Appendix A: Mathematical Reference for Quantum Mechanics

This appendix collects the most frequently needed mathematical results in one place. Rather than deriving each result, we state it cleanly, note conventions, and point back to the chapter where the physics context first appears. Keep this appendix open whenever you are grinding through a problem set; it will save you from hunting through three different textbooks for that one integral identity.

A.1 Gaussian Integrals and Generalizations

The Gaussian integral is the single most useful integral in quantum mechanics. Every path-integral calculation, every coherent-state overlap, and every wave-packet propagation ultimately reduces to a Gaussian.

Basic Gaussian integral:

$$ \int_{-\infty}^{\infty} e^{-ax^2}\,dx = \sqrt{\frac{\pi}{a}}, \qquad \text{Re}(a) > 0 $$

With a linear term (completing the square):

$$ \int_{-\infty}^{\infty} e^{-ax^2 + bx}\,dx = \sqrt{\frac{\pi}{a}}\,e^{b^2/(4a)} $$

Moments — even powers only survive:

$$ \int_{-\infty}^{\infty} x^{2n}\,e^{-ax^2}\,dx = \frac{(2n)!}{n!\,2^{2n}}\sqrt{\frac{\pi}{a}}\;\frac{1}{a^n} $$

Odd powers vanish by symmetry. The first few special cases:

$$ \int_{-\infty}^{\infty} x^{2}\,e^{-ax^2}\,dx = \frac{1}{2a}\sqrt{\frac{\pi}{a}} $$

$$ \int_{-\infty}^{\infty} x^{4}\,e^{-ax^2}\,dx = \frac{3}{4a^2}\sqrt{\frac{\pi}{a}} $$

Complex Gaussian (Fresnel-type):

$$ \int_{-\infty}^{\infty} e^{iax^2}\,dx = \sqrt{\frac{\pi}{a}}\,e^{i\pi/4}, \qquad a > 0 $$

This appears in free-particle propagators and in the stationary-phase approximation (Chapter 9).

Multidimensional Gaussian:

For a real symmetric positive-definite matrix A of dimension $n$:

$$ \int_{\mathbb{R}^n} e^{-\frac{1}{2}\mathbf{x}^T \mathbf{A}\,\mathbf{x}}\,d^n x = \frac{(2\pi)^{n/2}}{\sqrt{\det \mathbf{A}}} $$

A.2 The Dirac Delta Function

The Dirac delta is not a function in the ordinary sense but a distribution. It is defined by its action under an integral:

$$ \int_{-\infty}^{\infty} f(x)\,\delta(x - a)\,dx = f(a) $$

Key identities:

Integral representations:

$$ \delta(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{ikx}\,dk $$

$$ \delta(x) = \lim_{\epsilon \to 0^+} \frac{1}{\sqrt{2\pi\epsilon}}\,e^{-x^2/(2\epsilon)} $$

$$ \delta(x) = \lim_{\epsilon \to 0^+} \frac{1}{\pi}\frac{\epsilon}{x^2 + \epsilon^2} \quad \text{(Lorentzian representation)} $$

Three-dimensional delta function:

$$ \delta^3(\mathbf{r} - \mathbf{r}') = \delta(x-x')\,\delta(y-y')\,\delta(z-z') $$

In spherical coordinates:

$$ \delta^3(\mathbf{r} - \mathbf{r}') = \frac{\delta(r - r')}{r^2}\,\delta(\cos\theta - \cos\theta')\,\delta(\phi - \phi') $$

Completeness relation (general):

If $\{|n\rangle\}$ is a complete orthonormal set, then $\sum_n |n\rangle\langle n| = \hat{I}$ in Dirac notation, which in position representation becomes:

$$ \sum_n \psi_n(x)\,\psi_n^*(x') = \delta(x - x') $$

A.3 Special Functions

A.3.1 Hermite Polynomials

Hermite polynomials $H_n(x)$ arise in the harmonic oscillator (Chapter 6). Two conventions exist; we use the "physicist's" convention throughout this book.

Rodrigues formula:

$$ H_n(x) = (-1)^n\,e^{x^2}\,\frac{d^n}{dx^n}\,e^{-x^2} $$

First several:

Recurrence relation:

$$ H_{n+1}(x) = 2x\,H_n(x) - 2n\,H_{n-1}(x) $$

Orthogonality:

$$ \int_{-\infty}^{\infty} H_m(x)\,H_n(x)\,e^{-x^2}\,dx = \sqrt{\pi}\,2^n\,n!\;\delta_{mn} $$

Generating function:

$$ e^{2xt - t^2} = \sum_{n=0}^{\infty} \frac{H_n(x)}{n!}\,t^n $$

A.3.2 Associated Laguerre Polynomials

Associated Laguerre polynomials $L_n^k(x)$ appear in the radial hydrogen wave functions (Chapter 10).

Rodrigues formula:

$$ L_n^k(x) = \frac{e^x\,x^{-k}}{n!}\,\frac{d^n}{dx^n}\left(e^{-x}\,x^{n+k}\right) $$

First several (for $k=0$):

Orthogonality:

$$ \int_0^{\infty} x^k\,e^{-x}\,L_m^k(x)\,L_n^k(x)\,dx = \frac{(n+k)!}{n!}\,\delta_{mn} $$

Recurrence:

$$ (n+1)\,L_{n+1}^k(x) = (2n + k + 1 - x)\,L_n^k(x) - (n+k)\,L_{n-1}^k(x) $$

A.3.3 Legendre Polynomials and Associated Legendre Functions

These arise in the angular part of central-force problems (Chapter 8).

Rodrigues formula for Legendre polynomials:

$$ P_l(x) = \frac{1}{2^l\,l!}\,\frac{d^l}{dx^l}(x^2 - 1)^l $$

First several:

Associated Legendre functions:

$$ P_l^m(x) = (-1)^m\,(1-x^2)^{m/2}\,\frac{d^m}{dx^m}P_l(x), \qquad 0 \le m \le l $$

For negative $m$: $P_l^{-m}(x) = (-1)^m \frac{(l-m)!}{(l+m)!}P_l^m(x)$

Orthogonality:

$$ \int_{-1}^{1} P_l^m(x)\,P_{l'}^m(x)\,dx = \frac{2}{2l+1}\,\frac{(l+m)!}{(l-m)!}\,\delta_{ll'} $$

A.3.4 Spherical Harmonics

The spherical harmonics $Y_l^m(\theta,\phi)$ form a complete orthonormal basis on the unit sphere (Chapter 8).

$$ Y_l^m(\theta,\phi) = (-1)^m\sqrt{\frac{2l+1}{4\pi}\,\frac{(l-m)!}{(l+m)!}}\;P_l^m(\cos\theta)\;e^{im\phi} $$

Orthonormality:

$$ \int_0^{2\pi}d\phi\int_0^{\pi}\sin\theta\,d\theta\;Y_l^m(\theta,\phi)^*\,Y_{l'}^{m'}(\theta,\phi) = \delta_{ll'}\delta_{mm'} $$

Useful values:

Addition theorem:

$$ P_l(\cos\gamma) = \frac{4\pi}{2l+1}\sum_{m=-l}^{l} Y_l^{m*}(\theta',\phi')\,Y_l^m(\theta,\phi) $$

where $\gamma$ is the angle between directions $(\theta,\phi)$ and $(\theta',\phi')$.

Parity:

$$ Y_l^m(\pi - \theta, \phi + \pi) = (-1)^l\,Y_l^m(\theta,\phi) $$

A.3.5 Spherical Bessel Functions

Spherical Bessel functions appear in scattering theory (Chapter 27) and the free-particle solution in spherical coordinates.

$$ j_l(x) = (-x)^l \left(\frac{1}{x}\frac{d}{dx}\right)^l \frac{\sin x}{x} $$

$$ n_l(x) = -(-x)^l \left(\frac{1}{x}\frac{d}{dx}\right)^l \frac{\cos x}{x} $$

First several:

Asymptotic behavior ($x \to \infty$):

$$ j_l(x) \to \frac{1}{x}\sin\!\left(x - \frac{l\pi}{2}\right), \qquad n_l(x) \to -\frac{1}{x}\cos\!\left(x - \frac{l\pi}{2}\right) $$

A.4 Fourier Transform Conventions

Multiple conventions exist. We use the "physics" convention throughout this textbook:

Forward transform:

$$ \tilde{f}(k) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} f(x)\,e^{-ikx}\,dx $$

Inverse transform:

$$ f(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \tilde{f}(k)\,e^{ikx}\,dk $$

This symmetric convention places $1/\sqrt{2\pi}$ on both transforms.

Parseval's theorem (Plancherel):

$$ \int_{-\infty}^{\infty} |f(x)|^2\,dx = \int_{-\infty}^{\infty} |\tilde{f}(k)|^2\,dk $$

Convolution theorem:

$$ \widetilde{(f*g)}(k) = \sqrt{2\pi}\;\tilde{f}(k)\,\tilde{g}(k) $$

Useful transform pairs:

Three-dimensional Fourier transform:

$$ \tilde{f}(\mathbf{k}) = \frac{1}{(2\pi)^{3/2}}\int f(\mathbf{r})\,e^{-i\mathbf{k}\cdot\mathbf{r}}\,d^3r $$

A.5 Useful Series Expansions

Taylor expansion of the exponential:

$$ e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!} $$

Baker-Campbell-Hausdorff (BCH) formula (to second order):

$$ e^{\hat{A}}e^{\hat{B}} = e^{\hat{A}+\hat{B}+\frac{1}{2}[\hat{A},\hat{B}]+\cdots} $$

If $[\hat{A},[\hat{A},\hat{B}]] = [\hat{B},[\hat{A},\hat{B}]] = 0$, this truncates exactly:

$$ e^{\hat{A}}e^{\hat{B}} = e^{\hat{A}+\hat{B}+\frac{1}{2}[\hat{A},\hat{B}]} $$

This exact case appears constantly with creation and annihilation operators (Chapter 6).

Operator expansion (Hadamard lemma):

$$ e^{\hat{A}}\hat{B}\,e^{-\hat{A}} = \hat{B} + [\hat{A},\hat{B}] + \frac{1}{2!}[\hat{A},[\hat{A},\hat{B}]] + \cdots $$

Useful binomial-type expansion:

$$ (1+x)^n = \sum_{k=0}^{\infty}\binom{n}{k}x^k, \qquad |x|<1 $$

A.6 Clebsch-Gordan Coefficients

Clebsch-Gordan coefficients $\langle j_1 m_1; j_2 m_2 | J M\rangle$ couple two angular momenta: $|J,M\rangle = \sum_{m_1,m_2} \langle j_1 m_1; j_2 m_2 | J M\rangle\,|j_1,m_1\rangle|j_2,m_2\rangle$.

Selection rules: A Clebsch-Gordan coefficient vanishes unless: 1. $M = m_1 + m_2$ 2. $|j_1 - j_2| \le J \le j_1 + j_2$ (triangle inequality) 3. $j_1 + j_2 + J$ is consistent with integer/half-integer rules

Table: $j_1 = \tfrac{1}{2} \otimes j_2 = \tfrac{1}{2}$

Coupled states $J = 0, 1$:

Here $|+\rangle \equiv |{\tfrac{1}{2}},+{\tfrac{1}{2}}\rangle$ and $|-\rangle \equiv |{\tfrac{1}{2}},-{\tfrac{1}{2}}\rangle$.

Table: $j_1 = 1 \otimes j_2 = \tfrac{1}{2}$

Coupled states $J = \tfrac{3}{2}, \tfrac{1}{2}$:

Symmetry properties of Clebsch-Gordan coefficients:

$$ \langle j_1 m_1; j_2 m_2 | J M\rangle = (-1)^{j_1+j_2-J}\langle j_2 m_2; j_1 m_1 | J M\rangle $$

$$ \langle j_1 m_1; j_2 m_2 | J M\rangle = (-1)^{j_1+j_2-J}\langle j_1,{-m_1}; j_2,{-m_2} | J,{-M}\rangle $$

A.7 Wigner 3j Symbols

The Wigner 3j symbol is a more symmetric alternative to Clebsch-Gordan coefficients:

$$ \begin{pmatrix} j_1 & j_2 & J \\ m_1 & m_2 & M \end{pmatrix} = \frac{(-1)^{j_1-j_2-M}}{\sqrt{2J+1}}\,\langle j_1 m_1; j_2 m_2 | J,{-M}\rangle $$

Symmetry properties:

• Even permutation of columns: unchanged.
• Odd permutation of columns: multiplied by $(-1)^{j_1+j_2+J}$.
• Sign flip of all $m$ values: multiplied by $(-1)^{j_1+j_2+J}$.

Orthogonality:

$$ \sum_{m_1,m_2} \begin{pmatrix} j_1 & j_2 & J \\ m_1 & m_2 & M \end{pmatrix}\begin{pmatrix} j_1 & j_2 & J' \\ m_1 & m_2 & M' \end{pmatrix} = \frac{\delta_{JJ'}\delta_{MM'}}{2J+1} $$

A.8 Key Linear Algebra Results

Eigenvalue Theorems

Spectral theorem (finite-dimensional): Every Hermitian matrix $A$ (i.e., $A = A^\dagger$) can be diagonalized by a unitary transformation:

$$ A = U\,\Lambda\,U^\dagger $$

where $\Lambda = \text{diag}(\lambda_1, \ldots, \lambda_n)$ with all $\lambda_i \in \mathbb{R}$, and $U$ is unitary ($U^\dagger U = I$). The columns of $U$ are the orthonormal eigenvectors.

Implications for quantum mechanics: - All observables (Hermitian operators) have real eigenvalues. - Eigenstates of a Hermitian operator corresponding to distinct eigenvalues are orthogonal. - The eigenstates form a complete basis (resolution of the identity).

Simultaneous diagonalization: Two Hermitian operators $\hat{A}$ and $\hat{B}$ can be simultaneously diagonalized if and only if $[\hat{A},\hat{B}] = 0$.

Trace Identities

The trace is cyclic: $\text{Tr}(ABC) = \text{Tr}(BCA) = \text{Tr}(CAB)$.

Key identities:

$$ \text{Tr}(\hat{A}\hat{B}) = \sum_{n}\langle n|\hat{A}\hat{B}|n\rangle = \sum_{n,m}\langle n|\hat{A}|m\rangle\langle m|\hat{B}|n\rangle $$

$$ \text{Tr}(|a\rangle\langle b|) = \langle b|a\rangle $$

$$ \text{Tr}(\hat{A}) = \sum_i \lambda_i \quad \text{(sum of eigenvalues)} $$

$$ \det(\hat{A}) = \prod_i \lambda_i \quad \text{(product of eigenvalues)} $$

Matrix exponential identity:

$$ \det(e^A) = e^{\text{Tr}(A)} $$

Commutator Algebra

Commutator identities used throughout the book:

$$ [\hat{A},\hat{B}\hat{C}] = [\hat{A},\hat{B}]\hat{C} + \hat{B}[\hat{A},\hat{C}] $$

$$ [\hat{A}\hat{B},\hat{C}] = \hat{A}[\hat{B},\hat{C}] + [\hat{A},\hat{C}]\hat{B} $$

Jacobi identity:

$$ [\hat{A},[\hat{B},\hat{C}]] + [\hat{B},[\hat{C},\hat{A}]] + [\hat{C},[\hat{A},\hat{B}]] = 0 $$

Canonical commutation relation and its consequences:

$$ [\hat{x},\hat{p}] = i\hbar $$

$$ [\hat{x},\hat{p}^n] = i\hbar\,n\,\hat{p}^{n-1} $$

$$ [\hat{x}^n,\hat{p}] = i\hbar\,n\,\hat{x}^{n-1} $$

$$ [\hat{x}, f(\hat{p})] = i\hbar\,\frac{\partial f}{\partial p} $$

A.9 Integral Tables for Quantum Mechanics

Radial hydrogen integrals (useful for perturbation theory):

$$ \langle r^s \rangle_{n,l} = \int_0^{\infty} r^{s+2}\,|R_{nl}(r)|^2\,dr $$

Key results (in units of Bohr radius $a_0$):

$$ \langle r \rangle = \frac{a_0}{2}\left[3n^2 - l(l+1)\right] $$

$$ \langle r^2 \rangle = \frac{a_0^2\,n^2}{2}\left[5n^2 + 1 - 3l(l+1)\right] $$

$$ \langle r^{-1} \rangle = \frac{1}{n^2\,a_0} $$

$$ \langle r^{-2} \rangle = \frac{1}{n^3\,a_0^2\,(l+\tfrac{1}{2})} $$

$$ \langle r^{-3} \rangle = \frac{1}{n^3\,a_0^3\,l(l+\tfrac{1}{2})(l+1)}, \qquad l \neq 0 $$

Integrals involving spherical Bessel functions (scattering):

$$ \int_0^{\infty} j_l(kr)\,j_l(k'r)\,r^2\,dr = \frac{\pi}{2k^2}\,\delta(k-k') $$

Angular integral (product of three spherical harmonics):

$$ \int Y_{l_1}^{m_1}\,Y_{l_2}^{m_2}\,Y_{l_3}^{m_3}\,d\Omega = \sqrt{\frac{(2l_1+1)(2l_2+1)(2l_3+1)}{4\pi}}\begin{pmatrix}l_1 & l_2 & l_3\\0&0&0\end{pmatrix}\begin{pmatrix}l_1 & l_2 & l_3\\m_1&m_2&m_3\end{pmatrix} $$

This result appears in the evaluation of matrix elements for the Stark effect (Chapter 19) and selection rules (Chapter 17).

A.10 Useful Miscellaneous Results

Stirling's approximation:

$$ \ln(n!) \approx n\ln n - n + \frac{1}{2}\ln(2\pi n) $$

Euler's formula:

$$ e^{i\theta} = \cos\theta + i\sin\theta $$

Pauli matrix algebra:

$$ \sigma_i\,\sigma_j = \delta_{ij}\,I + i\,\epsilon_{ijk}\,\sigma_k $$

$$ (\boldsymbol{\sigma}\cdot\mathbf{a})(\boldsymbol{\sigma}\cdot\mathbf{b}) = \mathbf{a}\cdot\mathbf{b}\;I + i\,\boldsymbol{\sigma}\cdot(\mathbf{a}\times\mathbf{b}) $$

$$ e^{i\theta\,\hat{n}\cdot\boldsymbol{\sigma}/2} = \cos\frac{\theta}{2}\,I + i\sin\frac{\theta}{2}\,\hat{n}\cdot\boldsymbol{\sigma} $$

These identities are indispensable for spin-1/2 problems (Chapters 12-13).

Levi-Civita symbol $\epsilon_{ijk}$:

$$ \epsilon_{ijk}\epsilon_{ilm} = \delta_{jl}\delta_{km} - \delta_{jm}\delta_{kl} $$

$$ \epsilon_{ijk}\epsilon_{ijm} = 2\,\delta_{km} $$

$$ \epsilon_{ijk}\epsilon_{ijk} = 6 $$

This appendix is a living reference. When in doubt about a sign convention or normalization, the definitions here are the ones used throughout this textbook. For derivations and physical context, follow the chapter references given in each section.