## Introduction to the Series

In 2004, SPIE launched a new book series, the SPIE * Field Guides*, focused on SPIE’s core areas of Optics and Photonics. The idea of these

*is to give concise presentations of the key subtopics of a subject area or discipline, typically covering each subtopic on a single page, using the figures, equations, and brief explanations that summarize the key concepts. The aim is to give readers a handy desk or portable reference that provides basic, essential information about principles, techniques, or phenomena, including definitions and descriptions, important equations, illustrations, application examples, design considerations, and additional resources.*

**Field Guides**The series has grown to an extensive collection that covers a range of topics from broad fundamental ones to more specialized areas. Community response to the SPIE * Field Guides* has been exceptional. The concise and easy-to-use format has made these small-format, spiral-bound books essential references for students and researchers. I have been told by some readers that they take their favorite

*with them wherever they go.*

**Field Guide**We are now pleased and excited to extend the SPIE * Field Guides* into subjects in general Physics. Each

*will be written to address a core undergraduate Physics topic, or in some cases presented at a first-year graduate level. The*

**Field Guide***are not teaching texts, but rather references that condense the textbooks and course notes into the fundamental equations and explanations needed on a routine basis. We truly hope that you enjoy using the*

**Field Guides***to Physics.*

**Field Guides**We are interested in your suggestions for new * Field Guide* topics as well as what could be added to an individual volume to make these

*more useful to you. Please contact us at fieldguides@SPIE.org.*

**Field Guides**John E. Greivenkamp, **Series Editor**

James C. Wyant College of Optical Sciences

University of Arizona

## The Field Guide Series

*:*

**Field Guides**, John E. Greivenkamp**Geometrical Optics**, Rüdiger Paschotta**Lasers**, Peter E. Powers**Nonlinear Optics**, Daniel G. Smith**Physical Optics**, Marek Wartak and C.Y. Fong**Solid State Physics**, Larry C. Andrews**Special Functions for Engineers**, David W. Ball**Spectroscopy**

, David L. Andrews and David S. Bradshaw**Introduction to Photon Science and Technology**, Manijeh Razeghi, Leo Esaki, and Klaus von Klitzing, eds.**The Wonder of Nanotechnology: Quantum Optoelectronic Devices and Applications**

## Field Guide to Quantum Mechanics

This * Field Guide* is a condensed reference to the concepts, definitions, formalism, equations, and problems of quantum mechanics. Many topics covered in quantum mechanics courses are included, while numerous details and derivations are necessarily omitted. This

*is envisioned to appeal to undergraduate and graduate students engaged in quantum mechanics research or courses; to professors, as an aid in teaching and research; and to professional physicists and engineers pursuing cutting-edge applications of quantum mechanics. The mathematical formalism used here involves Dirac notation, with which the reader should be (or become) familiar to make the most of this*

**Field Guide***Nevertheless, readers who are not yet familiar with this formalism should be able to utilize various aspects of this*

**Field Guide.***, especially with extra attention directed to the basic concepts addressed in the first few sections.*

**Field Guide**I owe sincere thanks to mentors, professors, colleagues, collaborators, and friends too numerous to single out by name who have taught, motivated, and encouraged me throughout more than three decades of studying quantum physics. Since joining the University of Arizona faculty, the unwavering support and partnership of local and international colleagues and collaborators has been indispensable in learning and appreciating many of the numerous facets of this fascinating subject.

I am especially grateful to two physicists in particular who set in motion the trajectory of my eventual career while I was still in high school: the late Jeff Chalk, who first introduced me to Schrödinger’s equation and quantum mechanics; and Al Rosenberger, my first laboratory mentor, who launched my interest in lasers, optics, and experimental physics.

This * Field Guide* is dedicated to Jeff and Al, and to the students who have worked in my labs, sat through my courses, and made my career as a mentor and educator profoundly fulfilling.

Brian P. Anderson

University of Arizona

June 2019

## Table of Contents

#### Glossary of Symbols

####
**Assorted symbols**

**B**

Magnetic field vector

$\mathcal{E}$

A quantum-mechanical state space

**g _{J}
**

* g*-Factor for an angular momentum

**J**
**g _{n}
**

Degree of degeneracy for a quantum state with quantum number **n**

**i**

$(\mathrm{i})\text{\hspace{0.17em}}i\equiv \sqrt{-1}$ ; (ii) a discrete index

* J*,

**J**

Probability current (scalar and vector). For angular momentum symbols, see page 101.

**m**

(i) Mass of a particle; (ii) with subscript * J*, a magnetic quantum number associated with an angular momentum quantum number

**J**$\mathcal{P}$

Probability

${\mathcal{P}}_{a\to b}(t)$

Time-dependent transition probability from state $|a\rangle $ to state $|b\rangle $

${\mathcal{R}}_{n,l}(r)$

(i) Radial part of energy eigenfunction for a central potential; (ii) hydrogen radial wavefunction

**t**

Time

${Y}_{l}^{m}(\mathrm{\theta},\text{\hspace{0.17em}}\mathrm{\phi})$

Spherical harmonic

####
**Position and momentum coordinates**

**d ^{n}
**

**p**

Differential volume element in * n*-dimensional momentum space

**d ^{n}
**

**r**

Differential volume element in * n*-dimensional position space

**p**

(i) 3D momentum vector:
$\mathbf{p}=({p}_{x},{p}_{y},{p}_{z})$
; (ii) * n*-dimensional momentum vector

**p**

(i) Momentum variable in a 1D coordinate system; (ii) magnitude of * n*-dimensional momentum vector
$\mathbf{p}:\text{\hspace{0.17em}}\text{\hspace{0.17em}}p=\left|\mathbf{p}\right|$

**p _{x}, p_{y}, p_{z}
**

Orthogonal momentum coordinates

**r**

(i) 3D position vector:
$\mathbf{r}=(x,y,z)$
; (ii) * n*-dimensional position vector

**r**

Magnitude of * n*-dimensional position vector
$\mathbf{r}:\phantom{\rule{0ex}{0ex}}r=\left|\mathbf{r}\right|$

**x, y, z**

Orthogonal spatial coordinates

####
**Greek letters**

α

(i) Phase-space displacement coordinate; (ii) an eigenvalue of the harmonic oscillator annihilation operator $\widehat{a}$ ; (iii) fine-structure constant

γ**
_{J}
**

Gyromagnetic ratio for an angular momentum **J**

δ_{
jk
}

Kronecker delta for discrete indices * j*,

**k**δ(* x*)

Dirac delta function over a continuous variable **x**

Δ

Detuning, a difference of angular frequencies

θ

(i) An arbitrary angle; (ii) the polar angle in a spherical coordinate system

λ

(i) Perturbation scale parameter; (ii) deBroglie wavelength; (iii) general scalar quantity, such as an eigenvalue

μ

Reduced mass: for masses **m**_{1} and **m**_{2},
$\mathrm{\mu}\equiv \text{\hspace{0.17em}}\frac{{m}_{1}{m}_{2}}{{m}_{1}+{m}_{2}}$
; for
$\widehat{\mathbf{\mu}}$
, see page xvi

σ

(i) Harmonic oscillator length: ${\mathrm{\sigma}}_{j}\equiv \sqrt{\mathrm{\hbar}/(m{\mathrm{\omega}}_{j})}$ ; (ii) a Pauli spin matrix; (iii) for $\widehat{\mathbf{\sigma}}$ , see page xvii

φ

(i) An arbitrary angle; (ii) the azimuthal angle in a spherical coordinate system (to be distinguished from ϕ)

ϕ

(i) Within a ket, denotes a quantum state vector; (ii) a wavefunction of a continuous parameter, such as $\mathrm{\varphi}(x)$ (to be distinguished from φ)

ψ

(i) Within a ket, denotes a quantum state vector; (ii) a wavefunction of a continuous parameter, such as $\mathrm{\psi}(x)$

Ψ

(i) Within a ket, denotes a time-dependent quantum state vector; (ii) a time-dependent wavefunction of a continuous parameter, such as $\mathrm{\Psi}(x,t)$

ω

An angular frequency

ω**
_{L}
**

Larmor frequency: ${\mathrm{\omega}}_{L}\equiv -\mathrm{\gamma}\left|\mathbf{B}\right|$

Ω

Rabi frequency: $\mathrm{\Omega}\text{\hspace{0.17em}}\equiv \text{\hspace{0.17em}}\sqrt{{\mathrm{\Delta}}^{2}+{|{\mathrm{\Omega}}_{0}|}^{2}}$

Ω_{0}

Bare or resonant Rabi frequency

####
**Acronyms**

1D

One-dimensional

2D

Two-dimensional

3D

Three-dimensional

AM

Angular momentum

CG

Clebsch–Gordan (coefficient)

CSCO

Complete set of commuting observables

OAM

Orbital angular momentum

SI

International System of Units

SPT

Stationary perturbation theory

TAM

Total angular momentum (basis)

TDPT

Time-dependent perturbation theory

TP

Tensor product (basis)

####
**Mathematical operations and symbols**

$\Vert \mathbf{v}\Vert $

Norm of vector **v**:
$\Vert \mathbf{v}\Vert =\sqrt{{\mathbf{v}}^{\u2020}\mathbf{v}}$

$\mathcal{F}\{\cdots \}$

Fourier transform

${\mathcal{F}}^{-1}\{\cdots \}$

Inverse Fourier transform

⊗

Denotes a tensor product

**∇**

3D vector differential operator (“del”). In Cartesian coordinates: $\mathbf{\nabla}=\text{\hspace{0.17em}}\widehat{\mathbf{x}}\frac{\partial}{\partial x}+\widehat{\mathbf{y}}\frac{\partial}{\partial y}+\widehat{\mathbf{z}}\frac{\partial}{\partial z}$

**∇ ^{2}
**

Laplacian operator: **∇ ^{2} = ∇** ·

**∇**

$\text{Re}\{\mathrm{\alpha}\}$

Real part of a complex scalar α

$\text{Im}\{\mathrm{\alpha}\}$

Imaginary part of a complex scalar α

$\mathbb{1}$

Identity matrix

$\sum _{k}$

Sum over all values of discrete index **k**

####
**Quantum mechanics symbols**

$|\cdots \rangle $

Ket vector (ket)

$\langle \cdots |$

Bra vector (bra)

$\langle \mathrm{\varphi}|\mathrm{\psi}\rangle $

Scalar or inner product of the ordered pair of kets $(|\mathrm{\varphi}\rangle ,|\mathrm{\psi}\rangle )$

$\widehat{A}$

The “hat” or caret (i) denotes an operator; (ii) a directional unit vector when used over a coordinate (in bold), as in
$\widehat{\mathbf{x}}$
; (iii)
$\widehat{A}$
is used throughout this * Field Guide* as an arbitrary operator

${\widehat{A}}^{\u2020}$

The “dagger” superscript denotes the Hermitian conjugate of operator $\widehat{A}$

${\widehat{A}}^{-1}$

The superscript denotes the inverse of operator $\widehat{A}$

$\langle \cdots |\widehat{A}|\cdots \rangle $

A matrix element of operator $\widehat{A}$

**A _{jk}
**

Matrix element of operator
$\widehat{A}$
associated with matrix row * j* and column

*in a discrete representation*

**k**$\langle \widehat{A}\rangle $

An expectation value of operator $\widehat{A}$

$\mathrm{\Delta}\widehat{A}$

Uncertainty or standard deviation of operator $\widehat{A}$ : $\mathrm{\Delta}\widehat{A}\equiv \sqrt{\langle {\widehat{A}}^{2}\rangle -{\langle \widehat{A}\rangle}^{2}}$

[ $\widehat{A}$ , $\widehat{B}$ ]

Commutator of operators $\widehat{A}$ and $\widehat{B}$ : $[\widehat{A},\widehat{B}]=\widehat{A}\widehat{B}-\widehat{B}\widehat{A}$

$\langle \widehat{\mathbf{\sigma}}\rangle $

Bloch vector: $\langle \widehat{\mathbf{\sigma}}\rangle =(\langle {\widehat{\mathrm{\sigma}}}_{x}\rangle ,\langle {\widehat{\mathrm{\sigma}}}_{y}\rangle ,\langle {\widehat{\mathrm{\sigma}}}_{z}\rangle )$

$\Vert |\mathrm{\psi}\rangle \Vert $

Norm of ket $|\mathrm{\psi}\rangle \text{:}\Vert |\mathrm{\psi}\rangle \Vert =\sqrt{\langle \mathrm{\psi}|\mathrm{\psi}\rangle}$

${|\mathrm{\psi}\rangle}_{\{v\}}$

A ket $|\mathrm{\psi}\rangle $ expressed as a column vector in the representation labeled by $\{v\}$

${\langle \mathrm{\psi}|}_{\{v\}}$

A bra $\langle \mathrm{\psi}|$ expressed as a row vector in the representation labeled by $\{v\}$

${A}_{\{v\}}$

An operator $\widehat{A}$ expressed as a matrix in the representation labeled by $\{v\}$

$\tilde{\mathrm{\psi}}(\mathbf{p})$

Momentum-space wavefunction associated (by Fourier transform) with position-space wavefunction $\mathrm{\psi}(\mathbf{r})$

#### Glossary of Operators

$\widehat{H}$

Hamiltonian

$\widehat{\mathbf{P}}=\text{\hspace{0.17em}}({\widehat{P}}_{x},\text{\hspace{0.17em}}{\widehat{P}}_{y},\text{\hspace{0.17em}}{\widehat{P}}_{z})$

Vector momentum operator

${\widehat{P}}_{x},\text{\hspace{0.17em}}{\widehat{P}}_{y},\text{\hspace{0.17em}}{\widehat{P}}_{z}$

Scalar momentum operators

$\widehat{\mathbf{R}}=\text{\hspace{0.17em}}(\widehat{X},\text{\hspace{0.17em}}\widehat{Y},\text{\hspace{0.17em}}\widehat{Z})=({\widehat{R}}_{x},\text{\hspace{0.17em}}{\widehat{R}}_{y},\text{\hspace{0.17em}}{\widehat{R}}_{z})$

Vector position operator

$\widehat{X},\text{\hspace{0.17em}}\widehat{Y},\text{\hspace{0.17em}}\widehat{Z}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{or}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\widehat{R}}_{x},\text{\hspace{0.17em}}{\widehat{R}}_{y},\text{\hspace{0.17em}}{\widehat{R}}_{z}$

Scalar position operators

$\widehat{W}\text{\hspace{0.17em}}\mathrm{or}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\lambda}\widehat{W}$

A perturbation Hamiltonian

$\widehat{\mathbf{\mu}}$

Magnetic dipole moment

####
**Angular momentum (AM):** Various vector, magnitude-squared, and **û**-component AM operators

$\widehat{\mathbf{F}},{\widehat{\mathbf{F}}}^{2},{\widehat{F}}_{u}$

Total atomic AM

$\widehat{\mathbf{I}},{\widehat{\mathbf{I}}}^{2},{\widehat{I}}_{u}$

Nuclear spin AM

$\widehat{\mathbf{J}},{\widehat{\mathbf{J}}}^{2},{\widehat{J}}_{u}$

(i) Any generalized AM; (ii) sum of electron spin AM and orbital AM

$\widehat{\mathbf{L}},{\widehat{\mathbf{L}}}^{2},{\widehat{L}}_{u}$

Orbital AM

$\widehat{\mathbf{S}},{\widehat{\mathbf{S}}}^{2},{\widehat{S}}_{u}$

Spin AM

####
**Harmonic oscillator:** where
$j\in \{x,y,z\}$
and
${\mathrm{\sigma}}_{j}\equiv \sqrt{\mathrm{\hbar}/(m{\mathrm{\omega}}_{j})}$

${\widehat{a}}_{j}=\frac{1}{\sqrt{2}}(\frac{1}{{\mathrm{\sigma}}_{j}}{\widehat{R}}_{j}+i\frac{{\mathrm{\sigma}}_{j}}{\mathrm{\hbar}}{\widehat{P}}_{j})$

Lowering operator

${\widehat{a}}_{j}^{\u2020}=\frac{1}{\sqrt{2}}(\frac{1}{{\mathrm{\sigma}}_{j}}{\widehat{R}}_{j}-i\frac{{\mathrm{\sigma}}_{j}}{\mathrm{\hbar}}{\widehat{P}}_{j})$

Raising operator

${\widehat{N}}_{j}=\text{\hspace{0.17em}}{\widehat{a}}_{j}^{\u2020}{\widehat{a}}_{j}$

Number operator

####
**Projectors**

${\widehat{\mathbb{P}}}_{\mathrm{\psi}}=|\mathrm{\psi}\rangle \langle \mathrm{\psi}|$

Projector onto $|\mathrm{\psi}\rangle $

${\widehat{\mathbb{P}}}_{q}=\sum _{k=1}^{q}|{v}_{k}\rangle \langle {v}_{k}|$

Projector onto the subspace ${\mathcal{E}}_{q}$ spanned by $\{|{v}_{k}\rangle \}$ , $k\in \{1,\dots ,q\}$

####
**Unitary time evolution from ****t**
_{0} to **t**

**t**_{0}to

**t**$\widehat{\mathbb{U}}(t,{t}_{0})={e}^{-\frac{i}{\mathrm{\hbar}}(t-{t}_{0})\widehat{H}}$

For time-independent $\widehat{H}$

$\widehat{\mathbb{U}}(t,{t}_{0})={e}^{-\hspace{0.17em}\frac{i}{\mathrm{\hbar}}{\int}_{{t}_{0}}^{t}d{t}^{\prime}\widehat{H}({t}^{\prime})}$

If and only if [
$\widehat{H}(t)$
,
$\widehat{H}(t\prime )$
] = 0 for arbitrary * t* and

*′*

**t**####
**Other unitary operators**

$\widehat{\mathbb{I}}$

Identity operator

$\widehat{\mathbb{S}}({x}^{\prime})={e}^{-i{x}^{\prime}{\widehat{P}}_{x}/\mathrm{\hbar}}$

Spatial translation by * x*′ in the
$\widehat{\mathbf{x}}$
direction

$\widehat{\mathbb{T}}({p}^{\prime})={e}^{i{p}^{\prime}\widehat{X}/\mathrm{\hbar}}$

Translation by * p′* of the

*component of momentum*

**x**$\widehat{\mathbb{D}}(\mathrm{\alpha})={e}^{\mathrm{\alpha}\text{\hspace{0.17em}}{\widehat{a}}_{x}^{\u2020}-\mathrm{\alpha}*{\widehat{a}}_{x}}$

Phase-space displacement (translation) in $\widehat{\mathbf{x}}$ -direction position and momentum by $\mathrm{\alpha}\equiv \frac{1}{\sqrt{2}}(x\prime /\mathrm{\sigma}+i\mathrm{\sigma}p\prime /\mathrm{\hbar})$ , where $\mathrm{\sigma}\equiv \sqrt{\mathrm{\hbar}/(m\mathrm{\omega})},\text{\hspace{0.17em}}{\mathrm{\phi}}_{0}=x\prime p\prime /(2\mathrm{\hbar})$

$={e}^{i({p}^{\prime}\widehat{X}-{x}^{\prime}{\widehat{P}}_{x})/\mathrm{\hbar}}$

$={e}^{i{\mathrm{\phi}}_{0}}\widehat{\mathbb{T}}({p}^{\prime})\widehat{\mathbb{S}}({x}^{\prime})$

${\widehat{\mathbb{R}}}_{u}(\mathrm{\theta})={e}^{-i\mathrm{\theta}{\widehat{J}}_{u}/\mathrm{\hbar}}$

Rotation through angle θ about a unit vector $\widehat{\mathbf{u}}$

${\widehat{\mathrm{\sigma}}}_{u}$

Pauli spin operator associated with the $\widehat{\mathbf{u}}$ direction

$\widehat{\mathbf{\sigma}}=({\widehat{\mathrm{\sigma}}}_{x},{\widehat{\mathrm{\sigma}}}_{y},{\widehat{\mathrm{\sigma}}}_{z})$

Vector of Pauli spin operators

####
**Commutation relations**

$[{\widehat{R}}_{j},{\widehat{P}}_{k}]=i\mathrm{\hbar}{\mathrm{\delta}}_{jk}$

$j,k\in \text{\hspace{0.17em}}\{x,y,z\}$

$[{\widehat{a}}_{j},{\widehat{a}}_{k}^{\u2020}]={\mathrm{\delta}}_{jk}$

Vector Operators | ||

Angular momentum: $\widehat{\mathbf{J}}=({\widehat{J}}_{x},{\widehat{J}}_{y},{\widehat{J}}_{z})$ | Pauli spin operators: $\widehat{\mathbf{\sigma}}=({\widehat{\mathrm{\sigma}}}_{x},{\widehat{\mathrm{\sigma}}}_{y},{\widehat{\mathrm{\sigma}}}_{z})$ | |

$[{\widehat{J}}_{x},{\widehat{J}}_{y}]=i\mathrm{\hbar}{\widehat{J}}_{z}$ | $[{\widehat{\mathrm{\sigma}}}_{x},{\widehat{\mathrm{\sigma}}}_{y}]=2i{\widehat{\mathrm{\sigma}}}_{z}$ | |

$[{\widehat{J}}_{y},{\widehat{J}}_{z}]=i\mathrm{\hbar}{\widehat{J}}_{x}$ | $[{\widehat{\mathrm{\sigma}}}_{y},{\widehat{\mathrm{\sigma}}}_{z}]=2i{\widehat{\mathrm{\sigma}}}_{x}$ | |

$[{\widehat{J}}_{z},{\widehat{J}}_{x}]=i\mathrm{\hbar}{\widehat{J}}_{y}$ | $[{\widehat{\mathrm{\sigma}}}_{z},{\widehat{\mathrm{\sigma}}}_{x}]=2i{\widehat{\mathrm{\sigma}}}_{y}$ |