Mathematical Language

Quantum mechanics is formulated with vectors, linear maps, dual vectors, inner products, and complex scalars. Calculations usually replace those abstract objects by coordinate columns, matrices, and indexed components. Both levels are indispensable, but confusing an object with one of its representations produces many of the subject’s most persistent errors.

This chapter establishes that grammar. It is intentionally narrower than the later Linear Algebra and Hilbert Spaces chapters: the pages here explain what the basic objects are, how their representations are related, and which conventions must be declared before a formula is meaningful.

The structural layers

The same calculation can be described at several levels. Keeping the levels visible makes it easier to tell which statements are intrinsic and which depend on choices.

Layer	Typical object	Representation after choices	Quantum-mechanical role
Set-theoretic	map $f:X\to Y$	rule, table, or formula	transformations and outcome maps
Linear	vector $v\in V$, map $T:V\to W$	column $[v]_{\mathcal E}$, matrix $[T]_{\mathcal F\leftarrow\mathcal E}$	states and operators
Dual	functional $\ell\in V^*$	row of components	amplitudes and bras
Metric	inner product $\langle u,v\rangle$	Gram matrix or conjugate transpose in an orthonormal basis	probabilities and orthogonality
Topological	norm $\lVert v\rVert$, metric $d(u,v)$	numerical length and distance	normalization, limits, and approximation

A statement is coordinate-free when it does not depend on a basis. For example, $Tv=w$ and $\ker T=\{0\}$ are coordinate-free. The entries of $[T]_{\mathcal F\leftarrow\mathcal E}$ are not: changing either basis changes the matrix while leaving $T$ itself unchanged.

A dependency map

There are two main paths through the chapter, and they meet in Dirac notation.

1. Algebraic path: Sets, Functions, and Maps $\to$ Vector Spaces and Dual Spaces $\to$ Linear Maps $\to$ Bases and Coordinates $\to$ Matrices as Linear Maps.
2. Geometric path: Inner Products $\to$ Norms and Metrics $\to$ completeness and Hilbert-space methods.
3. Quantum translation: Complex Vector Spaces combines with the algebraic and geometric paths in Dirac Notation as Linear Algebra.
4. Component calculations: Index Notation and Summation Conventions provides a compact language for matrices, tensors, and angular-momentum identities.

Readers already fluent in elementary linear algebra can begin with the distinctions below and use the page map as a diagnostic. Readers who find any of the distinctions uncertain should follow the algebraic path in order.

Object versus representation

Let $V$ and $W$ be vector spaces over the same field $\mathbb F$. A map $T:V\to W$ is linear when

$$T(av+bw)=aT(v)+bT(w)$$

for all $v,w\in V$ and $a,b\in\mathbb F$. This property belongs to $T$ before any basis is chosen.

Choose a basis $\mathcal E=(e_1,\ldots,e_n)$ of $V$ and a basis $\mathcal F=(f_1,\ldots,f_m)$ of $W$. Coordinates and matrix entries are then defined by

$$v=e_jv^j, \qquad T(e_j)=f_iT^i{}_j, \qquad [T(v)]^i=T^i{}_jv^j.$$

The final expression is matrix multiplication written in index notation. It is a coordinate description of the abstract equation $T(v)$. The basis vectors, coordinate column, and matrix must transform together; changing only one of them changes the mathematical problem.

For an endomorphism $A:V\to V$, suppose a new basis is related to the old one by the invertible matrix $S$. With the convention

$$[v]'=S^{-1}[v],$$

the matrix of the same operator becomes

$$[A]'=S^{-1}[A]S.$$

This similarity transformation does not create a new operator. It expresses one operator in a new coordinate system. The distinction is essential when moving among energy, position, momentum, spin, or computational bases.

Dual vectors, bras, and the inner product

The algebraic dual $V^*$ is the vector space of linear functionals $\ell:V\to\mathbb F$. A dual vector consumes a vector and returns a scalar. Without additional structure, there is no canonical identification of $V$ with $V^*$.

An inner product adds that structure. This volume uses the physics convention: $\langle u,v\rangle$ is conjugate-linear in $u$ and linear in $v$. Thus, for fixed $u$, the rule

$$v\longmapsto \langle u,v\rangle$$

is a linear functional. In a Hilbert space, the Riesz representation theorem says that every continuous linear functional has this form for a unique $u$. Dirac notation writes the vector as $|u\rangle$, the corresponding functional as $\langle u|$, and its action as $\langle u|v\rangle$.

The identification is conjugate-linear:

$$a|u\rangle+b|w\rangle \quad\longmapsto\quad a^*\langle u|+b^*\langle w|.$$

This is why taking an adjoint conjugates scalar coefficients. Saying merely that a bra is a “row vector” hides both the dual-space role and the inner-product structure used to obtain that row.

Inner product, norm, and metric

These notions are related but not interchangeable. An inner product determines a norm, and a norm determines a metric:

$$\lVert v\rVert=\sqrt{\langle v,v\rangle}, \qquad d(u,v)=\lVert u-v\rVert.$$

The inner product contains more information than length alone: it determines angles, orthogonality, and amplitudes. The Cauchy–Schwarz inequality,

$$\lvert\langle u,v\rangle\rvert \leq \lVert u\rVert\,\lVert v\rVert,$$

ensures that normalized amplitudes have magnitude at most one. A general norm need not arise from an inner product, and a general metric need not arise from a norm. Later pages use these distinctions to define convergence, completeness, boundedness, and approximation error without silently assuming finite dimension.

Complex linearity and phase

Quantum state spaces are complex vector spaces. A complex-linear map obeys $T(iv)=iT(v)$; a conjugate-linear map obeys $K(iv)=-iK(v)$. Complex conjugation is therefore not a complex-linear operator, even though it is linear over the real numbers.

Phase also has two different roles. Multiplying an entire nonzero state vector by $e^{i\chi}$ changes its representative but not the physical ray. Changing the phase of one component relative to another can alter interference and therefore changes the physical state. The Complex Vector Spaces page develops this distinction; State Vectors and Rays and Global Phase give its physical interpretation.

A two-level translation

Let $\{|0\rangle,|1\rangle\}$ be an orthonormal basis of a two-dimensional Hilbert space. The abstract vector

$$|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$$

has coordinate column $c=(\alpha,\beta)^{\mathsf T}$ in this basis. If $A$ is a linear operator, its matrix has entries

$$A_{ij}=\langle i|A|j\rangle, \qquad i,j\in\{0,1\}.$$

The expectation value can be written at either level:

$$\langle\psi|A|\psi\rangle=c^\dagger A c.$$

Now choose another orthonormal basis related by a unitary matrix $U$. With $c'=U^\dagger c$ and $A'=U^\dagger A U$,

$${c'}^\dagger A'c' =c^\dagger U U^\dagger A U U^\dagger c =c^\dagger A c.$$

The column and matrix changed, but the scalar did not. This is the basic pattern behind representation changes throughout quantum mechanics: coordinate data transform so that intrinsic predictions remain invariant.

Page map

Page	Central question	Continue when you can…
Sets, Functions, and Maps	What does a map do, and what are its domain, codomain, image, and preimage?	distinguish inverse maps from inverse images
Vector Spaces and Dual Spaces	What structure permits linear combinations, and what is a linear functional?	distinguish vectors from dual vectors
Linear Maps	Which maps preserve linear combinations?	use kernels, images, composition, and invertibility
Bases and Coordinates	How does a basis turn abstract vectors into components?	separate a vector from its coordinate column
Matrices as Linear Maps	How does a matrix represent a map?	interpret multiplication as composition and similarity as basis change
Inner Products	What defines amplitudes, lengths, and orthogonality?	state the slot-linearity convention and apply Cauchy–Schwarz
Norms and Metrics	What defines length, distance, and convergence?	distinguish inner products, norms, and metrics
Complex Vector Spaces	What changes when the scalar field is complex?	distinguish linear from conjugate-linear maps and global from relative phase
Dirac Notation as Linear Algebra	How do kets, bras, outer products, and matrix elements translate into linear algebra?	move reliably between abstract and component forms
Index Notation and Summation Conventions	How do free and summed indices encode contractions?	check index balance and expand a contraction explicitly

How to read a formula

Before manipulating a formula, ask six questions:

1. What are the spaces? Record the domain, codomain, scalar field, and relevant subspaces.
2. What kind of object is each symbol? Separate vectors, covectors, operators, scalars, coordinates, and matrices.
3. Which choices are implicit? Identify bases, normalization conventions, ordering conventions, and the inner-product convention.
4. Which slots are linear? Complex conjugation and adjoints make this question unavoidable.
5. Which indices are free and which are summed? Every free index must agree on both sides of an equation.
6. Which structure is being used? A conclusion about orthogonality needs an inner product; a conclusion about convergence needs a metric or topology.

This checklist is especially useful when a familiar finite-dimensional identity is carried into wave mechanics. In infinite-dimensional spaces, domains, continuity, completeness, and convergence can become part of the statement rather than technical afterthoughts.

Routes onward

• For finite-dimensional states, observables, projectors, and tensor products, continue to Finite-Dimensional Hilbert Spaces, Eigenvalues and Eigenvectors, and Tensor Products.
• For wavefunctions and unbounded operators, continue to Hilbert Spaces, $L^2$ Spaces, and Domains of Operators.
• For rotations, spin, and generators, combine Index Notation with Groups and Representations.
• For computation, combine bases, matrices, and norms with Matrix Diagonalization and Conditioning and Stability.
• For the physical interpretation of this language, continue to States and Representations and Observables and Operators.

Common mistakes

Mistake	Why it fails	Repair
Calling a coordinate column “the vector” without naming a basis	the column changes under a basis change	name the abstract vector and its basis-dependent coordinates separately
Treating a matrix as basis-independent	its entries depend on domain and codomain bases	write $[T]_{\mathcal F\leftarrow\mathcal E}$ when the bases matter
Identifying $V$ with $V^*$ without structure	no canonical identification exists for a bare vector space	state the inner product and invoke the finite-dimensional or Hilbert-space Riesz result
Forgetting which inner-product slot is linear	coefficients receive the wrong complex conjugation	declare the convention before expanding bras and kets
Assuming every norm comes from an inner product	general normed spaces need not satisfy the parallelogram identity	use inner-product identities only when an inner product is given
Treating complex conjugation as complex-linear	conjugation sends $iv$ to $-iK(v)$	classify it as conjugate-linear
Repeating a free index or leaving a dummy index unmatched	the expression no longer defines consistent components	audit every term for the same free indices and paired dummy indices

Exercises

1. Object or representation?

In a fixed orthonormal basis, a state is represented by $c$, an operator by $A$, and an expectation value by $c^\dagger A c$. Classify each expression as basis-dependent or basis-independent, and state what abstract object it represents.

Solution

The column $c$ and matrix $A$ are basis-dependent. They represent an abstract vector $|\psi\rangle$ and an abstract operator $\hat A$, respectively. The scalar $c^\dagger A c=\langle\psi|\hat A|\psi\rangle$ is basis-independent, provided the vector and operator representations are transformed consistently.

2. Basis invariance of a matrix element

Let $c'=U^\dagger c$, $d'=U^\dagger d$, and $A'=U^\dagger A U$, where $U$ is unitary. Show that ${d'}^\dagger A'c'=d^\dagger A c$.

Solution

Because ${d'}^\dagger=(U^\dagger d)^\dagger=d^\dagger U$ and $UU^\dagger=I$,

$${d'}^\dagger A'c' =d^\dagger U(U^\dagger A U)U^\dagger c =d^\dagger A c.$$

The calculation shows explicitly how the coordinate changes cancel in the scalar matrix element.

3. Why the ket-to-bra map is conjugate-linear

Using the physics convention for the inner product, determine the bra corresponding to $a|u\rangle+b|v\rangle$.

Solution

For every $|w\rangle$,

$$\langle au+bv|w\rangle =a^*\langle u|w\rangle+b^*\langle v|w\rangle.$$

Therefore

$$a|u\rangle+b|v\rangle \longmapsto a^*\langle u|+b^*\langle v|.$$

The functional is linear in $|w\rangle$, while the map from its representing ket to the bra is conjugate-linear.

4. Free and dummy indices

For $(AB)^i{}_j=A^i{}_kB^k{}_j$, identify the free and dummy indices. Explain why replacing the right-hand side by $A^i{}_jB^j{}_k$ would not be an equation for $(AB)^i{}_j$.

Solution

In the correct expression, $i$ and $j$ are free and $k$ is summed. The proposed replacement has $i$ and $k$ free, while $j$ is summed. Its free-index pattern therefore does not match the left-hand side. Renaming a dummy index is harmless, but changing which indices are free changes the tensor components being described.

References

• S. Axler, Linear Algebra Done Right, 3rd ed., Springer, 2015.
• P. R. Halmos, Finite-Dimensional Vector Spaces, Springer, 1974.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics, Volume I: Functional Analysis, revised and enlarged ed., Academic Press, 1980.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.