Appendix D: Notation Guide
Quantum mechanics has accumulated notation from multiple traditions: Schrodinger's wave mechanics, Dirac's abstract formalism, the angular-momentum machinery of atomic spectroscopy, and the operator algebra of quantum field theory. This appendix provides a single-source reference for every notational convention used in this textbook. When two conventions exist in the wider literature, we state our choice explicitly and note the alternative.
D.1 Wave Mechanics Notation
These symbols appear primarily in Chapters 1-11, wherever we work in position or momentum representation.
| Symbol | Meaning | First appears |
|---|---|---|
| $\psi(x, t)$ | Position-space wave function (one dimension) | Ch 1 |
| $\Psi(\mathbf{r}, t)$ | Position-space wave function (three dimensions) | Ch 8 |
| $\phi(k, t)$ or $\tilde{\psi}(k, t)$ | Momentum-space wave function | Ch 3 |
| $|\psi(x)|^2$ | Probability density (position) | Ch 1 |
| $|\phi(k)|^2$ | Probability density (momentum) | Ch 3 |
| $\psi^*(x)$ | Complex conjugate of $\psi$ | Ch 1 |
| $\hat{H}$ | Hamiltonian operator | Ch 1 |
| $\hat{p}$ | Momentum operator, $= -i\hbar\frac{\partial}{\partial x}$ in position rep | Ch 2 |
| $\hat{x}$ | Position operator, $= x$ in position representation | Ch 2 |
| $V(x)$ or $V(\mathbf{r})$ | Potential energy function | Ch 4 |
| $\psi_n(x)$ | Energy eigenstate with quantum number $n$ | Ch 4 |
| $E_n$ | Energy eigenvalue | Ch 4 |
| $R_{nl}(r)$ | Radial wave function for hydrogen | Ch 10 |
| $Y_l^m(\theta, \phi)$ | Spherical harmonic | Ch 8 |
| $j(x, t)$ | Probability current density | Ch 9 |
| $\rho(x, t)$ | Probability density, $= |\psi|^2$ | Ch 1 |
Operator notation convention: In this textbook, operators are always denoted with a hat ($\hat{A}$) when there is any risk of confusion with a number or a function. In purely abstract sections where everything is an operator, we sometimes drop the hat for readability and note this explicitly.
D.2 Dirac (Bra-Ket) Notation
Dirac notation is introduced in Chapter 2 and used throughout the rest of the textbook. It provides a representation-independent way to write quantum mechanics.
| Symbol | Name | Meaning |
|---|---|---|
| $\lvert\psi\rangle$ | Ket | A state vector in Hilbert space |
| $\langle\phi\rvert$ | Bra | The dual vector of $\lvert\phi\rangle$ |
| $\langle\phi\lvert\psi\rangle$ | Bracket (inner product) | Complex number; amplitude for $\lvert\psi\rangle$ to be found in state $\lvert\phi\rangle$ |
| $\lvert\psi\rangle\langle\phi\rvert$ | Outer product | An operator (projects onto or maps between states) |
| $\langle\phi\rvert\hat{A}\lvert\psi\rangle$ | Matrix element | The $(\phi, \psi)$ element of operator $\hat{A}$ |
| $\langle\hat{A}\rangle$ or $\langle\psi\rvert\hat{A}\lvert\psi\rangle$ | Expectation value | The average of observable $\hat{A}$ in state $\lvert\psi\rangle$ |
| $\lvert n\rangle$ | Eigenket | Eigenstate of some operator, labeled by quantum number $n$ |
| $\lvert\psi(t)\rangle$ | Time-dependent state | State vector at time $t$ in the Schrodinger picture |
Completeness (resolution of the identity):
For a discrete orthonormal basis:
$$ \hat{I} = \sum_n |n\rangle\langle n| $$
For a continuous basis (position):
$$ \hat{I} = \int |x\rangle\langle x|\,dx $$
Connecting Dirac and wave-function notation:
$$ \psi(x) = \langle x|\psi\rangle, \qquad \phi(k) = \langle k|\psi\rangle $$
$$ \langle\phi|\psi\rangle = \int_{-\infty}^{\infty} \phi^*(x)\,\psi(x)\,dx $$
D.3 Operator Notation
| Symbol | Meaning |
|---|---|
| $\hat{A}$ | A general operator |
| $\hat{A}^\dagger$ | Hermitian adjoint (conjugate transpose) of $\hat{A}$ |
| $[\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}$ | Commutator |
| $\{\hat{A}, \hat{B}\} = \hat{A}\hat{B} + \hat{B}\hat{A}$ | Anticommutator |
| $\hat{I}$ or $\mathbb{1}$ | Identity operator |
| $\hat{U}$ | Unitary operator ($\hat{U}^\dagger\hat{U} = \hat{I}$) |
| $\hat{P}_n = \lvert n\rangle\langle n\rvert$ | Projection operator onto state $\lvert n\rangle$ |
| $e^{i\hat{A}}$ | Operator exponential (defined via Taylor series) |
| $\hat{T}$ | Time-ordering operator (Chapter 19) |
| $\hat{a}$, $\hat{a}^\dagger$ | Annihilation and creation (ladder) operators |
| $\hat{N} = \hat{a}^\dagger\hat{a}$ | Number operator |
| $\hat{\rho}$ | Density operator (density matrix) |
Hermiticity: An operator $\hat{A}$ is Hermitian if $\hat{A} = \hat{A}^\dagger$, equivalently $\langle\phi|\hat{A}|\psi\rangle = \langle\hat{A}\phi|\psi\rangle$ for all states. All physical observables are represented by Hermitian operators.
Unitary operators: Time evolution, rotations, and basis changes are represented by unitary operators. The key property: $\hat{U}^\dagger = \hat{U}^{-1}$.
D.4 Angular Momentum Notation
Angular momentum is the most notation-heavy part of quantum mechanics. We follow the standard conventions of Sakurai and Griffiths.
Orbital Angular Momentum
| Symbol | Meaning |
|---|---|
| $\hat{\mathbf{L}} = \hat{\mathbf{r}} \times \hat{\mathbf{p}}$ | Orbital angular momentum vector operator |
| $\hat{L}_x, \hat{L}_y, \hat{L}_z$ | Cartesian components |
| $\hat{L}^2$ | Total orbital angular momentum squared |
| $\hat{L}_\pm = \hat{L}_x \pm i\hat{L}_y$ | Raising/lowering operators |
| $l$ | Orbital angular momentum quantum number ($l = 0, 1, 2, \ldots$) |
| $m$ or $m_l$ | Magnetic quantum number ($m = -l, -l+1, \ldots, l$) |
| $\lvert l, m\rangle$ | Simultaneous eigenstate of $\hat{L}^2$ and $\hat{L}_z$ |
Eigenvalue equations:
$$ \hat{L}^2|l, m\rangle = \hbar^2 l(l+1)|l, m\rangle, \qquad \hat{L}_z|l, m\rangle = \hbar m|l, m\rangle $$
Spin Angular Momentum
| Symbol | Meaning |
|---|---|
| $\hat{\mathbf{S}}$ | Spin angular momentum vector operator |
| $\hat{S}_x, \hat{S}_y, \hat{S}_z$ | Spin components |
| $\hat{S}^2$ | Total spin squared |
| $\hat{S}_\pm$ | Spin raising/lowering |
| $s$ | Spin quantum number ($s = 0, \tfrac{1}{2}, 1, \tfrac{3}{2}, \ldots$) |
| $m_s$ | Spin magnetic quantum number |
| $\lvert s, m_s\rangle$ | Spin eigenstate |
| $\lvert +\rangle$ or $\lvert\uparrow\rangle$ | Spin-up ($m_s = +\tfrac{1}{2}$) for spin-1/2 |
| $\lvert -\rangle$ or $\lvert\downarrow\rangle$ | Spin-down ($m_s = -\tfrac{1}{2}$) for spin-1/2 |
| $\chi$ | Spinor (two-component column vector for spin-1/2) |
| $\sigma_x, \sigma_y, \sigma_z$ | Pauli matrices |
| $\boldsymbol{\sigma} = (\sigma_x, \sigma_y, \sigma_z)$ | Vector of Pauli matrices |
Pauli matrices (explicit):
$$ \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} $$
Total and Coupled Angular Momentum
| Symbol | Meaning |
|---|---|
| $\hat{\mathbf{J}} = \hat{\mathbf{L}} + \hat{\mathbf{S}}$ | Total angular momentum |
| $j$ | Total angular momentum quantum number |
| $m_j$ | Total magnetic quantum number |
| $\lvert j, m_j\rangle$ | Coupled angular momentum eigenstate |
| $\langle j_1 m_1; j_2 m_2 \lvert J M\rangle$ | Clebsch-Gordan coefficient |
| $\lvert j_1, j_2; J, M\rangle$ | Coupled state in the total-$J$ basis |
Spectroscopic notation (used in atomic physics chapters):
- Letters for $l$: $s = 0$, $p = 1$, $d = 2$, $f = 3$, $g = 4$, ...
- Term symbol: ${}^{2S+1}L_J$. Example: ${}^2P_{3/2}$ means $S = 1/2$, $L = 1$, $J = 3/2$.
D.5 Matrix Element Notation
| Expression | Meaning |
|---|---|
| $\langle n\lvert\hat{A}\rvert m\rangle$ | Matrix element of $\hat{A}$ between states $\lvert m\rangle$ and $\lvert n\rangle$ |
| $A_{nm}$ | Same, written in component form |
| $\langle n\lvert\hat{A}\rvert n\rangle$ | Diagonal matrix element (expectation value in eigenstate $\lvert n\rangle$) |
| $\langle f\lvert\hat{H}'\rvert i\rangle$ | Perturbation matrix element between initial and final states |
Reduced matrix elements (Wigner-Eckart theorem, Chapter 16):
$$ \langle j', m'|\hat{T}_q^{(k)}|j, m\rangle = \frac{\langle j'||\hat{T}^{(k)}||j\rangle}{\sqrt{2j'+1}}\,\langle j, m; k, q|j', m'\rangle $$
The double-bar notation $\langle j'||\hat{T}^{(k)}||j\rangle$ denotes the reduced matrix element, which is independent of $m$, $m'$, and $q$.
D.6 Tensor Product Notation
| Symbol | Meaning |
|---|---|
| $\lvert\psi\rangle \otimes \lvert\phi\rangle$ | Tensor product of two state vectors |
| $\lvert\psi\rangle\lvert\phi\rangle$ | Shorthand for tensor product (when unambiguous) |
| $\lvert\psi, \phi\rangle$ | Another common shorthand |
| $\hat{A} \otimes \hat{B}$ | Tensor product of operators |
| $\hat{A} \otimes \hat{I}$ | Operator $\hat{A}$ acting on subsystem 1, identity on subsystem 2 |
| $\mathcal{H}_1 \otimes \mathcal{H}_2$ | Tensor product of Hilbert spaces |
Dimensions: If $\dim(\mathcal{H}_1) = d_1$ and $\dim(\mathcal{H}_2) = d_2$, then $\dim(\mathcal{H}_1 \otimes \mathcal{H}_2) = d_1 \cdot d_2$.
D.7 Density Matrix Notation
| Symbol | Meaning |
|---|---|
| $\hat{\rho}$ | Density operator (density matrix) |
| $\hat{\rho} = \lvert\psi\rangle\langle\psi\rvert$ | Pure-state density matrix |
| $\hat{\rho} = \sum_i p_i \lvert\psi_i\rangle\langle\psi_i\rvert$ | Mixed-state density matrix |
| $\text{Tr}(\hat{\rho}\hat{A})$ | Expectation value of $\hat{A}$ in state $\hat{\rho}$ |
| $\text{Tr}_B(\hat{\rho}_{AB})$ | Partial trace over subsystem $B$ |
| $S(\hat{\rho}) = -\text{Tr}(\hat{\rho}\ln\hat{\rho})$ | von Neumann entropy |
| $\gamma = \text{Tr}(\hat{\rho}^2)$ | Purity ($= 1$ for pure states, $< 1$ for mixed) |
D.8 Quantum Number Labels
The hydrogen atom and multi-electron atoms use the following quantum numbers:
| Symbol | Name | Range | Determines |
|---|---|---|---|
| $n$ | Principal | $1, 2, 3, \ldots$ | Energy (in hydrogen) |
| $l$ | Orbital angular momentum | $0, 1, \ldots, n-1$ | Shape of orbital |
| $m_l$ | Magnetic (orbital) | $-l, \ldots, +l$ | Orientation |
| $s$ | Spin | $\tfrac{1}{2}$ for electrons | Intrinsic angular momentum |
| $m_s$ | Magnetic (spin) | $\pm\tfrac{1}{2}$ | Spin orientation |
| $j$ | Total angular momentum | $\lvert l - s\rvert, \ldots, l + s$ | Spin-orbit coupled |
| $m_j$ | Total magnetic | $-j, \ldots, +j$ | Total orientation |
Multi-electron atoms additionally use:
| Symbol | Name | Meaning |
|---|---|---|
| $L$ | Total orbital | $\sum_i \hat{l}_i$ coupled |
| $S$ | Total spin | $\sum_i \hat{s}_i$ coupled |
| $J$ | Grand total | $\hat{L} + \hat{S}$ coupled |
| $M_J$ | Grand total magnetic | $-J, \ldots, +J$ |
D.9 Scattering Notation
| Symbol | Meaning |
|---|---|
| $f(\theta, \phi)$ | Scattering amplitude |
| $\frac{d\sigma}{d\Omega}$ | Differential cross section |
| $\sigma_{\text{tot}}$ | Total cross section |
| $\delta_l$ | Phase shift for partial wave $l$ |
| $S_l = e^{2i\delta_l}$ | S-matrix element for partial wave $l$ |
| $T_{fi}$ or $\mathcal{T}$ | Transition matrix (T-matrix) element |
| $k = |\mathbf{k}|$ | Wave number of incident particle |
| $\mathbf{k}, \mathbf{k}'$ | Initial and final wave vectors |
| $\mathbf{q} = \mathbf{k}' - \mathbf{k}$ | Momentum transfer |
D.10 Time Evolution Notation
| Symbol | Meaning |
|---|---|
| $\hat{U}(t, t_0) = e^{-i\hat{H}(t-t_0)/\hbar}$ | Time-evolution operator (time-independent $\hat{H}$) |
| $\lvert\psi(t)\rangle_S$ | Schrodinger-picture state |
| $\hat{A}_H(t)$ | Heisenberg-picture operator |
| $\lvert\psi(t)\rangle_I$ | Interaction-picture state |
| $\hat{A}_I(t)$ | Interaction-picture operator |
Pictures of quantum mechanics:
| Picture | States evolve? | Operators evolve? |
|---|---|---|
| Schrodinger | Yes | No |
| Heisenberg | No | Yes |
| Interaction | Yes (via $\hat{H}'$) | Yes (via $\hat{H}_0$) |
D.11 Miscellaneous Notation
| Symbol | Meaning |
|---|---|
| $\delta_{mn}$ | Kronecker delta ($= 1$ if $m = n$, else $0$) |
| $\delta(x)$ | Dirac delta function |
| $\epsilon_{ijk}$ | Levi-Civita totally antisymmetric symbol |
| $\theta(x)$ | Heaviside step function ($= 1$ for $x > 0$, $0$ for $x < 0$) |
| $\text{Re}(z)$, $\text{Im}(z)$ | Real and imaginary parts of complex number $z$ |
| $|z|$ | Modulus of complex number; also absolute value |
| $z^*$ | Complex conjugate |
| $\nabla^2$ | Laplacian operator |
| $\mathcal{H}$ | Hilbert space |
| $\dim(\mathcal{H})$ | Dimension of Hilbert space |
| $\otimes$ | Tensor product |
| $\oplus$ | Direct sum |
| $\propto$ | Proportional to |
| $\sim$ | Of the order of; scales as |
| $\equiv$ | Defined as; identically equal to |
| $\text{c.c.}$ | Complex conjugate of the preceding term |
| $\text{h.c.}$ | Hermitian conjugate of the preceding term |
D.12 Common Abbreviations Used in This Textbook
| Abbreviation | Meaning |
|---|---|
| QM | Quantum mechanics |
| SHO | Simple harmonic oscillator |
| TISE | Time-independent Schrodinger equation |
| TDSE | Time-dependent Schrodinger equation |
| WKB | Wentzel-Kramers-Brillouin (semiclassical approximation) |
| CG | Clebsch-Gordan (coefficient) |
| EPR | Einstein-Podolsky-Rosen |
| CHSH | Clauser-Horne-Shimony-Holt (inequality) |
| DM | Density matrix |
| RHS / LHS | Right-hand side / Left-hand side |
| QED | Quantum electrodynamics |
| QFT | Quantum field theory |
| a.u. | Atomic units |
If you encounter a symbol in the text that is not listed here, check the paragraph immediately preceding its first use — we always define notation at the point of introduction. For mathematical identities involving these symbols, see Appendix A.