Chapter 1

Mathematical Tools of Quantum Mechanics

1.1Introduction

In classical mechanics the state of a particle is specified by its position $\vec r(t)$ and momentum $\vec p(t)$ as functions of time. These quantities are real numbers that can, in principle, be measured simultaneously with arbitrary precision. Equations of motion, such as Newton's second law or Hamilton's equations, are ordinary differential equations for these real variables.

In quantum mechanics the situation is fundamentally different:

A physical state is described by a state vector $\ket{\psi}$ in an abstract complex Hilbert space $\mathcal{H}$ .
Observable quantities such as position, momentum, and energy are represented by linear operators acting on $\mathcal{H}$ .
The time evolution of a state is governed by the (linear) Schrödinger equation.

Because of this linear structure, the language of Hilbert spaces, Dirac bra–ket notation, and linear operators is the natural mathematical framework of quantum mechanics.

The goals of this chapter are:

to formulate the basic postulates of quantum mechanics;
to introduce the Hilbert–space structures that realize each postulate (states, operators, and representations);
to organize the material so that different representations (matrix, position, momentum) are seen as the same physics in different bases.

1.2Postulate I: States and Hilbert Space

1.2.1Statement of the Postulate

Postulate I (State Postulate). Every isolated physical system is associated with a complex Hilbert space $\mathcal{H}$ . The physical state of the system at a given time is represented by a normalized ket $\ket{\psi}\in\mathcal{H}$ :

\begin{equation} \braket{\psi}{\psi} = 1. \end{equation}

The ket $\ket{\psi}$ encodes all physically accessible information about the system. It is not directly observable; only probabilities and expectation values derived from it are measurable.

1.2.2Hilbert Space and Inner Product

Definition - Complex vector space

A (complex) vector space $$V$$ is a set whose elements are kets $\ket{v},\ket{w},\dots$ , equipped with vector addition and multiplication by complex scalars $\alpha\in\mathbb{C}$ satisfying the usual linear algebra axioms (closure, associativity, commutativity of addition, distributivity, and existence of $$0$$ and $-\ket{v}$ ).

Definition - Inner product

An inner product on $$V$$ is a map

\begin{equation} (\ket{u},\ket{v}) \;\longmapsto\; \braket{u}{v}\in\mathbb{C} \end{equation}

such that, for all kets $\ket{u},\ket{v},\ket{w}$ and all $\alpha,\beta\in\mathbb{C}$ :

Conjugate symmetry: $\braket{u}{v} = \braket{v}{u}^{\ast}$ .
Linearity in the second argument: $\braket{u}{\alpha v+\beta w} = \alpha\braket{u}{v}+\beta\braket{u}{w}$ .
Positive definiteness: $\braket{v}{v} \ge 0$ and $\braket{v}{v}=0$ implies $\ket{v}=0$ .

The inner product defines the norm

\begin{equation} \|\ket{v}\| = \sqrt{\braket{v}{v}}, \end{equation}

and two kets are orthogonal if $\braket{u}{v}=0$ .

Definition - Hilbert space

A Hilbert space is a complex vector space $$H$$ with an inner product $\braket{\cdot}{\cdot}$ which is complete with respect to the norm $\|\ket{v}\| = \sqrt{\braket{v}{v}}$ ; i.e., every Cauchy sequence of kets in $$H$$ converges to a ket in $$H$$ .

Example - Typical Hilbert spaces

$\mathbb{C}^n$ with $\braket{\vec u}{\vec v}=\sum_{j=1}^{n}u_j^{\ast}v_j$ is a finite-dimensional Hilbert space (spin systems, few-level atoms).
$L^2(\mathbb{R}^3)$ , the space of square-integrable functions $\psi(\vec r)$ with
$\begin{equation} \int_{\mathbb{R}^3}|\psi(\vec r)|^2 d^3r < \infty, \end{equation}$
and inner product
$\begin{equation} \braket{\phi}{\psi} = \int_{\mathbb{R}^3}\phi^{\ast}(\vec r)\,\psi(\vec r)\,d^3r, \end{equation}$
describes nonrelativistic wave functions in position space.

1.2.3Bases, Orthonormality, and Completeness

Definition - Basis

A set of kets $\{\ket{e_n}\}$ in $$H$$ is a basis if:

the kets are linearly independent;
every $\ket{\psi}\in H$ can be written uniquely as
$\begin{equation} \ket{\psi} = \sum_n c_n \ket{e_n}. \end{equation}$

If a basis has $$N$$ elements, $$H$$ is $$N$$ –dimensional; otherwise $$H$$ is infinite–dimensional.

Definition - Orthonormal basis

A set of kets $\{\ket{n}\}$ is orthonormal if

\begin{equation} \braket{m}{n} = \delta_{mn}, \end{equation}

and complete if any $\ket{\psi}\in H$ can be expanded as

\begin{equation} \ket{\psi} = \sum_n c_n \ket{n},\qquad c_n = \braket{n}{\psi}. \end{equation}

A complete orthonormal set $\{\ket{n}\}$ is called an orthonormal basis.

Completeness can be written in operator form as

\begin{equation} \sum_n \ket{n}\bra{n} = \hat I, \end{equation}

where $\hat I$ is the identity operator on $$H$$ .

For a continuum of labels $\alpha$ we instead have

\begin{equation} \braket{\alpha}{\alpha'} = \delta(\alpha-\alpha'), \qquad \int d\alpha\,\ket{\alpha}\bra{\alpha} = \hat I. \end{equation}

1.2.4Dirac Notation and State Representations

A physical state is represented by a ket $\ket{\psi}\in\mathcal{H}$ . To each ket we associate a bra $\bra{\psi}$ via Hermitian conjugation:

\begin{equation} \ket{\psi} \longleftrightarrow \bra{\psi} = (\ket{\psi})^{\dagger}. \end{equation}

The inner product is written as

\begin{equation} \braket{\phi}{\psi} = \bra{\phi}\ket{\psi}. \end{equation}

Course notation convention. From this point onward (including atomic and molecular chapters), the abstract state vector is always written as a ket. Coordinate-space objects such as $\psi(\vec r)$ , $\phi(\vec p)$ , or molecular forms like $\Psi(\vec r,\vec R)$ are treated as representations of kets, never as a separate formalism:

\begin{equation} \psi(\vec r)=\braket{\vec r}{\psi}, \qquad \phi(\vec p)=\braket{\vec p}{\psi}, \qquad \Psi(\vec r,\vec R)=\braket{\vec r,\vec R}{\Psi}. \end{equation}

We will switch representations when useful, but the state itself remains $\ket{\psi}$ or $\ket{\Psi}$ .

Matrix (abstract) representation

In a discrete orthonormal basis $\{\ket{n}\}$ , we define the components

\begin{equation} c_n = \braket{n}{\psi}. \end{equation}

We can represent the state $\ket{\psi}$ as a column vector

\begin{equation} \ket{\psi} \;\widehat{=}\; \begin{pmatrix} c_1 \\ c_2 \\ \vdots \end{pmatrix}, \end{equation}

and bras as row vectors (complex conjugate transpose)

\begin{equation} \bra{\psi} \;\widehat{=}\; \bigl(c_1^{\ast},\,c_2^{\ast},\,\dots\bigr). \end{equation}

This is the usual matrix representation used in finite-dimensional quantum systems (two-level atoms, spin- $\tfrac12$ , etc.).

Position representation

The position basis $\{\ket{\vec r}\}$ satisfies

\begin{align} \braket{\vec r}{\vec r'} &= \delta(\vec r-\vec r'), \label{eq:pos-ortho}\\ \int d^3r\,\ket{\vec r}\bra{\vec r} &= \hat I. \label{eq:pos-complete} \end{align}

The position-space wave function of $\ket{\psi}$ is

\begin{equation} \psi(\vec r) = \braket{\vec r}{\psi}, \end{equation}

and

\begin{equation} \ket{\psi} = \int d^3r\,\ket{\vec r}\,\psi(\vec r). \end{equation}

Normalization $\braket{\psi}{\psi}=1$ becomes

\begin{equation} \int d^3r\,|\psi(\vec r)|^2 = 1, \end{equation}

so $|\psi(\vec r)|^2 d^3r$ is the probability to find the particle near $\vec r$ .

Momentum representation

Similarly, the momentum basis $\{\ket{\vec p}\}$ satisfies

\begin{align} \braket{\vec p}{\vec p'} &= \delta(\vec p-\vec p'), \label{eq:mom-ortho}\\ \int d^3p\,\ket{\vec p}\bra{\vec p} &= \hat I. \label{eq:mom-complete} \end{align}

The momentum-space wave function is

\begin{equation} \phi(\vec p) = \braket{\vec p}{\psi}, \qquad \ket{\psi} = \int d^3p\,\ket{\vec p}\,\phi(\vec p), \end{equation}

with normalization

\begin{equation} \int d^3p\,|\phi(\vec p)|^2 = 1. \end{equation}

Connection between representations

The position and momentum bases are related by

\begin{equation} \braket{\vec r}{\vec p} = \frac{1}{(2\pi\hbar)^{3/2}} \exp\!\left(\frac{i}{\hbar}\,\vec p\cdot\vec r\right), \end{equation}

so that

\begin{align} \psi(\vec r) &= \braket{\vec r}{\psi} = \frac{1}{(2\pi\hbar)^{3/2}} \int d^3p\,e^{\frac{i}{\hbar}\vec p\cdot\vec r}\,\phi(\vec p), \label{eq:fourier-x-from-p}\\ \phi(\vec p) &= \braket{\vec p}{\psi} = \frac{1}{(2\pi\hbar)^{3/2}} \int d^3r\,e^{-\frac{i}{\hbar}\vec p\cdot\vec r}\,\psi(\vec r). \label{eq:fourier-p-from-x} \end{align}

Thus:

Matrix representation: column vector of coefficients $$c_n$$ .
Position representation: wave function $\psi(\vec r)$ .
Momentum representation: wave function $\phi(\vec p)$ .

They are simply different representations of the same abstract state $\ket{\psi}$ .

1.3Postulate II: Observables as Operators

1.3.1Statement of the Postulate

Postulate II (Observable Postulate). Every physical observable $$A$$ is represented by a linear Hermitian operator $\hat{A}$ acting on the Hilbert space $\mathcal{H}$ .

The mathematical objects we need here are:

linear operators on $\mathcal{H}$ ;
adjoint (Hermitian conjugate) of an operator;
special classes of operators: Hermitian, unitary, projection, identity;
eigenvalues and eigenkets, and the spectral decomposition.

1.3.2Linear Operators and the Adjoint

An operator $\hat{A}$ is a map that sends kets to kets:

\begin{equation} \ket{\psi} \mapsto \hat{A}\ket{\psi}. \end{equation}

It is linear if, for all complex numbers $\alpha,\beta$ and kets $\ket{\psi_1},\ket{\psi_2}$ ,

\begin{equation} \hat{A}\bigl(\alpha\ket{\psi_1}+\beta\ket{\psi_2}\bigr) = \alpha\,\hat{A}\ket{\psi_1} + \beta\,\hat{A}\ket{\psi_2}. \end{equation}

Definition - Adjoint operator

Given an operator $\hat{A}$ , its adjoint $\hat{A}^{\dagger}$ is defined by

\begin{equation} \bra{\phi}\hat{A}\ket{\psi} = \bigl(\bra{\psi}\hat{A}^{\dagger}\ket{\phi}\bigr)^{\ast} \qquad\text{for all } \ket{\phi},\ket{\psi}\in\mathcal{H}. \end{equation}

In a discrete orthonormal basis $\{\ket{n}\}$ with matrix elements $A_{mn}=\bra{m}\hat{A}\ket{n}$ , the adjoint corresponds to the conjugate transpose:

\begin{equation} (\hat{A}^{\dagger})_{mn} = A_{nm}^{\ast}. \end{equation}

1.3.3Hermitian Operators (Observables)

Definition - Hermitian operator

An operator $\hat{A}$ is Hermitian (or self–adjoint) if

\begin{equation} \hat{A}^{\dagger} = \hat{A}. \end{equation}

For Hermitian operators:

all eigenvalues are real;
eigenkets with different eigenvalues are orthogonal;
(under suitable conditions) eigenkets form a complete basis.

Theorem - Spectral decomposition (discrete case)

If $\hat{A}$ has a discrete, nondegenerate spectrum $\{a_n\}$ with orthonormal eigenkets $\{\ket{a_n}\}$ , then

\begin{equation} \hat{A} = \sum_n a_n \ket{a_n}\bra{a_n}, \qquad \sum_n \ket{a_n}\bra{a_n} = \hat{I}. \end{equation}

In the presence of a continuous spectrum $\{a\}$ we write instead

\begin{equation} \hat{A} = \sum_n a_n\ket{a_n}\bra{a_n} + \int da\,a\,\ket{a}\bra{a}, \end{equation}

with the corresponding completeness relation

\begin{equation} \sum_n \ket{a_n}\bra{a_n} + \int da\,\ket{a}\bra{a} = \hat{I}. \end{equation}

The Hamiltonian $\hat{H}$ , which governs time evolution, is one of the most important Hermitian operators.

1.3.4Identity and Projection Operators

Definition - Identity operator

The identity operator $\hat{I}$ is defined by

\begin{equation} \hat{I}\ket{\psi} = \ket{\psi} \qquad\text{for all } \ket{\psi}\in\mathcal{H}. \end{equation}

In an orthonormal basis $\{\ket{n}\}$ we have

\begin{equation} \hat{I} = \sum_n \ket{n}\bra{n}, \end{equation}

and in a continuous basis $\{\ket{\alpha}\}$ ,

\begin{equation} \hat{I} = \int d\alpha\,\ket{\alpha}\bra{\alpha}. \end{equation}

Definition - Projection operator

An operator $\hat{P}$ is a projection operator (projector) if

\begin{equation} \hat{P}^2 = \hat{P}, \qquad \hat{P}^{\dagger} = \hat{P}. \end{equation}

Projectors represent “yes/no’’ questions. For example, if $\hat{P}$ projects onto a subspace $S\subset\mathcal{H}$ , then

$\hat{P}\ket{\psi}$ is the component of $\ket{\psi}$ inside $$S$$ ;
$\hat{I}-\hat{P}$ projects onto the orthogonal complement.

For a nondegenerate eigenvalue $$a_n$$ of $\hat{A}$ , the projector onto the eigenspace spanned by $\ket{a_n}$ is

\begin{equation} \hat{P}_n = \ket{a_n}\bra{a_n}. \end{equation}

The spectral decomposition can be written in terms of projectors:

\begin{equation} \hat{A} = \sum_n a_n \hat{P}_n. \end{equation}

Projectors will appear naturally in the measurement postulate (state reduction) and in decomposing dynamics into discrete and continuum contributions (Rydberg vs. ionization channels).

1.3.5Unitary Operators

Definition - Unitary operator

An operator $\hat{U}$ is unitary if

\begin{equation} \hat{U}^{\dagger}\hat{U} = \hat{U}\hat{U}^{\dagger} = \hat{I}. \end{equation}

Properties:

Unitary operators preserve inner products:
$\begin{equation} \braket{\phi}{\psi} = \braket{\phi}{\hat{U}^{\dagger}\hat{U}\psi} = \braket{\phi'}{\psi'}, \quad \ket{\phi'} = \hat{U}\ket{\phi},\, \ket{\psi'} = \hat{U}\ket{\psi}. \end{equation}$
They preserve norms and probabilities: $\|\hat{U}\ket{\psi}\| = \|\ket{\psi}\|$ .
They represent reversible transformations: change of basis, time evolution of closed systems, symmetry transformations (rotations, translations).

Important examples:

Time-evolution operator $\hat{U}(t,t_0)$ generated by the Hamiltonian (Sec. 1.6);
Rotation operators $e^{-i\theta\hat{J}_z/\hbar}$ generated by angular momentum;
Change-of-basis operators between different orthonormal bases.

1.3.6Representations of Operators

Just like states, the same abstract operator $\hat{A}$ can be represented in different bases.

Matrix representation in a discrete basis

In an orthonormal basis $\{\ket{n}\}$ , the matrix elements of $\hat{A}$ are

\begin{equation} A_{mn} = \bra{m}\hat{A}\ket{n}. \end{equation}

If $\ket{\psi}=\sum_n c_n\ket{n}$ , then

\begin{equation} \bigl(\hat{A}\ket{\psi}\bigr)_m = \bra{m}\hat{A}\ket{\psi} = \sum_n A_{mn}c_n. \end{equation}

In matrix form,

\begin{equation} \vec{c}' = A\,\vec{c}, \end{equation}

where $\vec{c}$ and $\vec{c}'$ are column vectors of components and $$A$$ is the matrix representing $\hat{A}$ .

Kernels in continuous bases

In the position basis $\{\ket{\vec r}\}$ , the kernel of $\hat{A}$ is

\begin{equation} A(\vec r,\vec r') = \braket{\vec r}{\hat{A}\vec r'}, \end{equation}

and

\begin{equation} (\hat{A}\psi)(\vec r) = \braket{\vec r}{\hat{A}\psi} = \int d^3r'\,A(\vec r,\vec r')\,\psi(\vec r'). \end{equation}

Examples:

Potential operator $\hat{V}=V(\hat{\vec r})$ :
$\begin{equation} A(\vec r,\vec r') = V(\vec r)\,\delta(\vec r-\vec r'), \quad (\hat{V}\psi)(\vec r) = V(\vec r)\psi(\vec r). \end{equation}$
Momentum operator $\hat{p}_x=-i\hbar\,\tfrac{d}{dx}$ :
$\begin{equation} (\hat{p}_x\psi)(x) = -i\hbar\,\frac{d}{dx}\psi(x). \end{equation}$

In the momentum basis $\{\ket{\vec p}\}$ we similarly define

\begin{equation} A(\vec p,\vec p') = \braket{\vec p}{\hat{A}\vec p'}, \qquad (\hat{A}\phi)(\vec p) = \int d^3p'\,A(\vec p,\vec p')\,\phi(\vec p'). \end{equation}

Thus:

Matrix representation: $A_{mn}$ in a discrete basis;
Position representation: kernel $A(\vec r,\vec r')$ ;
Momentum representation: kernel $A(\vec p,\vec p')$ .

1.3.7Functions of Operators

We often encounter functions of operators, such as $f(\hat{H})$ or $e^{-i\hat{H}t/\hbar}$ . For a Hermitian operator with spectral decomposition

\begin{equation} \hat{A} = \sum_n a_n\ket{a_n}\bra{a_n} + \int da\,a\,\ket{a}\bra{a}, \end{equation}

we define

\begin{equation} f(\hat{A}) = \sum_n f(a_n)\ket{a_n}\bra{a_n} + \int da\,f(a)\ket{a}\bra{a}. \end{equation}

Important special case:

\begin{equation} e^{-i\hat{H}t/\hbar} = \sum_n e^{-iE_n t/\hbar}\,\ket{E_n}\bra{E_n} + \int dE\, e^{-iE t/\hbar}\,\ket{E}\bra{E}, \end{equation}

which is the time-evolution operator for a time-independent Hamiltonian.

This spectral calculus is crucial later when we work with mixed discrete (“Rydberg’’) and continuum (“ionization’’) states.

1.4Commutators and the Heisenberg Uncertainty Principle

1.4.1Commutators and Compatible Observables

Given two operators $\hat{A}$ and $\hat{B}$ , their commutator is

\begin{equation} [\hat{A},\hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}. \end{equation}

If $[\hat{A},\hat{B}]=0$ , we say that $\hat{A}$ and $\hat{B}$ are compatible: there exists a basis of common eigenkets $\{\ket{a,b}\}$ such that

\begin{equation} \hat{A}\ket{a,b} = a\ket{a,b}, \qquad \hat{B}\ket{a,b} = b\ket{a,b}. \end{equation}

Then $$A$$ and $$B$$ can be simultaneously measured with arbitrary precision.

If $[\hat{A},\hat{B}]\neq 0$ , no such complete common eigenbasis exists. The corresponding observables cannot, in general, be simultaneously sharp; this is quantified by the uncertainty principle.

Important example in one dimension:

\begin{equation} [\hat{x},\hat{p}_x] = i\hbar\,\hat{I}. \end{equation}

1.4.2Uncertainties and Standard Deviations

For a normalized state $\ket{\psi}$ and Hermitian operator $\hat{A}$ , define

\begin{align*} \langle A \rangle &= \bra{\psi}\hat{A}\ket{\psi},\\[4pt] (\Delta A)^2 &= \bra{\psi}(\hat{A}-\langle A\rangle)^2\ket{\psi} = \langle A^2 \rangle - \langle A \rangle^2. \end{align*}

$\Delta A$ is the standard deviation of measurement outcomes of $$A$$ in state $\ket{\psi}$ .

Similarly for $\hat{B}$ :

\begin{equation} (\Delta B)^2 = \bra{\psi}(\hat{B}-\langle B\rangle)^2\ket{\psi}. \end{equation}

1.4.3Heisenberg (Robertson) Uncertainty Relation

Theorem - Heisenberg–Robertson uncertainty relation

For any Hermitian operators $\hat{A}$ and $\hat{B}$ and any normalized state $\ket{\psi}$ ,

\begin{equation} \Delta A\,\Delta B \;\ge\; \frac{1}{2}\,\bigl|\bra{\psi}[\hat{A},\hat{B}]\ket{\psi}\bigr|. \end{equation}

Thus, the commutator $[\hat{A},\hat{B}]$ sets a fundamental lower bound on the product of uncertainties in $$A$$ and $$B$$ .

For position and momentum in one dimension:

\begin{equation} [\hat{x},\hat{p}_x] = i\hbar\,\hat{I} \quad\Rightarrow\quad \Delta x\,\Delta p_x \;\ge\; \frac{\hbar}{2}. \end{equation}

This is the standard Heisenberg uncertainty relation. It is a direct consequence of:

the Hilbert–space structure (inner products and norms),
non-commutativity of the corresponding operators.

Later, similar commutator structures appear in:

angular momentum algebra,
ladder operators for the harmonic oscillator,
generators of rotations and other symmetry transformations.

These will be built on top of the operator concepts introduced here.

1.5Postulate III and IV: Measurement and State Reduction

1.5.1Measurement Outcomes and Probabilities

Postulate III (Measurement Postulate). Let $\hat{A}$ be the Hermitian operator associated with an observable $$A$$ , with spectral decomposition

\begin{equation} \hat{A} = \sum_n a_n\ket{a_n}\bra{a_n} \quad\text{or}\quad \hat{A} = \int da\,a\,\ket{a}\bra{a}. \end{equation}

If the system is in state $\ket{\psi}$ , then a measurement of $$A$$ yields:

in the discrete case, outcome $$a_n$$ with probability $P(a_n)=|\braket{a_n}{\psi}|^2$ ;
in the continuous case, outcome in $$(a,a+da)$$ with probability $P(a)\,da = |\braket{a}{\psi}|^2\,da$ .

1.5.2State Reduction (Collapse)

Postulate IV (State Reduction). Immediately after an ideal measurement of $$A$$ yielding the value $$a_n$$ , the state of the system becomes the corresponding normalized eigenstate $\ket{a_n}$ (or, more generally, the projection onto the eigensubspace for a degenerate eigenvalue).

This postulate introduces intrinsically probabilistic, non-unitary evolution associated with measurement, in contrast to the unitary evolution governed by the Schrödinger equation.

1.6Postulate V: Time Evolution

1.6.1Schrödinger Equation and Unitary Evolution

Postulate V (Time Evolution). The time evolution of a closed quantum system is governed by the time-dependent Schrödinger equation

\begin{equation} i\hbar\,\frac{\partial}{\partial t}\ket{\psi(t)} = \hat{H}(t)\ket{\psi(t)}, \end{equation}

where $\hat{H}(t)$ is the Hermitian Hamiltonian of the system.

The formal solution can be written as

\begin{equation} \ket{\psi(t)} = \hat{U}(t,t_0)\ket{\psi(t_0)}, \end{equation}

where $\hat{U}(t,t_0)$ is a unitary operator satisfying

\begin{equation} i\hbar\,\frac{\partial}{\partial t}\hat{U}(t,t_0) = \hat{H}(t)\,\hat{U}(t,t_0), \qquad \hat{U}(t_0,t_0) = \hat{I}, \end{equation}

and

\begin{equation} \hat{U}^{\dagger}(t,t_0)\hat{U}(t,t_0) = \hat{I}. \end{equation}

For time-independent $\hat{H}$ ,

\begin{equation} \hat{U}(t,t_0) = \exp\!\left[-\frac{i}{\hbar}\hat{H}(t-t_0)\right]. \end{equation}

In the energy eigenbasis $\hat{H}\ket{E_n}=E_n\ket{E_n}$ ,

\begin{equation} \ket{\psi(t)} = \sum_n c_n e^{-\tfrac{i}{\hbar}E_n(t-t_0)}\ket{E_n}, \end{equation}

so each energy component acquires only a phase factor.

Expectation values become time-dependent through

\begin{equation} \langle A \rangle(t) = \bra{\psi(t)}\hat{A}\ket{\psi(t)}. \end{equation}

1.7Postulate VI: Composite Systems

Postulate VI (Composite Systems). For a composite system with subsystems $$1$$ and $$2$$ , whose state spaces are $\mathcal{H}_1$ and $\mathcal{H}_2$ , the state space of the total system is the tensor product

\begin{equation} \mathcal{H}_{\text{tot}} = \mathcal{H}_1 \otimes \mathcal{H}_2. \end{equation}

A general state of the composite system is a ket $\ket{\Psi}$ in $\mathcal{H}_{\text{tot}}$ . Only a subset of such states can be written as simple products of subsystem states.

1.7.1Tensor Product and Product Basis

Let $\{\ket{i}\}$ be an orthonormal basis for $\mathcal{H}_1$ and $\{\ket{\alpha}\}$ an orthonormal basis for $\mathcal{H}_2$ . The tensor product space

\begin{equation} \mathcal{H}_{\text{tot}} = \mathcal{H}_1 \otimes \mathcal{H}_2 \end{equation}

has a natural product basis

\begin{equation} \{\ket{i}\otimes\ket{\alpha}\}, \end{equation}

which we often abbreviate as

\begin{equation} \ket{i}\otimes\ket{\alpha} \equiv \ket{i,\alpha} \equiv \ket{i}\ket{\alpha}. \end{equation}

If $\dim\mathcal{H}_1 = d_1$ and $\dim\mathcal{H}_2 = d_2$ , then

\begin{equation} \dim\mathcal{H}_{\text{tot}} = d_1 d_2. \end{equation}

For example, two spin- $\tfrac{1}{2}$ particles each have a two-dimensional state space; the composite system has dimension $2\times 2 = 4$ .

The inner product on $\mathcal{H}_{\text{tot}}$ is defined by

\begin{equation} \braket{i,\alpha}{j,\beta} = \braket{i}{j}\,\braket{\alpha}{\beta} = \delta_{ij}\,\delta_{\alpha\beta}, \end{equation}

and extended by linearity to arbitrary superpositions.

Any state $\ket{\Psi}\in\mathcal{H}_{\text{tot}}$ can be expanded as

\begin{equation} \ket{\Psi} = \sum_{i,\alpha} C_{i\alpha}\,\ket{i,\alpha}, \end{equation}

with complex coefficients $C_{i\alpha}$ satisfying

\begin{equation} \sum_{i,\alpha}|C_{i\alpha}|^2 = 1 \end{equation}

for a normalized state.

1.7.2Product States vs. Entangled States

A state of the composite system is called a product state (separable state) if it can be written as

\begin{equation} \ket{\Psi} = \ket{\psi_1}\otimes\ket{\psi_2}, \end{equation}

where $\ket{\psi_1}\in\mathcal{H}_1$ and $\ket{\psi_2}\in\mathcal{H}_2$ are normalized subsystem kets.

Writing

\begin{equation} \ket{\psi_1} = \sum_i a_i\ket{i}, \qquad \ket{\psi_2} = \sum_{\alpha} b_{\alpha}\ket{\alpha}, \end{equation}

we obtain

\begin{equation} \ket{\Psi} = \left(\sum_i a_i\ket{i}\right)\otimes \left(\sum_{\alpha} b_{\alpha}\ket{\alpha}\right) = \sum_{i,\alpha} a_i b_{\alpha}\,\ket{i,\alpha}. \end{equation}

Thus, for a product state, the coefficient matrix $C_{i\alpha}$ factorizes as

\begin{equation} C_{i\alpha} = a_i\,b_{\alpha}. \end{equation}

A state that cannot be written in this form is called entangled. For such states the coefficients $C_{i\alpha}$ do not factorize into a product of a purely $$i$$ -dependent piece and a purely $\alpha$ -dependent piece.

Example - Two spin-

\tfrac{1}{2}

particles

Let subsystem $$1$$ and $$2$$ each have basis $\{\ket{\uparrow},\ket{\downarrow}\}$ . A product basis for the composite system is

\begin{equation} \{\ket{\uparrow\uparrow}, \ket{\uparrow\downarrow}, \ket{\downarrow\uparrow}, \ket{\downarrow\downarrow}\}, \end{equation}

with the shorthand $\ket{\uparrow\downarrow}=\ket{\uparrow}\otimes\ket{\downarrow}$ , etc.

The state
$\begin{equation} \ket{\Psi_{\text{prod}}} = \ket{\uparrow}\otimes \Bigl(\tfrac{1}{\sqrt{2}} (\ket{\uparrow}+\ket{\downarrow})\Bigr) = \tfrac{1}{\sqrt{2}}\ket{\uparrow\uparrow} + \tfrac{1}{\sqrt{2}}\ket{\uparrow\downarrow} \end{equation}$
is a product state.
The state
$\begin{equation} \ket{\Psi_{\text{ent}}} = \tfrac{1}{\sqrt{2}} \bigl(\ket{\uparrow\downarrow} + \ket{\downarrow\uparrow}\bigr) \end{equation}$
cannot be written as $\ket{\psi_1}\otimes\ket{\psi_2}$ : it is an entangled state.

In later chapters, tensor-product structures appear when we treat:

multi-electron atoms (electrons $1,2,\dots$ );
molecule + radiation field (molecular states $\otimes$ photon modes);
nuclear motion + electronic motion (Born–Oppenheimer separation).

1.7.3Operators on Composite Systems

If $\hat{A}_1$ is an operator on $\mathcal{H}_1$ and $\hat{B}_2$ an operator on $\mathcal{H}_2$ , we can define an operator on $\mathcal{H}_{\text{tot}}=\mathcal{H}_1\otimes\mathcal{H}_2$ by

\begin{equation} \hat{A}_1\otimes\hat{B}_2. \end{equation}

Its action on a product state is

\begin{equation} (\hat{A}_1\otimes\hat{B}_2) \bigl(\ket{\psi_1}\otimes\ket{\psi_2}\bigr) = (\hat{A}_1\ket{\psi_1})\otimes(\hat{B}_2\ket{\psi_2}), \end{equation}

and extended by linearity to superpositions.

Important special cases:

An observable acting only on subsystem $$1$$ :
$\begin{equation} \hat{A}_1 \otimes \hat{I}_2, \end{equation}$
where $\hat{I}_2$ is the identity on $\mathcal{H}_2$ .
An observable acting only on subsystem $$2$$ :
$\begin{equation} \hat{I}_1 \otimes \hat{B}_2. \end{equation}$

In the product basis $\{\ket{i,\alpha}\}$ ,

\begin{equation} \bigl(\hat{A}_1\otimes\hat{B}_2\bigr)_{(i,\alpha),(j,\beta)} = \bra{i,\alpha}\hat{A}_1\otimes\hat{B}_2\ket{j,\beta} = \bra{i}\hat{A}_1\ket{j}\,\bra{\alpha}\hat{B}_2\ket{\beta}. \end{equation}

At the matrix level, this corresponds to a Kronecker product of matrices.

Expectation values of composite observables are computed as

\begin{equation} \langle A_1\otimes B_2 \rangle = \bra{\Psi}\hat{A}_1\otimes\hat{B}_2\ket{\Psi}. \end{equation}

1.7.4Reduced Descriptions and Partial Trace (Minimal Use)

In many situations we are interested only in one part of a composite system (e.g. electron vs. ionized core, molecule vs. radiation field). To describe subsystem $$1$$ alone, we use a reduced state obtained by tracing over the unobserved subsystem $$2$$ .

A convenient language for this is the density operator. For a normalized pure state $\ket{\Psi}$ of the composite system,

\begin{equation} \hat{\rho}_{\text{tot}} = \ket{\Psi}\bra{\Psi}. \end{equation}

The reduced density operator of subsystem $$1$$ is defined as the partial trace over subsystem $$2$$ :

\begin{equation} \hat{\rho}_1 = \mathrm{Tr}_2\bigl(\hat{\rho}_{\text{tot}}\bigr). \end{equation}

In a product basis $\{\ket{i,\alpha}\}$ this can be written as

\begin{equation} \hat{\rho}_1 = \sum_{\alpha} \bra{\alpha}\hat{\rho}_{\text{tot}}\ket{\alpha}, \end{equation}

where the bra and ket refer only to subsystem $$2$$ .

Example - Partial trace for a two-qubit state

Let

\begin{equation} \ket{\Psi} = \tfrac{1}{\sqrt{2}} \bigl(\ket{\uparrow\downarrow} + \ket{\downarrow\uparrow}\bigr), \end{equation}

and $\hat{\rho}_{\text{tot}}=\ket{\Psi}\bra{\Psi}$ . Using the basis $\{\ket{\uparrow},\ket{\downarrow}\}$ for each subsystem, show that

\begin{equation} \hat{\rho}_1 = \frac{1}{2}\ket{\uparrow}\bra{\uparrow} + \frac{1}{2}\ket{\downarrow}\bra{\downarrow}. \end{equation}

Thus, although the total state is pure, the reduced state of subsystem $$1$$ is a statistical mixture.

For the present notes, we will use these concepts mainly in simple finite-dimensional examples (e.g. two-level systems, two spin- $\tfrac12$ particles) to:

construct product bases,
distinguish product and entangled states,
interpret measurements on one part of a composite system.

A more systematic treatment of density operators and mixed states appears in advanced courses on quantum information or open quantum systems.

minimally in the present notes.

1.8Homework

The following problems are designed to test your understanding of the material in this chapter. They range from basic exercises to more conceptual questions. Unless stated otherwise, work in Dirac notation.

Hilbert Space, Inner Products, and Bases

Let $\mathcal{H}$ be a complex vector space with inner product $\braket{\cdot}{\cdot}$ .
1. Show that for any kets $\ket{\phi}$ and $\ket{\psi}$ and any $\alpha\in\mathbb{C}$ ,
  $\begin{equation} \braket{\phi}{\alpha\psi} = \alpha\,\braket{\phi}{\psi}. \end{equation}$
2. Prove the Cauchy–Schwarz inequality:
  $\begin{equation} |\braket{\phi}{\psi}| \le \|\ket{\phi}\|\,\|\ket{\psi}\|. \end{equation}$
3. Use the Cauchy–Schwarz inequality to show the triangle inequality for the norm:
  $\begin{equation} \|\ket{\phi} + \ket{\psi}\| \le \|\ket{\phi}\| + \|\ket{\psi}\|. \end{equation}$
Let $\{\ket{n}\}$ be an orthonormal basis of a Hilbert space $\mathcal{H}$ .
1. Show that any normalized state $\ket{\psi}$ can be written as
  $\begin{equation} \ket{\psi} = \sum_n c_n \ket{n}, \qquad \sum_n |c_n|^2 = 1, \end{equation}$
  and identify $$c_n$$ in terms of $\ket{\psi}$ and $\ket{n}$ .
2. Prove the completeness relation
  $\begin{equation} \sum_n \ket{n}\bra{n} = \hat{I} \end{equation}$
  by acting on an arbitrary state $\ket{\psi}$ .
3. Suppose $\ket{\phi} = \sum_n d_n \ket{n}$ is orthogonal to $\ket{\psi}$ . Express the orthogonality condition $\braket{\phi}{\psi}=0$ in terms of the coefficients $$c_n$$ and $$d_n$$ .
Consider a normalized state $\ket{\psi}$ in $L^2(\mathbb{R})$ with wave functions

\begin{equation} \psi(x) = \braket{x}{\psi}, \qquad \phi(p) = \braket{p}{\psi}. \end{equation}
1. Write explicitly the Fourier transforms relating $\psi(x)$ and $\phi(p)$ using $\braket{x}{p}$ .
2. Show that normalization is preserved under the Fourier transform, i.e.,
  $\begin{equation} \int_{-\infty}^{\infty} | \psi(x) |^2 dx = \int_{-\infty}^{\infty} | \phi(p) |^2 dp = 1. \end{equation}$
3. For the Gaussian wave function
  $\begin{equation} \psi(x) = \frac{1}{(\pi a^2)^{1/4}} \exp\!\left(-\frac{x^2}{2a^2}\right), \end{equation}$
  compute $\phi(p)$ up to an overall phase and check normalization.
Let $\{\ket{1},\ket{2},\ket{3}\}$ be an orthonormal basis of a three-dimensional Hilbert space.
1. Write the column-vector representation of a general state
  $\begin{equation} \ket{\psi} = \alpha\ket{1} + \beta\ket{2} + \gamma\ket{3}, \end{equation}$
  in this basis.
2. Suppose another orthonormal basis $\{\ket{a},\ket{b},\ket{c}\}$ is related by
  $\begin{equation} \ket{a} = \tfrac{1}{\sqrt{2}}(\ket{1}+\ket{2}),\quad \ket{b} = \tfrac{1}{\sqrt{2}}(\ket{1}-\ket{2}),\quad \ket{c} = \ket{3}. \end{equation}$
  Find the unitary matrix $$U$$ that transforms components from the $\{\ket{1},\ket{2},\ket{3}\}$ basis to the $\{\ket{a},\ket{b},\ket{c}\}$ basis.
3. Express the components of $\ket{\psi}$ in the $\{\ket{a},\ket{b},\ket{c}\}$ basis using $$U$$ .
Operators: Hermitian, Unitary, Projectors, and Functions
Let $\hat{A}$ be a Hermitian operator and $\ket{\psi}$ a normalized state.
1. Show that the expectation value $\langle A \rangle = \bra{\psi}\hat{A}\ket{\psi}$ is always real.
2. Suppose $\ket{\psi}$ is an eigenket of $\hat{A}$ with eigenvalue $$a$$ . Show that
  $\begin{equation} \Delta A = 0, \end{equation}$
  where $(\Delta A)^2 = \langle A^2\rangle - \langle A\rangle^2$ .
3. Conversely, show that if $\Delta A = 0$ in a normalized state $\ket{\psi}$ , then $\ket{\psi}$ must be an eigenket of $\hat{A}$ .
In the basis $\{\ket{1},\ket{2},\ket{3}\}$ , an operator $\hat{A}$ is represented by the matrix

\begin{equation} A = \begin{pmatrix} 1 & i & 0 \\ -i & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}. \end{equation}
1. Check whether $\hat{A}$ is Hermitian.
2. Find its eigenvalues.
3. For each distinct eigenvalue, find a corresponding normalized eigenket in the given basis.
Let $\ket{\phi}$ be a normalized state in $\mathcal{H}$ and define the operator

\begin{equation} \hat{P} = \ket{\phi}\bra{\phi}. \end{equation}
1. Show that $\hat{P}$ is Hermitian and that $\hat{P}^2 = \hat{P}$ .
2. For an arbitrary normalized state $\ket{\psi}$ , interpret $\hat{P}\ket{\psi}$ physically, and compute $\|\hat{P}\ket{\psi}\|^2$ .
3. Show that $\hat{I}-\hat{P}$ is also a projector and interpret the subspace onto which it projects.
Let $\hat{U}$ be a unitary operator, $\hat{U}^\dagger\hat{U}=\hat{I}$ .
1. Show that $\hat{U}$ preserves inner products, i.e.,
  $\begin{equation} \braket{\phi}{\psi} = \braket{\phi'}{\psi'}, \quad \ket{\phi'}=\hat{U}\ket{\phi},\; \ket{\psi'}=\hat{U}\ket{\psi}. \end{equation}$
2. Prove that if $\ket{a}$ is an eigenket of a Hermitian operator $\hat{A}$ with eigenvalue $$a$$ , then $\hat{U}\ket{a}$ is an eigenket of $\hat{A}' = \hat{U}\hat{A}\hat{U}^\dagger$ with the same eigenvalue $$a$$ .
3. Explain how this shows that unitary transformations correspond to changes of representation that preserve the spectrum of observables.
Let $\hat{A}$ be a Hermitian operator with discrete spectrum $\hat{A}=\sum_n a_n\ket{a_n}\bra{a_n}$ .
1. Define $f(\hat{A})$ for a real-valued function $$f(x)$$ .
2. Show that if $$f$$ is real-valued on the spectrum $\{a_n\}$ , then $f(\hat{A})$ is Hermitian.
3. For the Hamiltonian $\hat{H}$ with eigenvalues $$E_n$$ , write the spectral decomposition of the time-evolution operator $\hat{U}(t) = e^{-i\hat{H}t/\hbar}$ .
Commutators and the Heisenberg Uncertainty Principle
For operators $\hat{A}$ , $\hat{B}$ and scalar $\lambda\in\mathbb{C}$ , show:
1. $[\hat{A},\hat{B}] = -[\hat{B},\hat{A}]$ .
2. $[\hat{A},\lambda\hat{B}] = \lambda[\hat{A},\hat{B}]$ .
3. $[\hat{A},\hat{B}+\hat{C}] = [\hat{A},\hat{B}]+[\hat{A},\hat{C}]$ .
4. $[\hat{A},\hat{B}\hat{C}] = [\hat{A},\hat{B}]\,\hat{C} + \hat{B}[\hat{A},\hat{C}]$ .
In one dimension, the position and momentum operators act in the position representation as

\begin{equation} (\hat{x}\psi)(x) = x\psi(x), \qquad (\hat{p}\psi)(x) = -i\hbar\frac{d}{dx}\psi(x). \end{equation}
1. Compute $(\hat{x}\hat{p}\,\psi)(x)$ and $(\hat{p}\hat{x}\,\psi)(x)$ explicitly.
2. Use the results to show that
  $\begin{equation} [\hat{x},\hat{p}] = i\hbar\,\hat{I}. \end{equation}$
Let $\hat{A}$ and $\hat{B}$ be Hermitian operators and $\ket{\psi}$ a normalized state.
1. Define $\Delta A$ and $\Delta B$ and show that $\Delta A\ge 0$ , $\Delta B\ge 0$ .
2. Consider the kets
  $\begin{equation} \ket{\alpha} = (\hat{A}-\langle A\rangle)\ket{\psi}, \quad \ket{\beta} = (\hat{B}-\langle B\rangle)\ket{\psi}. \end{equation}$
  Use the Cauchy–Schwarz inequality to derive the Robertson inequality
  $\begin{equation} \Delta A\,\Delta B \ge \frac{1}{2}\bigl|\bra{\psi}[\hat{A},\hat{B}]\ket{\psi}\bigr|. \end{equation}$
3. Apply this to $\hat{A}=\hat{x}$ and $\hat{B}=\hat{p}_x$ to obtain the standard Heisenberg uncertainty relation.
Measurement, Time Evolution, and Composite Systems
Let $\hat{A}$ be an observable with nondegenerate discrete spectrum $\hat{A}=\sum_n a_n\ket{a_n}\bra{a_n}$ and normalized state $\ket{\psi}=\sum_n c_n\ket{a_n}$ .
1. Show that the probability to obtain the result $$a_n$$ in a measurement of $$A$$ is $$P(a_n)=|c_n|^2$$ .
2. Show that immediately after obtaining the result $$a_n$$ , the state becomes $\ket{a_n}$ (up to an overall phase).
3. Suppose instead that $\hat{A}$ has a twofold degenerate eigenvalue $$a$$ with orthonormal eigenkets $\{\ket{u_1},\ket{u_2}\}$ . Write the projector onto the degenerate eigenspace and describe the post-measurement state after observing $$a$$ .
Assume a time-independent Hamiltonian $\hat{H}$ with discrete eigenvalues $$E_n$$ and eigenkets $\ket{E_n}$ :

\begin{equation} \hat{H}\ket{E_n} = E_n\ket{E_n}. \end{equation}

A normalized initial state at $$t=0$$ is

\begin{equation} \ket{\psi(0)} = \sum_n c_n \ket{E_n}. \end{equation}
1. Find $\ket{\psi(t)}$ for $$t>0$$ using the time-evolution operator.
2. Show that $$|c_n|^2$$ is time-independent and interpret this physically.
3. Write the expectation value $\langle H\rangle(t)$ and verify that it is constant in time.
Consider a composite system made of two spin- $\tfrac12$ particles. The single-particle Hilbert space is spanned by $\{\ket{\uparrow},\ket{\downarrow}\}$ .
1. Write an orthonormal product basis for the two-particle Hilbert space $\mathcal{H}_1\otimes\mathcal{H}_2$ .
2. Consider the state
  $\begin{equation} \ket{\Psi} = \tfrac{1}{\sqrt{2}} \bigl(\ket{\uparrow}\otimes\ket{\downarrow} + \ket{\downarrow}\otimes\ket{\uparrow}\bigr). \end{equation}$
  Show that $\ket{\Psi}$ cannot be written as a product $\ket{\psi_1}\otimes\ket{\psi_2}$ of single-particle states.
3. Interpret the physical meaning of such a state in terms of quantum correlations (entanglement).