Mathematical Preliminaries

This section collects the core mathematical objects used to formulate quantum mechanics. Familiarity with linear algebra over the complex numbers is assumed.

Hilbert Space

A Hilbert space $\mathcal{H}$ is a complex vector space equipped with an inner product $⟨\cdot |\cdot ⟩:\mathcal{H}\times \mathcal{H}\to \mathbb{C}$ that is complete with respect to the norm $\Vert \psi \Vert =\sqrt{⟨\psi |\psi ⟩}$. The inner product satisfies:

• Conjugate symmetry: $⟨\varphi |\psi ⟩=\overline{⟨\psi |\varphi ⟩}$.
• Linearity in the second argument: $⟨\varphi |\alpha {\psi }_1+\beta {\psi }_2⟩=\alpha ⟨\varphi |{\psi }_1⟩+\beta ⟨\varphi |{\psi }_2⟩$ (physics convention).
• Positive-definiteness: $⟨\psi |\psi ⟩\ge 0$, with equality iff $|\psi ⟩=0$.

In quantum mechanics, $\mathcal{H}$ is taken to be separable (it admits a countable orthonormal basis). Examples: ${\mathbb{C}}^n$ for finite-dimensional systems (e.g. spin), and $L^2({\mathbb{R}}^3)$ — square-integrable wavefunctions — for a particle in three-dimensional space.

Notation Key for Common Hilbert Spaces

The symbols ${\mathbb{C}}^n$ and $L^2$ recur throughout these notes. Their meaning:

$\mathbb{C}$ and ${\mathbb{C}}^n$

• $\mathbb{C}$ is the field of complex numbers; $\mathbb{R}$, $\mathbb{Z}$, $\mathbb{Q}$ are the reals, integers, rationals.
• ${\mathbb{C}}^n$ is the $n$-dimensional complex vector space: $n$-tuples $(z_1,\dots ,z_n)$ with $z_i\in \mathbb{C}$, componentwise addition and complex scalar multiplication.
• It comes with the standard inner product $⟨z,w⟩={\sum }_i{z}_i^{\ast }{w}_i$, making it a finite-dimensional Hilbert space.
• Typical uses: ${\mathbb{C}}^2$ for a single qubit / electron spin, ${\mathbb{C}}^4$ for a Dirac spinor, ${\mathbb{C}}^n$ for any internal/spin/flavor index space.

$L^p$ and $L^2$

$L^p(X,d\mu )$ is the space of complex-valued functions $f:X\to \mathbb{C}$ that are $p$-integrable with respect to the measure $\mu$:

$${\int }_X|f{|}^{p}d\mu <\infty .$$

The "$p$" is just the exponent inside the integrand — different choices give different function spaces:

So "$p$-integrable" is the umbrella term, of which "integrable" ($p=1$) and "square-integrable" ($p=2$) are the most common special cases. A function being in $L^p$ for one value of $p$ does not imply it is in $L^q$ for another — e.g. $f(x)=1/x$ on $[1,\infty )$ is in $L^2$ and $L^3$ but not in $L^1$.

The "$L$" honors Henri Lebesgue (the integral is the Lebesgue integral, not the Riemann integral). The case $p=2$ is special:

• $L^2$ is the only $L^p$ that is a Hilbert space, with inner product $⟨f,g⟩=\int f^{\ast }gd\mu$.
• It is the natural space for wavefunctions, since $|\psi {|}^2$ is the probability density (Born rule), and the normalization condition $\int |\psi {|}^2=1$ is precisely the $p=2$ condition.

The argument specifies the underlying measure space:

Two technicalities usually glossed over in physics: elements of $L^2$ are equivalence classes of functions agreeing almost everywhere (the "value at a single point" is not really meaningful), and position eigenstates ${\delta }^3(\mathbf{x}-{\mathbf{x}}_0)$ are not in $L^2$ — they are distributions in a larger rigged-Hilbert-space construction.

Tensor Products and Direct Sums

The symbols $\otimes$ and $\oplus$ are used to compose Hilbert spaces (see also Tensor Product below):

• ${\mathcal{H}}_A\otimes {\mathcal{H}}_B$ — both kinds of data simultaneously (e.g. spatial × spin, particle 1 × particle 2). A vector is a "product" $|{\varphi }_A⟩\otimes |{\varphi }_B⟩$ or a sum of such products.
• ${\mathcal{H}}_A\oplus {\mathcal{H}}_B$ — either kind of data (orthogonal direct sum). A vector is a pair $(|{\varphi }_A⟩,|{\varphi }_B⟩)$, with norm $\Vert {\varphi }_A{\Vert }^2+\Vert {\varphi }_B{\Vert }^2$.

Common Hilbert-Space Building Blocks

These building blocks are composed via $\otimes$ and $\oplus$ to give every Hilbert space encountered in QM and QFT. In particular, Fock space (see QFT/fock-space-inventory.md) is built as an orthogonal direct sum of (anti)symmetrized tensor powers of a one-particle space.

Bra–Ket (Dirac) Notation

• A ket $|\psi ⟩$ denotes a vector in $\mathcal{H}$.
• A bra $⟨\varphi |$ denotes the corresponding dual vector (a continuous linear functional on $\mathcal{H}$), so that $⟨\varphi |\psi ⟩$ is the inner product.
• The outer product $|\psi ⟩⟨\varphi |$ is a linear operator acting as $(|\psi ⟩⟨\varphi |)|\chi ⟩=⟨\varphi |\chi ⟩|\psi ⟩$.

State Vector

A state vector is a unit vector $|\psi ⟩\in \mathcal{H}$ (i.e. $⟨\psi |\psi ⟩=1$) representing the physical state of a system. State vectors are defined only up to a global phase: $|\psi ⟩$ and $e^{i\theta }|\psi ⟩$ describe the same physical state.

Linear Operators

A linear operator $\hat{A}:\mathcal{H}\to \mathcal{H}$ satisfies $\hat{A}(\alpha |\psi ⟩+\beta |\varphi ⟩)=\alpha \hat{A}|\psi ⟩+\beta \hat{A}|\varphi ⟩$.

• The adjoint ${\hat{A}}^{†}$ is defined by $⟨\varphi |\hat{A}\psi ⟩=⟨{\hat{A}}^{†}\varphi |\psi ⟩$ for all $|\varphi ⟩,|\psi ⟩\in \mathcal{H}$.
• $\hat{A}$ is Hermitian (self-adjoint) if ${\hat{A}}^{†}=\hat{A}$.
• $\hat{U}$ is unitary if ${\hat{U}}^{†}\hat{U}=\hat{U}{\hat{U}}^{†}=\hat{1}$. Unitary operators preserve the inner product: $⟨\hat{U}\varphi |\hat{U}\psi ⟩=⟨\varphi |\psi ⟩$.
• $\hat{P}$ is a projector if ${\hat{P}}^2=\hat{P}$ and ${\hat{P}}^{†}=\hat{P}$.

Eigenvalues, Eigenstates, and Eigenspaces

For an operator $\hat{A}$ on $\mathcal{H}$:

• An eigenvalue is a scalar $a\in \mathbb{C}$ for which there exists a nonzero vector $|a⟩$ such that $\hat{A}|a⟩=a|a⟩$.
• An eigenstate (or eigenvector) is such a vector $|a⟩$.
• The eigenspace associated with eigenvalue $a$ is the subspace $${\mathcal{H}}_a=\{|\psi ⟩\in \mathcal{H}:\hat{A}|\psi ⟩=a|\psi ⟩\}.$$ Its dimension is the degeneracy of $a$.
• The spectrum of $\hat{A}$ is the set of its eigenvalues (more generally, the set of $\lambda$ for which $\hat{A}-\lambda \hat{1}$ is not invertible).

For a Hermitian operator, all eigenvalues are real, eigenstates corresponding to distinct eigenvalues are orthogonal, and the eigenstates form a complete orthonormal basis of $\mathcal{H}$ (spectral theorem).

Observable

An observable is a Hermitian operator $\hat{A}$ representing a measurable physical quantity. By the spectral theorem it admits the decomposition

$$\hat{A}=\sum _n{a}_n{\hat{P}}_n,$$

where $a_n$ are the (real) eigenvalues and ${\hat{P}}_n$ is the projector onto the eigenspace ${\mathcal{H}}_{a_n}$. The projectors satisfy ${\hat{P}}_n{\hat{P}}_m={\delta }_{nm}{\hat{P}}_n$ and ${\sum }_n{\hat{P}}_n=\hat{1}$ (completeness relation).

Expectation Value

The expectation value of an observable $\hat{A}$ in the state $|\psi ⟩$ is

$$⟨\hat{A}{⟩}_{\psi }=⟨\psi |\hat{A}|\psi ⟩=\sum _n{a}_{n}P(a_n).$$

It is the statistical mean of measurement outcomes over many identically prepared systems.

Commutator

The commutator of two operators is $[\hat{A},\hat{B}]=\hat{A}\hat{B}-\hat{B}\hat{A}$. Two observables can be measured simultaneously with arbitrary precision (they share a common eigenbasis) if and only if $[\hat{A},\hat{B}]=0$.

Tensor Product

For Hilbert spaces ${\mathcal{H}}_A$ and ${\mathcal{H}}_B$ with bases $\{|i{⟩}_A\}$ and $\{|j{⟩}_B\}$, the tensor product ${\mathcal{H}}_A\otimes {\mathcal{H}}_B$ has basis $\{|i{⟩}_A\otimes |j{⟩}_B\}$. A general element is

$$|\Psi ⟩=\sum _{i,j}{c}_{ij}|i{⟩}_A\otimes |j{⟩}_B.$$

A state is separable if it can be written as $|{\psi }_A⟩\otimes |{\psi }_B⟩$, and entangled otherwise.

Density Operator (brief)

A more general description of a quantum state — including statistical mixtures — is given by a density operator $\hat{\rho }$: a Hermitian, positive semidefinite operator with $\mathrm{Tr}(\hat{\rho })=1$. A pure state corresponds to $\hat{\rho }=|\psi ⟩⟨\psi |$; a mixed state is a convex combination $\hat{\rho }={\sum }_k{p}_k|{\psi }_k⟩⟨{\psi }_k|$ with $p_k\ge 0$ and ${\sum }_k{p}_k=1$.