Formula Sheet

This formula sheet is a lookup aid, not a derivation page. Use it to check conventions and locate the canonical home for each result. For the meaning, dimensions, and common collisions of individual notation, use Common Symbols.

Conventions used here:

$$\begin{gathered} H=H_0+\lambda V, \\ E_n^{(0)} \text{ is an unperturbed energy}, \\ k=\frac{\sqrt{2mE}}{\hbar}. \end{gathered}$$

Time-Independent Perturbation Theory

Canonical homes:

• Nondegenerate Perturbation Theory
• Degenerate Perturbation Theory

For an isolated unperturbed level,

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

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 the first-order energy splittings inside the degenerate subspace.

Time-Dependent Transitions

Canonical homes:

• First-Order Transition Probability
• Fermi’s Golden Rule

In the interaction picture,

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

The leading transition probability is

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

For a continuum of final states in the long-time weak-coupling regime,

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

Variational Methods

Canonical homes:

• Variational Principle
• Rayleigh-Ritz Method

For a Hamiltonian bounded below,

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

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 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

Canonical homes:

• 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:

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

Forbidden-region momentum:

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

Forbidden-region WKB form:

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

WKB validity indicator:

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

For two smooth turning points,

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

For barrier tunneling,

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

Scattering

Canonical homes:

• Scattering Amplitude
• Differential and Total Cross Sections
• Lippmann-Schwinger Equation
• First Born Approximation
• Partial-Wave Expansion
• Phase Shifts
• Partial-Wave Cross Sections
• Optical Theorem
• Breit–Wigner Form

Asymptotic scattering state:

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

Elastic differential cross section:

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

Lippmann-Schwinger equation:

$$\lvert\psi^{(\pm)}\rangle = \lvert\phi\rangle + G_0^{(\pm)}(E) V \lvert\psi^{(\pm)}\rangle.$$

First Born amplitude:

$$f_{\mathrm B}(\mathbf q) = - \frac{m}{2\pi\hbar^2} \int d^3r\, e^{-i\mathbf q\cdot\mathbf r} V(\mathbf r).$$

Partial-wave amplitude:

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

Elastic partial-wave total cross section:

$$\sigma_{\mathrm{el}} = \frac{4\pi}{k^2} \sum_{\ell=0}^\infty (2\ell+1) \sin^2\delta_\ell.$$

Optical theorem:

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

Isolated elastic Breit–Wigner factor:

$$S_\ell(E) = e^{2i\delta_{\ell,\mathrm{bg}}(E)} \frac{ E-E_R-i\Gamma/2 }{ E-E_R+i\Gamma/2 }.$$

Quick Validity Reminders

• Small perturbative corrections require small matrix elements relative to relevant energy gaps.
• Degenerate or nearly degenerate levels require degenerate or effective-Hamiltonian methods.
• Golden-rule rates require weak coupling, long times, and a continuum or dense final spectrum.
• Variational energies are upper bounds for the ground state, not two-sided error bars.
• WKB fails at turning points unless connection formulas are used.
• Born approximation fails for strong, resonant, low-energy, or long-range scattering without extra care.
• Unitarity constraints such as the optical theorem are powerful checks on scattering approximations.

Use the Error-Estimate Checklist to turn these reminders into a complete validity statement.

References

• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• 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, Dover, 2006.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Butterworth-Heinemann, 1981.