Mathematical Notation Used in This Volume

This page summarizes the notation used by the Mathematical Toolkit. It is a local orientation guide, not a replacement for the canonical Conventions Overview or the detailed tool pages linked below.

The guiding rule is simple: mathematical notation should make the object clear before it makes a calculation compact. When notation is overloaded, the page should say which meaning is being used.

Vectors, Coordinates, and Kets

An abstract vector may be written as $v\in V$, where $V$ is a vector space. Once a basis $\{e_i\}$ is chosen, the same vector can be represented by coordinates:

$$v=\sum_i v^i e_i.$$

In quantum-mechanical contexts, state vectors are usually written as kets:

$$\lvert\psi\rangle\in\mathcal H.$$

Coordinates depend on the basis; the abstract vector does not. The ket $\lvert\psi\rangle$ and the column of coefficients representing it in a chosen basis are therefore not the same object, even when they are often identified in finite-dimensional calculations.

When the scalar field matters, the default quantum-mechanical setting is a complex vector space.

Bras, Duals, and Inner Products

The dual space $V^*$ consists of linear maps from $V$ to the scalar field. In Dirac notation, a bra $\langle\phi\rvert$ acts on a ket $\lvert\psi\rangle$ to give an inner product:

$$\langle\phi\vert\psi\rangle\in\mathbb C.$$

The convention used here is the standard physics convention:

$$\langle\phi\vert\psi\rangle \text{ is conjugate-linear in } \phi \text{ and linear in } \psi.$$

Thus

$$\langle a\phi_1+b\phi_2\vert\psi\rangle = a^*\langle\phi_1\vert\psi\rangle + b^*\langle\phi_2\vert\psi\rangle.$$

For the canonical statement of bra-ket conventions, see Bra-Ket Notation. For the mathematical object behind the notation, see Dirac Notation as Linear Algebra and Inner Products.

Matrices, Operators, and Linear Maps

The Toolkit uses:

Notation	Meaning
$A:V\to W$	a linear map from $V$ to $W$
$A_{ij}$	matrix elements in a specified basis
$A^\dagger$	adjoint of $A$
$I$ or $I_{\mathcal H}$	identity operator
$P$	projector, usually satisfying $P^2=P=P^\dagger$
$U$	unitary operator, usually satisfying $U^\dagger U=I$
$H$	Hamiltonian or a general Hermitian operator, depending on context

An operator and its matrix are distinct. The operator is basis-independent; the matrix elements are coordinates for that operator after bases have been chosen:

$$A_{ij} = \langle e_i\vert A\vert e_j\rangle.$$

If the basis changes, the matrix changes, but the linear map does not. The finite-dimensional transformation law is summarized in Change of Basis.

Function Spaces and Wavefunctions

For wave mechanics, the main Hilbert space is often an $L^2$ space. A wavefunction may be written as

$$\psi\in L^2(\mathbb R^d),$$

meaning that $\psi$ is square-integrable up to the usual equivalence of functions that differ only on sets of measure zero. In physics notation, one also writes

$$\psi(x)=\langle x\vert\psi\rangle,$$

where $\lvert x\rangle$ is a generalized eigenket rather than a normalizable Hilbert-space vector.

Use L2 Spaces for the functional-analytic meaning and Position and Momentum Representations for the bridge to wavefunctions.

Norms and Absolute Values

The norm of a vector is written $\lVert v\rVert$. For a normalized quantum state,

$$\lVert\psi\rVert^2 = \langle\psi\vert\psi\rangle = 1.$$

The absolute value or modulus of a complex number is written $\lvert z\rvert$. In probability formulas,

$$\lvert\langle\phi\vert\psi\rangle\rvert^2$$

is the squared modulus of a complex amplitude, not the square of a real number.

For the distinction between vector norms, induced metrics, $L^2$ norms, and ray distances, see Norms and Metrics.

Tensor Products

Tensor products are written with $\otimes$. For two vector spaces,

$$V\otimes W$$

is the space generated by formal products $v\otimes w$ subject to bilinearity. In quantum mechanics, a bipartite Hilbert space is often written

$$\mathcal H_{AB} = \mathcal H_A\otimes\mathcal H_B.$$

The ordering matters. A compact basis label such as $\lvert01\rangle$ has meaning only after the page declares which subsystem label is written first. For the mathematical construction, see Tensor Products; for the physical convention in composite systems, see Bipartite Systems.

Index Notation

Indices label components, basis vectors, tensor factors, or summation variables. The meaning should be declared locally.

Typical finite-dimensional notation is

$$(Av)^i = \sum_j A^i{}_j v^j.$$

Repeated indices are not automatically summed unless a page explicitly declares Einstein summation convention. In most introductory Toolkit pages, sums are written explicitly to avoid ambiguity. The detailed convention guide is Index Notation and Summation Conventions.

For basis expansions, lower and upper index placement is used only when it helps distinguish vectors, dual vectors, or covariant and contravariant components. Many finite-dimensional quantum pages use simpler matrix notation.

Common Symbols

Symbol	Usual meaning
$\mathbb R$	real numbers
$\mathbb C$	complex numbers
$V,W$	vector spaces
$\mathcal H$	Hilbert space
$\mathcal B(\mathcal H)$	bounded operators on $\mathcal H$
$I$	identity operator or identity matrix
$A^\dagger$	adjoint or conjugate transpose
$\operatorname{Tr}A$	trace of $A$
$\det A$	determinant of $A$
$\ker A$	kernel or null space of $A$
$\operatorname{im} A$	image or range of $A$
$[A,B]$	commutator $AB-BA$
$\{A,B\}$	anticommutator $AB+BA$

The Symbol Map is the best place to compare notation across the Core Formalism pages. The Representation Translation Table is the quickest way to translate among bra-ket, matrix, wavefunction, and density-operator notation.

Common Mistakes

• Treating a coordinate column as the abstract vector itself.
• Forgetting complex conjugation when turning kets into bras.
• Using $\lvert z\rvert^2$ and $z^2$ interchangeably for complex amplitudes.
• Suppressing tensor-product identity factors before the subsystem ordering is clear.
• Assuming repeated indices are summed without checking the page convention.
• Mixing Fourier transform conventions from different references without translating them.

Cross-Links

• Conventions Overview
• Bra-Ket Notation
• Fourier Transform Conventions
• Diagnostic Checklist
• Sets, Functions, and Maps
• Complex Vector Spaces
• Dirac Notation as Linear Algebra
• Index Notation and Summation Conventions
• Bases and Coordinates
• Matrices as Linear Maps
• Change of Basis
• Vector Spaces and Dual Spaces
• Inner Products
• Norms and Metrics
• Tensor Products
• L2 Spaces
• Representation Translation Table
• Symbol Map

References

• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• G. Strang, Linear Algebra and Its Applications, 4th ed., Brooks/Cole, 2006.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.

Exercises

1. In the basis $\{e_1,e_2\}$, let $v=2e_1-ie_2$. Write the coordinate column for $v$.

Solution

The coordinate column is

$$\begin{pmatrix} 2\\ -i \end{pmatrix}.$$

This column represents $v$ only after the basis has been chosen.

2. With the physics inner-product convention, compute $\langle\psi\rvert$ for

$$\lvert\psi\rangle = \frac{1}{\sqrt2} \left( \lvert0\rangle+i\lvert1\rangle \right).$$

Solution

Taking the adjoint conjugates the coefficients:

$$\langle\psi\rvert = \frac{1}{\sqrt2} \left( \langle0\rvert-i\langle1\rvert \right).$$

3. Explain why $\lvert01\rangle$ is ambiguous if no tensor-product ordering has been declared.

Solution

The label could mean $\lvert0\rangle_A\otimes\lvert1\rangle_B$, but it could also mean $\lvert0\rangle_B\otimes\lvert1\rangle_A$ if the opposite ordering is used. The compact label becomes meaningful only after the subsystem order has been fixed.