points, a necessary local WKB diagnostic is:

$$\left| \hbar \frac{p'}{p^2} \right| \ll1.$$

For two smooth turning points,

$$\int_{x_1}^{x_2}p(x)\,dx = \pi\hbar \left( n+\frac12 \right).$$

For an opaque smooth barrier with two well-separated turning points, the leading exponential factor is

$$T \approx \exp\left[ - \frac{2}{\hbar} \int_{x_1}^{x_2} \sqrt{2m(V(x)-E)} \,dx \right].$$

Effective Hamiltonians and Time Evolution

For complementary orthogonal projectors $P+Q=I$, eliminating $Q$ gives

$$H_{\mathrm{eff}}(E)=PHP+PHQ(E-QHQ)^{-1}QHP.$$

The inverse must exist, or a specified scattering boundary value must be used. This is an exact energy-dependent identity; a low-energy expansion needs separation from the eliminated spectrum. See Feshbach Projection.

For $\dot U=A(t)U$ with $A=-iH/\hbar$, the first two Magnus terms are

$$\Omega_1=\int_0^t A(t_1)dt_1,\qquad \Omega_2=\frac12\int_0^t dt_1\int_0^{t_1}dt_2[A(t_1),A(t_2)].$$

Use $U\approx e^{\Omega_1+\Omega_2}$ only in a justified convergence or asymptotic regime. For bounded generators, $\int_0^t\|A(s)\|ds<\pi$ is a sufficient convergence condition, not a necessary one; Blanes and collaborators discuss this bound and practical truncation. Each finite anti-Hermitian truncation exponentiates to a unitary operator, but unitarity alone does not establish accuracy.

Applying a Formula

Identify its small parameter, boundaries, and observable before substitution. A turning point requires matching, a nearly degenerate pair requires subspace diagonalization, and an energy upper bound does not bound every observable. The Error-Estimate Checklist explains how to distinguish a proved bound from a useful numerical check.

References

• S. Blanes, F. Casas, J. A. Oteo, and J. Ros, “The Magnus Expansion and Some of Its Applications,” Physics Reports 470, 151–238 (2009), arXiv:0810.5488, doi:10.1016/j.physrep.2008.11.001.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Butterworth-Heinemann, 1981.
• 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.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.