Mathematical Prerequisites for Quantum Mechanics
This reference collects every canonical integral identity, transform pair, and special-function result you will meet across quantum mechanics -- from undergraduate wave mechanics through graduate-level path integrals and quantum field theory. Each formula is presented with the precise conditions under which it holds and a note on where it appears in the physics.
Quick Jump
§1 — Gaussian Integrals
The Gaussian integral is arguably the single most important integral in all of quantum physics. It underpins the quantum harmonic oscillator, coherent states, the path-integral formulation, statistical mechanics, and quantum field theory.
$$\int_{-\infty}^{+\infty} e^{-ax^2}\,dx = \sqrt{\frac{\pi}{a}}, \qquad a > 0$$
The foundational result. Proved by squaring the integral and switching to polar coordinates. Every Gaussian calculation in QM ultimately reduces to this identity.
$$\int_{-\infty}^{+\infty} x^2\, e^{-ax^2}\,dx = \frac{1}{2}\sqrt{\frac{\pi}{a^3}}$$
Obtained by differentiating the fundamental Gaussian with respect to a. Gives the variance of the ground-state wave function of the harmonic oscillator.
$$\int_{-\infty}^{+\infty} x^{2n}\,e^{-ax^2}\,dx = \frac{(2n-1)!!}{2^n\, a^n}\sqrt{\frac{\pi}{a}}, \qquad n = 0,1,2,\ldots$$
The double factorial $(2n-1)!! = 1 \cdot 3 \cdot 5 \cdots (2n-1)$. Odd moments vanish by symmetry. These moments appear in perturbation theory and Wick contractions.
$$\int_{-\infty}^{+\infty} e^{-ax^2 + bx}\,dx = \sqrt{\frac{\pi}{a}}\;\exp\!\left(\frac{b^2}{4a}\right), \qquad \mathrm{Re}(a)>0$$
The workhorse of the path integral. The shift $x \to x + b/(2a)$ completes the square. The exponential prefactor generates the classical action in semiclassical approximations.
$$\left\langle x^{2n}\right\rangle = \left(-\frac{\partial}{\partial a}\right)^n \sqrt{\frac{\pi}{a}} \bigg/ \sqrt{\frac{\pi}{a}} = \frac{(2n-1)!!}{(2a)^n}$$
Differentiation with respect to the parameter a generates all even moments. This is the prototype of the source-field technique used throughout quantum field theory to generate correlation functions.
$$\int_0^{\infty} x^{n}\,e^{-ax^2}\,dx = \frac{\Gamma\!\left(\frac{n+1}{2}\right)}{2\,a^{(n+1)/2}}, \qquad a>0,\; n>-1$$
The half-line version connects Gaussian integrals to the Gamma function. For integern it reproduces factorials or double factorials; for half-integer nit gives results involving $\sqrt{\pi}$. Used in radial integrals of the harmonic oscillator.
$$\int_{\mathbb{R}^n} \exp\!\left(-\tfrac{1}{2}\,\mathbf{x}^T \mathbf{A}\,\mathbf{x} + \mathbf{b}^T\mathbf{x}\right) d^n x = \sqrt{\frac{(2\pi)^n}{\det \mathbf{A}}}\;\exp\!\left(\tfrac{1}{2}\,\mathbf{b}^T \mathbf{A}^{-1}\mathbf{b}\right)$$
Requires A symmetric positive-definite. This is the fundamental identity of free bosonic field theory: the source term b generates alln-point correlation functions via Wick's theorem.
$$\int \mathcal{D}[\bar\psi,\psi]\;\exp\!\left(-\bar\psi\,\mathbf{A}\,\psi\right) = \det \mathbf{A}$$
The fermionic counterpart: anticommuting Grassmann variables givedet A instead of (det A)-1/2. This sign difference is the algebraic origin of the Pauli exclusion principle and Fermi-Dirac statistics in QFT.
$$\int \frac{d^n z\;d^n \bar{z}}{(2\pi i)^n}\;\exp\!\left(-\bar{\mathbf{z}}^T \mathbf{A}\,\mathbf{z} + \bar{\mathbf{j}}^T\mathbf{z} + \bar{\mathbf{z}}^T\mathbf{j}\right) = \frac{1}{\det \mathbf{A}}\;\exp\!\left(\bar{\mathbf{j}}^T \mathbf{A}^{-1}\mathbf{j}\right)$$
The complex bosonic Gaussian appears in the coherent-state path integral and in the functional integral for charged scalar fields. Note the full inverse determinant (no square root) compared with the real case.
§2 — Fourier Transforms
Fourier analysis is the bridge between position and momentum representations in quantum mechanics. The conventions below use the physicist's symmetric normalization with$\,\hbar\,$.
$$\Delta x\;\Delta k \;\geq\; \frac{1}{2} \qquad\Longrightarrow\qquad \Delta x\;\Delta p \;\geq\; \frac{\hbar}{2}$$
A purely mathematical consequence of Fourier analysis: a function and its Fourier transform cannot both be arbitrarily narrow. With the quantum identification$p = \hbar k$, this becomes the Heisenberg uncertainty principle. Equality holds for Gaussian wave packets.
$$\tilde{f}(k) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} f(x)\,e^{-ikx}\,dx$$
The symmetric convention places a factor of $1/\sqrt{2\pi}$ in both the forward and inverse transforms.
$$f(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} \tilde{f}(k)\,e^{+ikx}\,dk$$
Together with the forward transform, these form a unitary pair preserving the inner product.
$$\psi(p) = \frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{+\infty} \psi(x)\,e^{-ipx/\hbar}\,dx$$
The momentum-space wave function. The factor of $\hbar$ ensures that$\int|\psi(p)|^2\,dp = 1$ when the position-space wave function is normalised.
$$e^{-ax^2} \;\longleftrightarrow\; \frac{1}{\sqrt{2a}}\,e^{-k^2/(4a)}$$
$$\delta(x-a) \;\longleftrightarrow\; \frac{1}{\sqrt{2\pi}}\,e^{-ika}$$
$$e^{ik_0 x} \;\longleftrightarrow\; \sqrt{2\pi}\;\delta(k-k_0)$$
$$\frac{1}{x^2+a^2} \;\longleftrightarrow\; \frac{\pi}{a\sqrt{2\pi}}\,e^{-a|k|}$$
$$\theta(t)\,e^{-\gamma t}\,e^{-i\omega_0 t} \;\longleftrightarrow\; \frac{1}{\sqrt{2\pi}}\,\frac{1}{\gamma + i(\omega-\omega_0)}$$
$$H_n(x)\,e^{-x^2/2} \;\longleftrightarrow\; (-i)^n\,H_n(k)\,e^{-k^2/2}$$
Gaussian, delta, plane wave, Lorentzian, causal propagator, and Hermite function transform pairs. The Hermite pair shows that QHO eigenfunctions are eigenstates of the Fourier transform.
$$\int_{-\infty}^{+\infty} |f(x)|^2\,dx = \int_{-\infty}^{+\infty} |\tilde{f}(k)|^2\,dk$$
Conservation of probability in quantum mechanics: the norm of the wave function is the same in position and momentum space. This is a direct consequence of the unitarity of the Fourier transform on $L^2(\mathbb{R})$.
$$\mathcal{F}\{f * g\} = \sqrt{2\pi}\;\tilde{f}\,\tilde{g}, \qquad (f*g)(x) \equiv \int f(x-y)\,g(y)\,dy$$
Convolution in position space becomes multiplication in momentum space (and vice versa). Used heavily in scattering theory and in computing propagator convolutions for time evolution.
§3 — Dirac Delta Function
The delta function is the backbone of continuous-spectrum quantum mechanics. It provides the orthonormality condition for position and momentum eigenstates and appears in Fermi's golden rule, propagator calculations, and scattering theory.
$$\int_{-\infty}^{+\infty} f(x)\,\delta(x-a)\,dx = f(a)$$
The defining property of the Dirac delta. Extracts the value of a function at a point. This is used every time we expand a state in the position basis:$\psi(a) = \langle a|\psi\rangle$.
$$\delta(x) = \frac{1}{2\pi}\int_{-\infty}^{+\infty} e^{ikx}\,dk$$
The plane-wave representation. This is the statement that a complete set of plane waves resolves the identity. It is the continuous analogue of the Kronecker delta for discrete bases.
$$\delta\!\big(f(x)\big) = \sum_{i}\frac{\delta(x-x_i)}{|f'(x_i)|}$$
Where the sum runs over all simple zeros $x_i$ of $f$. Applied in relativistic kinematics for energy-momentum conserving delta functions, e.g.$\delta(p^2 - m^2)$.
$$\int_{-\infty}^{+\infty} f(x)\,\delta'(x-a)\,dx = -f'(a)$$
More generally, $\int f(x)\,\delta^{(n)}(x-a)\,dx = (-1)^n f^{(n)}(a)$. Derivative deltas appear when computing matrix elements of the momentum operator in position space: $\langle x|\hat{p}|x'\rangle = -i\hbar\,\delta'(x-x')$.
$$\delta^3(\mathbf{r}-\mathbf{r}') = \frac{\delta(r-r')}{r^2}\,\delta(\cos\theta-\cos\theta')\,\delta(\varphi-\varphi')$$
The 3D delta in spherical coordinates. The Fourier representation is$\delta^3(\mathbf{r}) = \frac{1}{(2\pi)^3}\int e^{i\mathbf{k}\cdot\mathbf{r}}\,d^3k$. Essential for the Green's function of the Coulomb problem.
$$\delta(x) = \lim_{\epsilon\to 0^+}\frac{1}{\pi}\frac{\epsilon}{x^2+\epsilon^2}$$
The Lorentzian (Breit-Wigner) nascent delta. In scattering theory, finite$\epsilon$ corresponds to the natural linewidth of a resonance with lifetime $\tau = \hbar/\epsilon$.
$$\lim_{\epsilon\to 0^+}\frac{1}{x \pm i\epsilon} = \mathrm{P.V.}\frac{1}{x} \mp i\pi\,\delta(x)$$
Splits a singular denominator into a principal value and a delta-function piece. This identity is the key to deriving Fermi's golden rule from the retarded Green's function $G^R(E) = (E - H + i\epsilon)^{-1}$.
$$\delta(x) = \lim_{\sigma\to 0}\frac{1}{\sigma\sqrt{2\pi}}\,\exp\!\left(-\frac{x^2}{2\sigma^2}\right)$$
The Gaussian representation of the delta function. As the width$\sigma \to 0$, the normalised Gaussian becomes infinitely peaked. This regularisation is particularly natural in quantum optics and coherent-state calculations.
$$\delta(\alpha x) = \frac{1}{|\alpha|}\,\delta(x), \qquad \alpha \neq 0$$
A special case of the composition rule. Frequently used when changing variables in energy-conserving delta functions, e.g. converting$\delta(E - \hbar\omega)$ between different unit systems.
$$\frac{1}{2\pi}\int_{-\infty}^{+\infty} e^{i(k-k')x}\,dx = \delta(k-k')$$
The orthogonality relation for plane waves -- the continuous analogue of the Kronecker delta for discrete Fourier modes. This is the fundamental reason that momentum eigenstates$|k\rangle$ satisfy $\langle k|k'\rangle = \delta(k-k')$.
§4 — Spherical & Radial Integrals
Central-force problems (hydrogen atom, nuclear shell model, partial-wave scattering) separate into radial and angular parts. The integrals below govern both.
$$\int_0^{\infty} r^n\,e^{-\alpha r}\,dr = \frac{n!}{\alpha^{n+1}}, \qquad \alpha>0,\; n=0,1,2,\ldots$$
The basic radial integral for hydrogenic wave functions. All expectation values$\langle r^k \rangle$ in the hydrogen atom reduce to this form.
$$\int_0^{\infty} r^{s-1}\,e^{-\alpha r}\,dr = \frac{\Gamma(s)}{\alpha^{s}}, \qquad \mathrm{Re}(\alpha)>0,\;\mathrm{Re}(s)>0$$
The continuous extension of the factorial integral to non-integer powers. Required when evaluating radial integrals involving fractional powers or performing dimensional regularisation.
$$\int_0^{\infty} r^{2\ell+2}\,e^{-\alpha r^2}\,dr = \frac{(2\ell+1)!!}{2^{\ell+2}\,\alpha^{\ell+3/2}}\,\sqrt{\pi}$$
Governs the radial integrals of the isotropic harmonic oscillator in 3D. The angular momentum quantum number $\ell$ enters through the power of r.
$$\int d^3 r = \int_0^{\infty} r^2\,dr \int_0^{\pi}\sin\theta\,d\theta \int_0^{2\pi} d\varphi, \qquad \int d\Omega = 4\pi$$
The spherical volume element factorises into radial and angular parts. The total solid angle integrates to $4\pi$ steradians.
$$\int Y_\ell^{m*}(\theta,\varphi)\,Y_{\ell'}^{m'}(\theta,\varphi)\,d\Omega = \delta_{\ell\ell'}\,\delta_{mm'}$$
Spherical harmonics form a complete orthonormal set on the unit sphere. This is why angular momentum eigenstates are labelled by the quantum numbers $\ell$ and $m$.
$$\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell} Y_\ell^{m*}(\theta',\varphi')\,Y_\ell^m(\theta,\varphi) = \delta(\cos\theta-\cos\theta')\,\delta(\varphi-\varphi')$$
The angular completeness relation. Any square-integrable function on the sphere can be expanded in spherical harmonics. This is the angular analogue of the plane-wave completeness of the delta function.
$$\int Y_{\ell_1}^{m_1}\,Y_{\ell_2}^{m_2}\,Y_{\ell_3}^{m_3}\,d\Omega = \sqrt{\frac{(2\ell_1+1)(2\ell_2+1)(2\ell_3+1)}{4\pi}} \begin{pmatrix} \ell_1 & \ell_2 & \ell_3 \\ 0 & 0 & 0\end{pmatrix} \begin{pmatrix}\ell_1 & \ell_2 & \ell_3 \\ m_1 & m_2 & m_3\end{pmatrix}$$
The parenthesised objects are Wigner 3j-symbols (related to Clebsch-Gordan coefficients). Selection rules: $m_1+m_2+m_3=0$, triangle inequality on$\ell_i$, and $\ell_1+\ell_2+\ell_3$ even. Governs dipole, quadrupole, and higher multipole transition matrix elements.
$$P_\ell(\cos\gamma) = \frac{4\pi}{2\ell+1}\sum_{m=-\ell}^{\ell} Y_\ell^{m*}(\theta',\varphi')\,Y_\ell^m(\theta,\varphi)$$
Here $\gamma$ is the angle between the directions$(\theta,\varphi)$ and $(\theta',\varphi')$. Appears whenever the Coulomb potential $1/|\mathbf{r}-\mathbf{r}'|$ is expanded in multipoles.
§5 — Orthogonal Polynomial Integrals
Bound-state wave functions in exactly solvable potentials are built from classical orthogonal polynomials. Their orthogonality and recursion relations underpin selection rules, matrix elements, and the algebraic structure of quantum mechanics.
$$\int_{-\infty}^{+\infty} H_m(x)\,H_n(x)\,e^{-x^2}\,dx = \sqrt{\pi}\;2^n\,n!\;\delta_{mn}$$
The physicists' Hermite polynomials $H_n(x)$ with weight function$e^{-x^2}$. These are the building blocks of the quantum harmonic oscillator eigenfunctions.
The Hermite Functions: A Complete Orthonormal Basis
The Hermite functions form a particular orthonormal basis that will reappear as the eigenstates of the quantum harmonic oscillator Hamiltonian. They are constructed from the Rodrigues formula with a specific normalization choice.
DEFINITION
$$\phi_n(x) = a_n\,e^{x^2/2}\,\frac{d^n}{dx^n}\!\left(e^{-x^2}\right), \qquad n = 0, 1, 2, \ldots$$
The first few Hermite functions are:
$$\phi_0(x) \propto e^{-x^2/2}$$
$$\phi_1(x) \propto x\,e^{-x^2/2}$$
$$\phi_2(x) \propto (2x^2 - 1)\,e^{-x^2/2}$$
NORMALIZATION CHOICE
The coefficient $a_n$ is chosen so that the functions are orthonormal:
$$a_n = \frac{(-1)^n}{(\sqrt{\pi}\,2^n\,n!)^{1/2}} \qquad \Longrightarrow \qquad \int_{-\infty}^{+\infty} \phi_m(x)\,\phi_n(x)\,dx = \delta_{m,n}$$
Orthonormal set → eigenstates of the harmonic oscillator Hamiltonian
This orthonormal set will be recovered as the eigenstates of the quantum harmonic oscillator. Any square-integrable function can be expanded in this basis, exactly as in a finite-dimensional vector space.
EXPANSION IN THE HERMITE BASIS
Given any state $\psi(x)$, compute the expansion coefficients:
$$C_n = \langle \phi_n | \psi \rangle = \int_{-\infty}^{+\infty} \phi_n(x)\,\psi(x)\,dx$$
Then the function is reconstructed as:
$$\psi(x) = \sum_{n=0}^{\infty} C_n\,\phi_n(x)$$
This expression manipulates exactly like a vector in a finite-dimensional space — the coefficients $C_n$ play the role of components, and the Hermite functions $\phi_n$ play the role of basis vectors. This is the gateway to the abstract Hilbert space formulation of quantum mechanics.
$$H_n(x) = (-1)^n\,e^{x^2}\frac{d^n}{dx^n}e^{-x^2}$$
The Rodrigues representation generates all Hermite polynomials by successive differentiation. It is the fastest way to derive explicit forms: $H_0=1$,$H_1=2x$,$H_2=4x^2-2$,$H_3=8x^3-12x$,$H_4=16x^4-48x^2+12$.
$$e^{2xt - t^2} = \sum_{n=0}^{\infty} H_n(x)\,\frac{t^n}{n!}$$
The generating function packages all Hermite polynomials into a single exponential. Expanding the left side in powers of $t$ and comparing coefficients yields each $H_n$. This is the most efficient route to deriving the Mehler kernel and coherent-state overlaps.
$$H_{n+1}(x) = 2x\,H_n(x) - 2n\,H_{n-1}(x)$$
$$H_n'(x) = 2n\,H_{n-1}(x)$$
The three-term recurrence builds $H_{n+1}$ from the two preceding polynomials. The derivative relation shows that differentiation lowers the index by one — the classical analogue of the annihilation operator$\hat{a}$. Together these relations underpin the ladder-operator algebra of the QHO.
$$H_n'' - 2x\,H_n' + 2n\,H_n = 0$$
The Hermite ODE arises when the time-independent Schrödinger equation for the harmonic oscillator $-\frac{\hbar^2}{2m}\psi'' + \frac{1}{2}m\omega^2 x^2\psi = E\psi$ is written in dimensionless units $\xi = x\sqrt{m\omega/\hbar}$ and the Gaussian factor $e^{-\xi^2/2}$ is separated out. Solutions that remain finite require $E_n = \hbar\omega(n+\tfrac{1}{2})$ — quantization emerges from the boundary condition.
$$\sum_{n=0}^{\infty}\frac{H_n(x)\,H_n(y)}{2^n\,n!}\;t^n = \frac{1}{\sqrt{1-t^2}}\exp\!\left(\frac{2xyt - (x^2+y^2)t^2}{1-t^2}\right)$$
The Mehler kernel (bilinear generating function). Setting $t = e^{-i\omega T}$ and multiplying by the Gaussian weight reproduces the exact harmonic-oscillator propagator$K(x,y;T)$. At $t \to 1$ it reduces to $\delta(x-y)$ (completeness). This single identity encodes the entire spectral decomposition of the QHO propagator.
$$\int_{-\infty}^{+\infty} H_n(x)\,e^{-x^2/2}\,e^{-ikx}\,dx = \sqrt{2\pi}\;(-i)^n\,H_n(k)\,e^{-k^2/2}$$
Hermite functions $\phi_n(x) = H_n(x)e^{-x^2/2}$ are eigenfunctions of the Fourier transform with eigenvalue $(-i)^n$. This is why the QHO ground state is a Gaussian in both position and momentum space — and why the uncertainty product $\Delta x\,\Delta p = \hbar(n + \tfrac{1}{2})$ is minimized for $n=0$.
$$\psi_n(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4}\frac{1}{\sqrt{2^n\,n!}}\;H_n\!\left(\sqrt{\frac{m\omega}{\hbar}}\,x\right)\exp\!\left(-\frac{m\omega x^2}{2\hbar}\right)$$
With these, the orthonormality relation reads$\int_{-\infty}^{+\infty}\psi_m^*(x)\,\psi_n(x)\,dx = \delta_{mn}$. The energy eigenvalues are $E_n = \hbar\omega(n + \tfrac{1}{2})$.
$$\langle n|\hat{x}|m\rangle = \sqrt{\frac{\hbar}{2m\omega}}\left(\sqrt{m}\;\delta_{n,m-1} + \sqrt{m+1}\;\delta_{n,m+1}\right)$$
The position operator connects only neighbouring energy levels ($\Delta n = \pm 1$). This selection rule follows from the three-term recursion of Hermite polynomials, or equivalently from $\hat{x} = \sqrt{\hbar/(2m\omega)}\,(\hat{a}+\hat{a}^\dagger)$.
$$\int_0^{\infty} x^{\alpha}\,e^{-x}\,L_n^{(\alpha)}(x)\,L_m^{(\alpha)}(x)\,dx = \frac{\Gamma(n+\alpha+1)}{n!}\;\delta_{mn}$$
The associated Laguerre polynomials $L_n^{(\alpha)}$ with weight$x^\alpha e^{-x}$. For the hydrogen atom,$\alpha = 2\ell+1$ and the polynomials appear in the radial wave functions$R_{n\ell}(r)$.
$$\langle r \rangle_{n\ell} = \frac{a_0}{2}\left[3n^2 - \ell(\ell+1)\right]$$
$$\langle r^2 \rangle_{n\ell} = \frac{a_0^2\,n^2}{2}\left[5n^2 + 1 - 3\ell(\ell+1)\right]$$
$$\langle r^{-1} \rangle_{n\ell} = \frac{1}{n^2\,a_0}$$
$$\langle r^{-2} \rangle_{n\ell} = \frac{1}{n^3\,a_0^2\left(\ell+\frac{1}{2}\right)}$$
Derived by integrating the radial probability density$|R_{n\ell}(r)|^2\,r^2$ against powers of r. The $\langle r^{-1}\rangle$ result gives the energy, while$\langle r^{-2}\rangle$ enters the spin-orbit coupling. Here $a_0$ is the Bohr radius.
§6 — Complex & Contour Integrals
Complex analysis provides the rigorous framework for Green's functions, propagators, and the causal structure of quantum theory. Contour deformation and the residue theorem are indispensable tools.
$$f(z_0) = \frac{1}{2\pi i}\oint_C \frac{f(z)}{z - z_0}\,dz$$
The value of an analytic function inside a contour is determined by its values on the contour. This is the prototype for all resolvent and Green's function identities in QM:$\hat{G}(z) = (z - \hat{H})^{-1}$.
$$\oint_C f(z)\,dz = 2\pi i \sum_{k}\mathrm{Res}(f, z_k)$$
Sums the residues at all poles enclosed by C. For a simple pole,$\mathrm{Res}(f, z_k) = \lim_{z\to z_k}(z-z_k)\,f(z)$. Used to evaluate propagator integrals, spectral sums, and Matsubara frequency sums in finite-temperature field theory.
$$G^R(t) = -\frac{i}{\hbar}\,\theta(t)\!\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}\,\frac{e^{-i\omega t}}{\omega - \omega_0 + i\epsilon}$$
The pole is shifted into the lower half-plane by $+i\epsilon$, ensuring causality: the contour closes in the lower half-plane for $t>0$ and in the upper half-plane for $t<0$ (giving zero).
$$G^A(t) = +\frac{i}{\hbar}\,\theta(-t)\!\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}\,\frac{e^{-i\omega t}}{\omega - \omega_0 - i\epsilon}$$
The pole sits in the upper half-plane. The advanced Green's function is nonzero only for$t < 0$ and satisfies$G^A(t) = [G^R(-t)]^*$.
$$\int_{-\infty}^{+\infty} e^{iax^2}\,dx = \sqrt{\frac{i\pi}{a}} = \sqrt{\frac{\pi}{a}}\;e^{i\pi/4}, \qquad a>0$$
The oscillatory Gaussian. The square root of i arises from a 90-degree rotation of the integration contour in the complex plane (Wick rotation). This is the integral at the heart of every path-integral time-slice calculation.
$$G_F(\omega) = \frac{1}{(\omega - \omega_0 + i\epsilon)(\omega + \omega_0 - i\epsilon)} \;=\; \frac{1}{\omega^2 - \omega_0^2 + i\epsilon}$$
The Feynman prescription places positive-energy poles below the real axis and negative-energy poles above, selecting the time-ordered (Feynman) propagator. This is the starting point for perturbative QFT.
$$\lim_{R\to\infty}\int_{C_R} f(z)\,e^{i\lambda z}\,dz = 0, \qquad \lambda > 0$$
Valid when $C_R$ is a semicircular arc in the upper half-plane and$|f(z)| \to 0$ uniformly as $|z|\to\infty$. Jordan's lemma justifies closing the contour at infinity and is used in virtually every propagator and Green's function calculation.
$$\mathrm{Re}\,\chi(\omega) = \frac{1}{\pi}\,\mathrm{P.V.}\!\int_{-\infty}^{+\infty}\frac{\mathrm{Im}\,\chi(\omega')}{\omega'-\omega}\,d\omega'$$
$$\mathrm{Im}\,\chi(\omega) = -\frac{1}{\pi}\,\mathrm{P.V.}\!\int_{-\infty}^{+\infty}\frac{\mathrm{Re}\,\chi(\omega')}{\omega'-\omega}\,d\omega'$$
Causality (analyticity in the upper half-plane) forces the real and imaginary parts of any response function to be Hilbert transforms of each other. These relations connect absorption to dispersion in quantum optics, and the spectral function to the self-energy in many-body physics.
$$\frac{1}{\beta}\sum_{n} g(i\omega_n) = -\oint_C \frac{dz}{2\pi i}\;n_F(z)\;g(z), \qquad \omega_n = \frac{(2n+1)\pi}{\beta}$$
Converts a discrete Matsubara sum into a contour integral using the Fermi-Dirac function$n_F(z) = 1/(e^{\beta z}+1)$ (whose poles are at the fermionic Matsubara frequencies). The bosonic version uses $n_B(z) = 1/(e^{\beta z}-1)$with even frequencies. Central to finite-temperature field theory.
§7 — Hilbert Space & Integral Identities
What is a Hilbert Space?
A Hilbert space $\mathcal{H}$ is a complete inner-product space — a (possibly infinite-dimensional) vector space equipped with a notion of “angle” and “length” between its elements, in which every Cauchy sequence converges. In quantum mechanics, the state of a system is a ray (unit vector up to phase) in a Hilbert space.
Inner Product
$$\langle \phi | \psi \rangle = \int_{-\infty}^{+\infty} \phi^*(x)\,\psi(x)\,dx \in \mathbb{C}$$
Generalises the dot product. Hermitian: $\langle\phi|\psi\rangle = \overline{\langle\psi|\phi\rangle}$. Positive-definite: $\langle\psi|\psi\rangle \geq 0$, equals zero iff $\psi = 0$.
Norm (Probability)
$$\|\psi\|^2 = \langle \psi | \psi \rangle = \int_{-\infty}^{+\infty} |\psi(x)|^2\,dx = 1$$
The norm squared is the total probability. Physical states must be normalizable: $\psi \in L^2(\mathbb{R})$.
Key Properties
- • Linearity: $|\psi\rangle = \alpha|\phi_1\rangle + \beta|\phi_2\rangle$ — superposition principle
- • Completeness: Every Cauchy sequence converges — no “holes” in the space
- • Separability: Admits a countable orthonormal basis $\{|n\rangle\}$
- • Operators: Observables are self-adjoint operators $\hat{A}^\dagger = \hat{A}$ on $\mathcal{H}$
The Analogy with Finite-Dimensional Vectors
FINITE DIM ($\mathbb{C}^n$)
- • Vectors: $\vec{v} = \sum_i v_i\,\hat{e}_i$
- • Inner product: $\vec{u}\cdot\vec{v} = \sum_i u_i^* v_i$
- • Basis: $\hat{e}_i \cdot \hat{e}_j = \delta_{ij}$
- • Components: $v_i = \hat{e}_i \cdot \vec{v}$
HILBERT SPACE ($L^2$)
- • States: $|\psi\rangle = \sum_n C_n\,|\phi_n\rangle$
- • Inner product: $\langle\phi|\psi\rangle = \int \phi^*\psi\,dx$
- • Basis: $\langle\phi_m|\phi_n\rangle = \delta_{mn}$
- • Components: $C_n = \langle\phi_n|\psi\rangle$
The Hermite functions $\{\phi_n\}$ from §5 form exactly such a countable orthonormal basis for $L^2(\mathbb{R})$. The coefficients $C_n = \langle\phi_n|\psi\rangle$ are the “coordinates” of the state in the energy eigenbasis. The Dirac bra-ket notation below packages this structure into a powerful symbolic calculus.
Eigenstates, Eigenvalues & Their Role in Quantum Physics
The eigenvalue problem is the central mathematical structure of quantum mechanics. Every observable quantity (energy, momentum, angular momentum, spin) is represented by a Hermitian operator, and the possible measurement outcomes are its eigenvalues.
$$\hat{A}\,|\psi_n\rangle = a_n\,|\psi_n\rangle$$
$\hat{A}$ is a Hermitian operator (observable),$|\psi_n\rangle$ is the eigenstate (the state in which the measurement gives a definite result), and$a_n \in \mathbb{R}$ is the eigenvalue (the measurement outcome). Hermiticity guarantees all eigenvalues are real — as physical measurements must be.
$$\hat{H}\,|\psi_n\rangle = E_n\,|\psi_n\rangle \qquad \Longleftrightarrow \qquad -\frac{\hbar^2}{2m}\frac{d^2\psi_n}{dx^2} + V(x)\,\psi_n = E_n\,\psi_n$$
The time-independent Schrödinger equation is the eigenvalue equation for the Hamiltonian. Its solutions $|\psi_n\rangle$ are the stationary states — they evolve as $|\psi_n(t)\rangle = e^{-iE_n t/\hbar}|\psi_n\rangle$, acquiring only a phase. The energies $E_n$ form the spectrum.
| Observable | Operator | Eigenstates | Eigenvalues |
|---|---|---|---|
| Position | $\hat{x}$ | $|x_0\rangle \;\to\; \delta(x-x_0)$ | $x_0 \in \mathbb{R}$ |
| Momentum | $\hat{p} = -i\hbar\frac{d}{dx}$ | $|p_0\rangle \;\to\; \frac{e^{ip_0 x/\hbar}}{\sqrt{2\pi\hbar}}$ | $p_0 \in \mathbb{R}$ |
| Energy (QHO) | $\hat{H} = \frac{\hat{p}^2}{2m}+\frac{1}{2}m\omega^2\hat{x}^2$ | $|n\rangle \;\to\; \phi_n(x)$ | $E_n = \hbar\omega(n+\tfrac{1}{2})$ |
| Energy (Hydrogen) | $\hat{H} = \frac{\hat{p}^2}{2m}-\frac{e^2}{4\pi\epsilon_0 r}$ | $|n,\ell,m\rangle \;\to\; R_{n\ell}(r)Y_\ell^m(\theta,\phi)$ | $E_n = -\frac{13.6\;\text{eV}}{n^2}$ |
| Angular momentum | $\hat{L}^2,\;\hat{L}_z$ | $|\ell,m\rangle \;\to\; Y_\ell^m(\theta,\phi)$ | $\hbar^2\ell(\ell+1),\;\hbar m$ |
| Spin-1/2 | $\hat{S}_z = \frac{\hbar}{2}\sigma_z$ | $|\!\uparrow\rangle,\;|\!\downarrow\rangle$ | $+\frac{\hbar}{2},\;-\frac{\hbar}{2}$ |
$$|\psi\rangle = \sum_n C_n\,|\psi_n\rangle, \qquad C_n = \langle\psi_n|\psi\rangle$$
$$P(a_n) = |C_n|^2 = |\langle\psi_n|\psi\rangle|^2$$
When observable $\hat{A}$ is measured on state $|\psi\rangle$, the result is always one of the eigenvalues $a_n$, with probability $|C_n|^2$. After measurement, the state collapses to the corresponding eigenstate $|\psi_n\rangle$. The expectation value is:
$$\langle\hat{A}\rangle = \langle\psi|\hat{A}|\psi\rangle = \sum_n a_n\,|C_n|^2$$
$$\hat{A} = \sum_n a_n\,|\psi_n\rangle\langle\psi_n| \qquad \text{(discrete)} \qquad \hat{A} = \int a\,|a\rangle\langle a|\,da \qquad \text{(continuous)}$$
Any Hermitian operator can be reconstructed from its eigenvalues and eigenstates. Functions of operators follow immediately:$f(\hat{A}) = \sum_n f(a_n)\,|\psi_n\rangle\langle\psi_n|$. This is how the time evolution operator$\hat{U}(t) = e^{-i\hat{H}t/\hbar} = \sum_n e^{-iE_n t/\hbar}|n\rangle\langle n|$is defined.
$$[\hat{A},\hat{B}] = 0 \quad \Longleftrightarrow \quad \exists\;\text{common eigenbasis}\;\{|\psi_n\rangle\}:\; \hat{A}|\psi_n\rangle = a_n|\psi_n\rangle,\;\hat{B}|\psi_n\rangle = b_n|\psi_n\rangle$$
Two observables can be measured simultaneously with definite outcomes if and only if they commute. For hydrogen: $[\hat{H},\hat{L}^2] = [\hat{H},\hat{L}_z] = [\hat{L}^2,\hat{L}_z] = 0$, so states can be labelled by $|n,\ell,m\rangle$ — a complete set of commuting observables (CSCO).
$$[\hat{A},\hat{B}] \neq 0 \quad \Longrightarrow \quad \Delta A\;\Delta B \geq \frac{1}{2}\left|\langle[\hat{A},\hat{B}]\rangle\right|$$
The generalised uncertainty relation. For position and momentum:$[\hat{x},\hat{p}] = i\hbar$ gives the Heisenberg bound$\Delta x\,\Delta p \geq \hbar/2$. Non-commuting operators cannot share a common eigenbasis — there is no state with simultaneously definite position and momentum.
Dirac Notation: Kets, Bras & Brackets
Introduced by P.A.M. Dirac (1926), this notation provides a compact symbolic calculus that works identically for finite and infinite-dimensional Hilbert spaces.
Ket (column vector)
$$|\psi\rangle \;\longleftrightarrow\; \begin{pmatrix} C_0 \\ C_1 \\ C_2 \\ \vdots \end{pmatrix}$$
Normalised vector in Hilbert space representing the quantum state.
Bra (row vector)
$$\langle\psi| = (C_0^*,\; C_1^*,\; C_2^*,\; \ldots)$$
Conjugate transpose of the ket. Maps kets to complex numbers.
Bracket (inner product)
$$\langle\psi_b|\psi_a\rangle = \sum_n D_n^* C_n$$
bracket = bra × ket = complex number (probability amplitude).
THREE WAYS TO CHARACTERISE A STATE
1. Wave function
$\psi(x)$
Uncountably many values
2. Fourier transform
$\varphi(p) = \frac{1}{\sqrt{2\pi\hbar}}\int\psi(x)e^{-ipx/\hbar}dx$
Uncountably many values
3. Basis expansion
$\psi(x) = \sum_n C_n\,\phi_n(x)$
Countably many coefficients $C_0, C_1, \ldots$
Example: Photon Polarization (Dimension 2)
The simplest non-trivial Hilbert space has dimension 2. Photon polarization provides a concrete physical realisation where all the abstract formalism can be verified with a polariser.
Basis states
$$|\leftrightarrow\rangle \;=\; \begin{pmatrix}1\\0\end{pmatrix} \qquad |\updownarrow\rangle \;=\; \begin{pmatrix}0\\1\end{pmatrix}$$
$$\langle\leftrightarrow|\leftrightarrow\rangle = 1, \quad \langle\updownarrow|\updownarrow\rangle = 1, \quad \langle\updownarrow|\leftrightarrow\rangle = 0$$
General polarization state
$$|\psi\rangle = C_h|\leftrightarrow\rangle + C_v|\updownarrow\rangle = \begin{pmatrix}C_h\\C_v\end{pmatrix}$$
$$\text{Linear at angle }\theta:\; |\psi_\theta\rangle = \cos\theta\,|\updownarrow\rangle + \sin\theta\,|\leftrightarrow\rangle$$
Malus's Law from QM
$$P(\text{transmission through vertical polariser}) = |\langle\updownarrow|\psi_\theta\rangle|^2 = \cos^2\theta$$
The classical Malus law emerges as a direct consequence of the measurement postulate applied to a 2D Hilbert space.
Circular polarization
$$|\psi_{L,R}\rangle = \frac{|\updownarrow\rangle \pm i\,|\leftrightarrow\rangle}{\sqrt{2}}$$
Complex coefficients $C_h, C_v$ are essential — they encode the phase between components, distinguishing linear from circular polarization.
Time Evolution & the Schrödinger Equation
$$i\hbar\frac{d|\psi(t)\rangle}{dt} = \hat{H}|\psi(t)\rangle$$
Between measurements, the state evolves unitarily. The norm is conserved because the Hamiltonian is Hermitian.
$$|\psi(0)\rangle = \sum_n C_n\,|\psi_n\rangle, \qquad C_n = \langle\psi_n|\psi(0)\rangle$$
$$\boxed{|\psi(t)\rangle = \sum_n C_n\,e^{-iE_n t/\hbar}\,|\psi_n\rangle}$$
Each energy eigenstate acquires a phase factor $e^{-iE_n t/\hbar}$. The expansion coefficients $|C_n|^2$ (probabilities of measuring energy $E_n$) are time-independent — energy is conserved. But probabilities for other observables oscillate in time if the system is in a superposition of energy eigenstates.
Application: Neutrino Flavour Oscillations
A beautiful example that demonstrates superposition, time evolution, and non-commuting observablesin a 3-dimensional (simplified to 2D) Hilbert space. The “missing solar neutrinos” are explained by quantum oscillation between flavour eigenstates.
Two non-commuting bases
FLAVOUR EIGENSTATES (what we detect)
$|\nu_e\rangle,\;|\nu_\mu\rangle,\;|\nu_\tau\rangle$
MASS EIGENSTATES (what propagates)
$|\nu_1\rangle,\;|\nu_2\rangle,\;|\nu_3\rangle \quad (m_1 \neq m_2 \neq m_3)$
At emission (t=0): electron neutrino in the mass basis:
$$|\psi(0)\rangle = |\nu_e\rangle = \cos\theta\,|\nu_1\rangle + \sin\theta\,|\nu_2\rangle$$
After propagation (time evolution via energy eigenstates):
$$|\psi(t)\rangle = e^{-iE_1 t/\hbar}\cos\theta\,|\nu_1\rangle + e^{-iE_2 t/\hbar}\sin\theta\,|\nu_2\rangle$$
Probability of detecting an electron neutrino:
$$\boxed{P_e(t) = |\langle\nu_e|\psi(t)\rangle|^2 = 1 - \sin^2 2\theta\;\sin^2\!\left(\frac{(E_1 - E_2)\,t}{\hbar}\right)}$$
For ultra-relativistic neutrinos ($p \gg mc$):$\;t \simeq L/c$ and$\;E_1 - E_2 \simeq (m_1^2 - m_2^2)c^3/(2p)$. The KamLand experiment confirmed these oscillations with$\;|m_1^2-m_2^2|c^4 = 8\times 10^{-5}\;\text{eV}^2$,$\;\tan^2\theta = 0.5$.
Why non-commutation matters
The Hamiltonian $\hat{H} = \text{diag}(E_1, E_2, E_3)$ in the mass basis does not commute with the flavour operator whose eigenstates are $|\nu_e\rangle, |\nu_\mu\rangle, |\nu_\tau\rangle$. This is why it is impossible to prepare a neutrino with both definite energy AND definite flavour — the same logic as $[\hat{x},\hat{p}] = i\hbar$ for position and momentum.
The Dirac bra-ket formalism unifies discrete and continuous spectra through completeness relations. These integral identities translate abstract operator equations into concrete wave-function integrals.
$$\int_{-\infty}^{+\infty} |x\rangle\langle x|\,dx = \hat{\mathbb{1}}$$
The resolution of identity in the position basis. Inserting this between operators is the standard technique for converting abstract operator equations into differential equations in wave mechanics.
$$\int_{-\infty}^{+\infty} |p\rangle\langle p|\,dp = \hat{\mathbb{1}}$$
The momentum-space analogue. Together with position completeness, it allows switching between representations at will.
$$\langle x | p \rangle = \frac{1}{\sqrt{2\pi\hbar}}\,e^{ipx/\hbar}$$
A momentum eigenstate in position representation is a plane wave. This single formula encodes the Fourier-transform relationship between position and momentum wave functions.
$$\psi(x) = \langle x|\psi\rangle = \int \langle x|p\rangle\langle p|\psi\rangle\,dp = \frac{1}{\sqrt{2\pi\hbar}}\int \tilde\psi(p)\,e^{ipx/\hbar}\,dp$$
Derived by inserting the momentum completeness relation. This demonstrates how the abstract state $|\psi\rangle$ is reconstructed from its momentum-space components.
$$\sum_n |n\rangle\langle n| + \int |E\rangle\langle E|\,dE = \hat{\mathbb{1}}$$
For Hamiltonians with both bound states (discrete) and scattering states (continuous), such as the hydrogen atom or the finite square well, both contributions are needed for a complete resolution of the identity.
$$\mathrm{Tr}[\hat{A}] = \int \langle x|\hat{A}|x\rangle\,dx$$
The trace of an operator in the position basis. This is how one computes the partition function in statistical mechanics:$Z = \mathrm{Tr}[e^{-\beta\hat{H}}] = \int \langle x|e^{-\beta\hat{H}}|x\rangle\,dx$, which connects to the Euclidean path integral.
$$\langle\phi|\psi\rangle = \int_{-\infty}^{+\infty} \phi^*(x)\,\psi(x)\,dx = \int_{-\infty}^{+\infty} \tilde\phi^*(p)\,\tilde\psi(p)\,dp$$
The inner product is representation-independent -- a consequence of the unitarity of the Fourier transform and the self-consistency of the completeness relations. This is the Hilbert-space version of Parseval's theorem.
$$\frac{1}{\pi}\int_{\mathbb{C}} |\alpha\rangle\langle\alpha|\,d^2\alpha = \hat{\mathbb{1}}$$
Coherent states $|\alpha\rangle = e^{-|\alpha|^2/2}\sum_n \frac{\alpha^n}{\sqrt{n!}}|n\rangle$ are overcomplete (non-orthogonal), yet they still resolve the identity. This overcompleteness is what makes the coherent-state path integral possible and is the foundation of the Glauber-Sudarshan P-representation in quantum optics.
$$\hat{A} = \sum_n a_n\,|a_n\rangle\langle a_n| + \int a\,|a\rangle\langle a|\,da$$
Any self-adjoint operator can be written as a weighted sum/integral over its eigenprojectors. The eigenvalues $a_n$ (discrete) and $a$ (continuous) are the possible measurement outcomes. This is the spectral theorem -- the mathematical core of the measurement postulate.
$$\rho(x,x') = \langle x|\hat\rho|x'\rangle = \sum_i p_i\,\psi_i(x)\,\psi_i^*(x'), \qquad \mathrm{Tr}[\hat\rho] = \int\rho(x,x)\,dx = 1$$
The density matrix for a mixed state, expressed in the position basis. The diagonal$\rho(x,x) = \sum_i p_i\,|\psi_i(x)|^2$ gives the position probability density. Off-diagonal elements $\rho(x,x')$ encode quantum coherence between positions.
§8 — Path Integrals
Feynman's path-integral formulation reformulates quantum mechanics as a sum over histories. It provides the most natural bridge to quantum field theory and statistical mechanics. All integrals below are exact closed-form results.
$$K(x_f,t_f;\,x_i,t_i) = \int \mathcal{D}[x(t)]\;\exp\!\left(\frac{i}{\hbar}\int_{t_i}^{t_f} L\!\left(x,\dot{x}\right)dt\right)$$
The quantum propagator as a functional integral over all paths from$(x_i,t_i)$ to $(x_f,t_f)$. The Lagrangian $L$ in the exponent weights each path by its classical action. In the classical limit $\hbar\to 0$, the integral is dominated by the stationary-phase (classical) path.
$$K_{\text{free}}(x_f,t_f;\,x_i,t_i) = \sqrt{\frac{m}{2\pi i\hbar\,T}}\;\exp\!\left(\frac{im(x_f-x_i)^2}{2\hbar\,T}\right), \quad T = t_f - t_i$$
Evaluating the path integral for $L = \tfrac{1}{2}m\dot{x}^2$ reduces to a product of Fresnel integrals over time slices. The result is a spreading Gaussian -- the quantum-mechanical counterpart of diffusion.
$$K_{\text{HO}} = \sqrt{\frac{m\omega}{2\pi i\hbar\sin\omega T}}\;\exp\!\left(\frac{im\omega}{2\hbar\sin\omega T}\left[(x_f^2+x_i^2)\cos\omega T - 2x_f x_i\right]\right)$$
The Mehler formula. Because the Lagrangian is quadratic, the path integral is Gaussian and can be evaluated exactly. The semiclassical approximation is exact for all quadratic Lagrangians. The caustic singularities at $\omega T = n\pi$ reflect classical focal points.
$$Z = \mathrm{Tr}\!\left[e^{-\beta\hat{H}}\right] = \oint \mathcal{D}[x(\tau)]\;\exp\!\left(-\frac{1}{\hbar}\int_0^{\hbar\beta} L_E\,d\tau\right)$$
Wick-rotating $t \to -i\tau$ turns the oscillatory Minkowski path integral into a damped Euclidean one. The trace enforces periodic boundary conditions$x(0) = x(\hbar\beta)$, connecting quantum mechanics at inverse temperature $\beta = 1/k_B T$ to classical statistical mechanics.
$$\langle z_f | e^{-i\hat{H}T/\hbar} | z_i \rangle = \int \mathcal{D}[\bar{z},z]\;\exp\!\left(i\int_0^T \left[\frac{i\hbar}{2}\frac{\bar{z}\dot{z}-\dot{\bar{z}}z}{\bar{z}z+1} - H(\bar{z},z)\right]dt\right)$$
The path integral in the coherent-state (holomorphic) representation. This form is the starting point for the functional integral in many-body quantum mechanics and for spin path integrals on the Bloch sphere.
§9 — Special Function Integrals
A collection of integral identities involving the classical special functions of mathematical physics. These arise throughout scattering theory, partial-wave analysis, semiclassical approximations, and dimensional regularisation.
$$\Gamma(n) = \int_0^{\infty} t^{n-1}\,e^{-t}\,dt, \qquad \mathrm{Re}(n)>0$$
The continuous generalisation of the factorial: $\Gamma(n) = (n-1)!$ for positive integers. Satisfies the recursion $\Gamma(n+1) = n\,\Gamma(n)$and the reflection formula $\Gamma(n)\,\Gamma(1-n) = \pi/\sin(n\pi)$. Ubiquitous in dimensional regularisation.
$$B(a,b) = \int_0^1 t^{a-1}(1-t)^{b-1}\,dt = \frac{\Gamma(a)\,\Gamma(b)}{\Gamma(a+b)}, \qquad \mathrm{Re}(a),\mathrm{Re}(b)>0$$
Appears in Feynman parameter integrals. The identity$\frac{1}{A^\alpha B^\beta} = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\int_0^1\frac{x^{\alpha-1}(1-x)^{\beta-1}}{[xA+(1-x)B]^{\alpha+\beta}}\,dx$ is the Feynman parametrisation used to combine denominators in loop integrals.
$$\int_0^{\infty} J_\nu(kr)\,J_\nu(k'r)\,r\,dr = \frac{1}{k}\,\delta(k-k'), \qquad \nu > -\tfrac{1}{2}$$
The Hankel-type orthogonality of Bessel functions of the first kind. Bessel functions arise in cylindrical-geometry quantum problems (e.g. the infinite cylindrical well, Aharonov-Bohm effect) and in partial-wave expansions of plane waves.
$$\int \frac{e^{i\mathbf{k}\cdot\mathbf{r}}}{r}\,d^3r = \frac{4\pi}{k^2}$$
The 3D Fourier transform of the Coulomb potential. This identity is used constantly in Born-approximation scattering, in deriving the Coulomb matrix elements in second quantisation, and in computing the electron self-energy in QED.
$$\int \frac{e^{-\mu r}}{r}\,e^{i\mathbf{k}\cdot\mathbf{r}}\,d^3r = \frac{4\pi}{k^2 + \mu^2}$$
The screened Coulomb (Yukawa) potential. The screening mass $\mu$ provides a natural IR regulator. Setting $\mu \to 0$ recovers the Coulomb result. This is the momentum-space propagator of a massive scalar field.
$$\int_{-1}^{+1} P_\ell(x)\,P_{\ell'}(x)\,dx = \frac{2}{2\ell+1}\;\delta_{\ell\ell'}$$
Legendre polynomials are the $m=0$ spherical harmonics (up to normalisation). Their orthogonality is used in partial-wave expansions of scattering amplitudes: $f(\theta) = \sum_\ell (2\ell+1)\,f_\ell\,P_\ell(\cos\theta)$.
$$\int g(x)\,e^{i\lambda\phi(x)}\,dx \;\approx\; \sum_{x_s}\sqrt{\frac{2\pi}{\lambda\,|\phi''(x_s)|}}\;g(x_s)\;\exp\!\left(i\lambda\,\phi(x_s) + \frac{i\pi}{4}\,\mathrm{sgn}\,\phi''(x_s)\right)$$
For large $\lambda$ (i.e. $\hbar \to 0$), the integral is dominated by stationary points $\phi'(x_s)=0$. This is the mathematical foundation of the WKB approximation, the semiclassical limit of the path integral, and the method of steepest descent.
$$e^{i\mathbf{k}\cdot\mathbf{r}} = 4\pi\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell} i^\ell\;j_\ell(kr)\;Y_\ell^{m*}(\hat{k})\;Y_\ell^m(\hat{r})$$
Expands a plane wave into spherical Bessel functions and spherical harmonics. This is the starting point for partial-wave scattering theory and the multipole expansion of electromagnetic fields.
$$\Gamma(n+1) \sim \sqrt{2\pi n}\;\left(\frac{n}{e}\right)^n, \qquad n \to \infty$$
The asymptotic expansion of the Gamma function for large argument. Essential in statistical mechanics (entropy of large systems) and in deriving the thermodynamic limit of quantum statistical ensembles.
$$\frac{1}{\pi}\int_{\mathbb{C}} \bar{z}^m\,z^n\,e^{-|z|^2}\,d^2z = n!\;\delta_{mn}$$
The orthogonality relation in the Bargmann-Segal-Fock space of entire functions. This Hilbert space is the natural home for coherent states and provides an elegant alternative to the occupation-number representation.