How to Read This Site · Quantum Mechanics (Phillips)

Six kinds of box, each with one job, and a colour that always means the same thing. Learning that grammar first is what makes everything after it skimmable.

This site rebuilds A. C. Phillips’ Introduction to Quantum Mechanics for a reader who has linear algebra, signal processing, numerical methods and code — and no physics background. That single fact shapes every page, so it is worth two minutes to see how the pages are built before you start chapter 1.

Nothing on this page is from the book. It is the toolkit: the conventions, the colours, the symbols, and one worked example of each kind of box you will meet.

The problem this site is trying to solve

Phillips’ book is short, careful and genuinely well written — but it was written for second-year physics undergraduates in Manchester in 2002. It assumes, without ever saying so, that you have already met classical mechanics, electromagnetism, thermodynamics and a year of laboratory physics. When it says “the classical oscillator has energy $E = \tfrac{1}{2}m\omega^2 a^2$ ” it expects you to nod.

If your background is computing rather than physics, that assumption is the whole difficulty. It is not that quantum mechanics is too hard — the mathematics is mostly eigenvalue problems and Fourier transforms, which you already own. It is that the physics vocabulary is missing, and no textbook stops to supply it.

So this site does three things the book does not:

Teaches the missing background wherever the book assumes it, in a box marked 📐, before it is used.
Translates every new idea into linear algebra and signal processing, in a box marked ⇄ — including, always, the point where the translation breaks down.
Makes every quantitative claim runnable. If a number appears in the prose, there is a short NumPy snippet on the page that prints it.

The shape of the book

Phillips lays out his own route in the preface, and it is worth having in your head before you start. Chapter 1 is the evidence — the experiments that make classical physics untenable. Chapters 2–4 build the theory those experiments force on you. Chapters 5 and 6 are that theory put to work on two concrete potentials. Chapter 7 goes back and formalises what the first six chapters used informally. Chapter 8 introduces angular momentum, and chapters 9–11 spend it on real atoms.

The route through the book, drawn from the author’s preface. Colour marks the part; the site’s sidebar uses the same five accents.

What is on a page

Every section page follows the same skeleton, so you learn where to look once:

The skeleton of every section page. An equation never appears alone: it is always followed by its symbol breakdown, a plain-language reading, and a number.

The six boxes

Each box below is a real one, doing its real job — so this section is both the legend and your first lesson.

Engineer's bridge — the master dictionary

Almost everything in this book is a statement about eigenvectors of Hermitian matrices. Here is the whole translation table; you will meet each row in context later, but it is worth reading once now.

The book says	You already know it as
a state $\Psi$	a complex vector
an observable $\hat{A}$	a Hermitian matrix
an eigenstate of $\hat{A}$	an eigenvector of that matrix
a value you can measure	an eigenvalue
“find the energy levels”	`numpy.linalg.eigh(H)`
$\psi(x) \leftrightarrow \phi(p)$	a Fourier transform pair
$\Delta x\,\Delta p \ge \hbar/2$	the time–bandwidth product
expanding in basis functions	a change of basis
normal modes of a string	DFT bins
time evolution	`expm(-1jHt/hbar)` — a norm-preserving LTI update
tunnelling through a barrier	an evanescent wave below cutoff
spin-½	a qubit
$\hat{A}\hat{B} \ne \hat{B}\hat{A}$	matrices that do not commute
degeneracy	repeated eigenvalues
selection rules	the sparsity pattern of a matrix
the Pauli exclusion principle	“this basis vector is forbidden”

Where the analogy breaks — four places, and they matter:

The space is infinite-dimensional, and the “vectors” are functions. The finite matrices you will build are a discretization you chose. Answers depend on your grid until they converge, and some states (a free particle of definite momentum) do not fit in the space at all.
Hermitian is not “symmetric”. It is conjugate transpose, $A^\dagger = A$ . With complex entries the distinction is the whole point: it is what forces eigenvalues to be real, which is what lets them be measurement outcomes.
Measurement is not a linear operation. The eigendecomposition gives you the possible outcomes and their probabilities. The collapse to one of them is not something a matrix does; it sits outside the linear-algebra picture, and that is exactly the part physicists still argue about.
The state is not really a vector — it is a ray. Multiply $\Psi$ by $e^{i\theta}$ and nothing observable changes. Overall phase is unphysical; relative phase between terms in a superposition is everything.

Physics you need first — what an “operator” is

Physicists write $\hat{A}$ and call it an operator . There is nothing mysterious in it: an operator is a rule that eats a function and returns a function.

Two you already use every day:

Differentiation. $\hat{D} = \dfrac{d}{dx}$ takes $f(x)$ to $f'(x)$ .
Multiplication by $x$ . $\hat{x}$ takes $f(x)$ to $x\,f(x)$ .

The hat is only a reminder that the thing is a rule, not a number — the difference between f and f(3).

The moment you put $x$ on a grid, both become matrices. Differentiation becomes a banded matrix of finite differences; multiplication by $x$ becomes a diagonal matrix. From there on, “operator” and “matrix” are the same word, and every question the book asks about an operator is a question numpy.linalg can answer.

What this really says — the whole subject in one paragraph

Classical physics says a particle has a position and a momentum, full stop. Quantum mechanics says that is the wrong kind of object. What a particle has is a complex-valued function spread over space — its wave function $\Psi(x,t)$ — and the only questions with answers are statistical ones: the probability of finding it here, the average momentum you would measure, the set of energies it is allowed to have.

The function evolves smoothly and deterministically — there is an equation for it, and it is linear — but what you see when you look is a random draw from a distribution the function defines. Every chapter of this book is either (a) working out what $\Psi$ is for some particular situation, or (b) working out what you are allowed to ask about it.

Put a number on it — how small is ħ?

Planck’s constant is what sets the scale of everything here:

\hbar = 1.055 \times 10^{-34}\ \mathrm{J\,s} = 6.586 \times 10^{-16}\ \mathrm{eV\,s}

That number is why you have never noticed quantum mechanics. Take a visible photon of wavelength $\lambda = 663\ \mathrm{nm}$ — orange-red light. Its energy is $E = hc/\lambda$ :

E = \frac{(6.626\times10^{-34}\ \mathrm{J\,s})(2.998\times10^{8}\ \mathrm{m\,s^{-1}})}{663\times10^{-9}\ \mathrm{m}} = 3.00\times10^{-19}\ \mathrm{J} = 1.87\ \mathrm{eV}

Three times $10^{-19}$ joules. A 100 W bulb radiates energy at a rate of about $3\times10^{20}$ such photons every second, which is why light arrives looking continuous rather than like hail.

This is also why the electronvolt exists as a unit: at $1.602\times10^{-19}$ joules apiece, it puts atomic-scale energies in the range 1–100 instead of $10^{-19}$ . Almost every energy in this book is quoted in eV, and you should develop a feel for it: visible light ≈ 2 eV, chemical bonds ≈ a few eV, hydrogen’s ground state = −13.6 eV, X-rays ≈ 10 keV.

Reproduce it in NumPy

Every quantitative claim on this site comes with the code that produced it. Run this and you get the same number the paragraph above quotes.

h    = 6.626e-34      # J s      Planck constant
c    = 2.998e8        # m/s      speed of light
eV   = 1.602e-19      # J        one electronvolt
lam  = 663e-9         # m        orange-red light

E = h * c / lam
print(f"E = {E:.3e} J = {E/eV:.2f} eV")

printsE = 2.996e-19 J = 1.87 eV

Common misconception — |ψ|² is a density, not a probability

You will read that $|\Psi(x,t)|^2$ “is the probability of finding the particle at $x$ ”. Taken literally that is false, and the error will bite you the first time you check units.

$|\Psi|^2$ is a probability density . In one dimension it carries units of $\mathrm{m^{-1}}$ , so $\Psi$ itself has units of $\mathrm{m^{-1/2}}$ — a fact worth keeping, because it is the fastest way to catch an algebra slip. A probability only appears once you integrate over an interval:

P(a \le x \le b) = \int_a^b |\Psi(x,t)|^2\,dx

The probability of finding a particle at exactly $x = 0.5$ is zero, the same way the probability that a Gaussian random variable equals exactly 0.5 is zero. If you have met probability density functions in statistics, this is the same object with the same rules — nothing quantum about it.

The colours mean something

Every diagram on this site uses one fixed colour per idea, from chapter 1 to chapter 11. Once you learn that gold means “an allowed energy” and violet-filled means “probability density”, you can read a new figure before you read its caption.

The last family is the one rule to memorise: anything dashed and slate-grey is what classical physics predicts. Everything the book is actually about is the difference between that dashed line and the solid one next to it.

The equation the whole book is about

You will not meet this properly until chapter 2, and you are not expected to understand it yet. It is here so you can see what the symbol breakdown does:

i\hbar\,\frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m}\,\frac{\partial^2 \Psi}{\partial x^2} + V(x)\,\Psi

Once you discretize $x$ onto a grid, the right-hand side becomes a matrix $\hat{H}$ acting on a vector, and the whole of chapters 3–6 turns into one eigenvalue problem you can solve in three lines. That is the subject of §0.3.

The notation reference

The book uses a consistent set of symbols and rarely restates them. This table is the complete list, and it grows as the site is built — each chapter adds its symbols the moment it introduces them, so it is never out of date. It is linked from the sidebar of every page; when you hit an unfamiliar symbol, come here.

Note especially the distinction the book relies on constantly and never belabours: capital $\Psi$ carries time, lower-case $\psi$ does not.

104 of 104

Symbol	Name	What it is	Units	Type & shape	First seen
$ℏ$	reduced Planck constant	Planck's constant divided by 2π — the quantum of action, and the conversion factor between wave language and particle language.	$J s$	scalar, 1.055 × 10⁻³⁴	toolkit
$Ψ (x, t)$	wave function	The full time-dependent state of a particle. Capital psi always carries the time dependence.	$m^{- 1/2} (1-D)$	complex-valued function of x and t; discretized, a complex vector of length N	toolkit
$ψ (x)$	eigenfunction / spatial wave function	The spatial part of a state of definite energy. Lower-case psi never carries time dependence.	$m^{- 1/2} (1-D)$	complex (often real-choosable) function of x; discretized, a vector of length N	toolkit
$\hat{H}$	Hamiltonian operator	The total-energy operator. Discretized on a grid it is a real symmetric tridiagonal matrix, and its eigenvalues are the allowed energies.	$J$	operator = N × N Hermitian matrix	toolkit
$E$	energy eigenvalue	An allowed total energy of the system — one eigenvalue of Ĥ.	$J or eV$	real scalar	toolkit
$α$	fine structure constant	e²/4πε₀ħc = 1/137.0. Dimensionless, and equal to the electron's orbital speed in hydrogen as a fraction of c — so α² measures how wrong it is to ignore relativity.	—	real scalar, ≈ 0.0073	toolkit
$a_{0}$	Bohr radius	The natural length of atomic physics, 4πε₀ħ²/mₑe². Atomic distances are quoted as multiples of it throughout chapters 9 and 11.	$m$	real scalar, 5.292 × 10⁻¹¹	toolkit
$E_{R}$	Rydberg energy	The ionization energy of hydrogen, ħ²/2mₑa₀² = ½α²mₑc² = 13.61 eV. The natural energy unit of atomic physics.	$eV$	real scalar, 13.61	toolkit
$h$	Planck's constant	The quantum of action, 6.626 × 10⁻³⁴ J s. Links wave language to particle language: E = hν and p = h/λ. If it were zero, matter would be purely classical.	$J s$	real scalar	ch. 1
$ν$	frequency	Cycles per second of a wave, in E = hν. Related to angular frequency by ω = 2πν.	$Hz$	real scalar	ch. 1
$λ$	wavelength	For a photon, the wavelength of the radiation; for matter, the de Broglie wavelength h/p. The same symbol for both is the point of the chapter.	$m$	real scalar	ch. 1
$p$	momentum	Classically mv. In quantum mechanics it is the Fourier conjugate of position: p = ħk where k = 2π/λ is the spatial frequency.	$kg m s^{- 1}$	real scalar (1-D); a vector in 3-D	ch. 1
$Δ x, Δ p$	uncertainties	The spread in position and in momentum — standard deviations, not measurement errors. They obey Δx Δp ≥ ħ/2. NOTE: lower-case δx is the numerical grid spacing (ch00), a different thing.	$m, kg m s^{- 1}$	real scalars	ch. 1
$n$	principal quantum number	The integer labelling a hydrogen energy level, Eₙ = −13.6/n² eV, n = 1, 2, 3, …	—	positive integer	ch. 1
$Z$	atomic number	The number of protons in the nucleus, and of electrons in a neutral atom. It alone fixes the chemical identity of the element.	—	positive integer	ch. 1
$k$	wave number	Spatial frequency, k = 2π/λ, in radians per metre. Momentum is ħk — this is where the wave and particle descriptions meet.	$m^{- 1}$	real scalar (a vector k in 3-D)	ch. 2
$ω$	angular frequency	Radians per second. TWO related uses: for a WAVE (ch02) ω = 2π/τ and energy is ħω, the time-domain counterpart of p = ħk; for an OSCILLATOR (ch06) ω = √(k/m) is the classical frequency of the spring, and the quantum levels come out spaced by exactly ħω. The second is the more remarkable — a classical frequency setting a quantum level spacing is what Planck assumed in 1900.	$s^{- 1}$	real scalar	ch. 2
$ω (k)$	dispersion relation	How frequency depends on wave number. Non-dispersive if ω = ck; for a free particle ω = ħk²/2m, which is what the Schrödinger equation encodes.	$s^{- 1}$	real function of k	ch. 2
$v_{group} = \frac{d ω}{d k}$	group velocity	The speed of a wave packet's envelope. Setting it equal to the particle's velocity is the requirement that fixes the Schrödinger equation.	$m s^{- 1}$	real scalar	ch. 2
$\nabla^{2}$	Laplacian	The sum of second derivatives, ∂²/∂x² + ∂²/∂y² + ∂²/∂z². Discretized, it is the 3-D version of the [1, −2, 1] stencil.	$m^{- 2}$	operator = a sparse matrix once discretized	ch. 2
$⟨ x ⟩$	expectation value	The average over infinitely many repeated measurements. Angle brackets are used for every observable from ch03 on — and the value need not be a possible outcome.	$t h oseo f t h eo b ser v ab l e$	real scalar	ch. 3
$ρ (x)$	probability density	Probability per unit length. For position it is \|Ψ\|²; only its integral over an interval is a probability.	$m^{- 1} (1-D)$	real non-negative function of x	ch. 3
$\tilde{Ψ} (p, t)$	momentum-space wave function	The Fourier transform of Ψ(x,t). It is the probability amplitude for momentum exactly as Ψ is for position — the tilde is the book’s marker for “transformed”.	$(kg m s^{- 1})^{- 1/2}$	complex function of p and t	ch. 3
$\overset{x}{^}, \overset{p}{^}$	position and momentum operators	x̂ = x (just multiply) and p̂ = −iħ∂/∂x (differentiate and scale). Sandwich them between Ψ* and Ψ and integrate to get an expectation value.	$m, kg m s^{- 1}$	operators = matrices once discretized	ch. 3
$j (x, t)$	probability current	The flow of probability, obeying ∂ρ/∂t = −∂j/∂x. Same bookkeeping as charge conservation in electromagnetism.	$s^{- 1} (1-D)$	real function of x and t	ch. 3
$δ x$	grid spacing	The distance between neighbouring sample points when x is discretized. Written lower-case δx: h is Planck’s constant everywhere in this book, and capital Δx is reserved for an uncertainty (ch01). The code calls it dx. Error is O((δx)²).	$m$	real scalar	toolkit
$t = \frac{ℏ ^{2}}{2 m ( δ x ) ^{2}}$	hopping energy	The whole kinetic term reduced to one number: how strongly neighbouring grid points are coupled. It is the off-diagonal of the discretized Hamiltonian.	$J$	real scalar	toolkit
$V (x)$	potential energy	The potential-energy landscape the particle moves in. Discretized, it is just the diagonal of Ĥ.	$J or eV$	real function of x; discretized, a real vector of length N	toolkit
$E_{n}$	the n-th energy eigenvalue	One rung of the spectrum. The subscript is a LABEL, not a factor — Eₙ is the n-th allowed energy, and which integers n runs over depends on the potential (from 1 for a box, from 0 for an oscillator).	$J or eV$	real scalar, one per level; as data, a sorted array	ch. 4
$ψ_{n} (r)$	the n-th energy eigenfunction	The spatial part of the n-th stationary state — lower-case ψ, so no time dependence. The full wave function is Ψ = ψₙ e^(−iEₙt/ħ), and the site keeps the upper/lower case distinction as strictly as the book does.	$m^{- 1/2} (1-D)$	function of position; discretized, one column of the eigenvector matrix	ch. 4
$T (t)$	the separated time factor	What is left of Ψ when the space part is divided out: T(t) = Ae^(−iEt/ħ). It has modulus 1, which is exactly why a state of definite energy has no observable time dependence. CAUTION: not a temperature, and not the transmission probability of ch05.	—	complex function of t, of unit modulus	ch. 4
$c_{n}$	energy probability amplitude	The coefficient of ψₙ in a superposition; \|cₙ\|² is the probability that a measurement of energy returns Eₙ. CAUTION: collides with c the wave speed on a string (§4.2) and with c the third box dimension (§4.4).	—	complex scalar, one per level; as data, the state vector in the energy basis	ch. 4
$δ t$	time scale for observable change	How long a non-stationary state takes to change appreciably, in δt ΔE ≈ ħ. The book is insistent that this is NOT an uncertainty in time — time is a parameter, not an observable, and has no operator. Distinct from Δx and Δp, which are genuine spreads.	$s$	real scalar	ch. 4
$τ (lifetime)$	mean lifetime of an excited state	The average time before an excited state decays. Combined with δt ΔE ≈ ħ it gives the natural line width of a spectral line. CAUTION: ch05 uses τ for a scattering TIME DELAY, an unrelated quantity.	$s$	real scalar	ch. 4
$\hat{A}, \hat{B}$	a general observable and its operator	Placeholders for any observable at all. The point of ch07 is that its arguments never need to know WHICH observable — linearity, Hermiticity and completeness are all that is used.	$w ha t e v er A m e a s u r es$	operator = Hermitian matrix; its eigenvalues are the possible measurement outcomes	ch. 7
$[\hat{A}, \hat{B}]$	commutator	ÂB̂ − B̂Â. Zero exactly when the two observables are compatible, i.e. can be sharp at once. As matrices, zero exactly when they share an eigenbasis — so this one bracket decides what can be known together.	$p r o d u c t o f t h e tw oo b ser v ab l e s^{'} u ni t s$	operator; usually a multiple of the identity or of a third operator	ch. 7
$δ (x - x^{'})$	Dirac delta function	Zero everywhere but x′, infinite there, unit area — defined by ∫f(x)δ(x−x′)dx = f(x′). It is the position eigenfunction. CAUTION: the THIRD δ in this book, after ch05's phase shift and ch06's Kronecker δₘₙ, and distinct again from ch04's δt and ch00's grid spacing δx.	$m^{- 1}$	not a function at all but a distribution — meaningful only inside an integral	ch. 7
$ψ_{x^{'}} (x), ψ_{p^{'}} (x)$	position and momentum eigenfunctions	ψ_{x′} = δ(x−x′) and ψ_{p′} = e^(ip′x/ħ)/√(2πħ). The PRIMED letter labels the eigenvalue, the unprimed one is the variable. Both are orthogonal but normalize to a delta rather than to 1, so neither is a physically acceptable state alone.	$m^{- 1/2}$	a distribution and a plane wave — neither square-integrable	ch. 7
$\hat{T}, \hat{V}$	kinetic and potential energy operators	The two halves of Ĥ = T̂ + V̂: T̂ = −(ħ²/2m)∇² is a second derivative, V̂ = V(r) is a multiplication. On a grid the first is tridiagonal and the second is diagonal, which is the whole of the ch00/0.3 recipe.	$J$	operators; tridiagonal and diagonal respectively	ch. 7
$Δ x, Δ p$	deviation operators	The operators x̂ − ⟨x⟩ and p̂ − ⟨p⟩ — NOT the numbers Δx and Δp, which are their root-mean-square values. The hat is doing real work: §7.4 derives the uncertainty principle by feeding these two operators, which obey the same commutator as x̂ and p̂ themselves ([Δ̂x, Δ̂p] = iħ, Eq. 7.19), into the Schwarz inequality. Subtracting a constant shifts an operator without changing how it fails to commute.	$m, kg m s^{- 1}$	operators; each is the original minus a scalar multiple of the identity	ch. 7
$δ_{ϵ} (x - x^{'})$	the box that becomes a delta	Fig. 7.1: a rectangle of width 2ε and height 1/2ε centred on x′, so its area is 1 for every ε. The Dirac delta is what it tends to as ε → 0 — which is a definition by limit, not by pointwise values, and is why δ only ever means something inside an integral.	$m^{- 1}$	a family of ordinary functions indexed by ε; the limit is not a function	ch. 7
$c (x^{'}, t), c (p^{'}, t)$	position and momentum probability amplitudes	The coefficients when Ψ is expanded in position or momentum eigenfunctions. \|c(x′,t)\|² is the position probability density and \|c(p′,t)\|² the momentum one — so §7.2 derives what ch03 Eq. (3.19) had to assume: the momentum amplitude IS the Fourier transform of Ψ.	$m^{- 1/2}, (kg m s^{- 1})^{- 1/2}$	complex functions of a continuous eigenvalue and t	ch. 7
$V_{0}$	well depth	How deep an attractive square well is. A positive number, but the potential inside the well is −V₀ — the opposite sign convention to V_B, deliberately.	$J or eV$	real scalar	ch. 5
$V_{B}$	barrier height	How high a repulsive square barrier stands above the outside potential. Also positive — and here the potential really is +V_B.	$J or eV$	real scalar	ch. 5
$ϵ$	binding energy	The energy needed to free a bound particle, ε = −E > 0. Positive while E is negative: deep binding means large ε and very negative E. CAUTION: not ε₀, the permittivity of free space, which appears in the same chapter inside the Coulomb barrier.	$J or eV$	real scalar, positive for a bound state	ch. 5
$w$	well-depth parameter	Defined by V₀ = ħ²w²/2m. It alone decides how many bound states a well supports: one appears each time w passes an odd multiple of π/2a.	$m^{- 1}$	real scalar	ch. 5
$E_{G}$	Gamow energy	E_G = (e²/4πε₀ħc)² 2π²μc², equal to 493 keV for two protons. It sets Coulomb-barrier tunnelling through T ≈ exp(−√(E_G/E)), and hence how slowly a star burns.	$J or keV$	real scalar, fixed by the particle pair	ch. 5
$k_{0}$	wave number inside a well	ħ²k₀²/2m = E + V₀ — the wave number in the region where the particle has fallen into an attractive well, so it is always larger than the outside k.	$m^{- 1}$	real scalar	ch. 5
$k_{B}$	wave number inside a barrier	The wave number in the barrier region when E > V_B, from E = ħ²k_B²/2m + V_B. Smaller than the outside k, because the barrier slows the particle — and it goes imaginary below the barrier top, where it becomes iβ.	$m^{- 1}$	real scalar when E > V_B; the analytic continuation to iβ below it is what unifies the two cases	ch. 5
$α, β$	decay constants in a forbidden region	The inverse decay length of ψ where E < V: α outside a well (E_B = ħ²α²/2m), β inside a barrier (V_B − E = ħ²β²/2m). Same physical role, two letters, because the book keeps the two situations apart. CAUTION: α is also the fine-structure constant in ch01 — the same glyph, an unrelated quantity.	$m^{- 1}$	real scalar; 1/α and 1/β are lengths, and they are the numbers to estimate first	ch. 5
$δ$	phase shift	The one thing a potential can change about an unbound wave of fixed energy: ψ = D sin(kx + δ) outside the well. What a scattering experiment actually measures.	$radians$	real scalar, defined only modulo π	ch. 5
$τ = 2ℏ d δ / d E$	time delay	The observable meaning of a phase shift: how much later (or earlier) a wave packet leaves the scatterer. Negative for an attractive well, because the particle speeds up crossing it. Exactly the group delay dφ/dω of a filter.	$s$	real scalar, signed	ch. 5
$A_{I}, A_{R}, A_{T}$	incident, reflected and transmitted amplitudes	The complex coefficients of the three travelling waves at a barrier. Only their ratios are ever observable, which is what lets scattering dodge the normalization problem entirely.	—	complex scalars	ch. 5
$R, T$	reflection and transmission probabilities	R = \|A_R\|²/\|A_I\|² and T = \|A_T\|²/\|A_I\|², with R + T = 1. Probabilities that a whole particle bounces or crosses — never fractions of a particle.	—	real scalars in [0, 1]	ch. 5
$r_{N}, r_{C}$	nuclear force range; classical closest approach	r_N ≈ 2 fm is where the strong attraction takes over; r_C = e²/4πε₀E is where a classical proton of energy E would stop. Between them lies the barrier that has to be tunnelled — at solar energies, 720 nuclear radii of it.	$m (fm)$	real scalars; r_C depends on E, r_N does not	ch. 5
$k (elastic constant)$	elastic / spring constant	The k of F = −kx, so V = ½kx² and ω = √(k/m). CAUTION: ch02-ch05 use k for a WAVE NUMBER, and the two appear in the same sentences in §6.4. The dimensions are the reliable tell: N m⁻¹ here, m⁻¹ there.	$N m^{- 1}$	real scalar	ch. 6
$a (oscillator length)$	oscillator length	a = √(ħ/mω), the only length a harmonic oscillator has. Every eigenfunction is a Gaussian of this width times a polynomial, and ⟨x²⟩ = (n+½)a². THIRD meaning of a in this book: ch04's box width and ch05's well width are the others.	$m$	real scalar	ch. 6
$H_{n}$	Hermite polynomial	The degree-n polynomial in ψₙ ∝ Hₙ(x/a)e^(−x²/2a²). Its n roots are the n nodes of the eigenfunction, which is why node-counting works as a check on numerics.	—	polynomial of degree n; as data, n+1 coefficients	ch. 6
$q, ϵ$	dimensionless position and energy	q = x/√(ħ/mω) and ε = E/ħω — the substitution that strips every constant out of the eigenvalue equation in §6.6, leaving −d²ψ/dq² + q²ψ = 2εψ, with εₙ = n + ½. CAUTION: the book prints this as a lowercase epsilon, the THIRD use of that glyph after ch05's binding energy and ε₀ the permittivity of free space.	—	dimensionless real scalars	ch. 6
$[q \mp \frac{d}{d q}]$	raising and lowering operators	Step one rung up (minus sign) or down (plus sign) the energy ladder. The lowering operator gives zero on the ground state, which is what bounds the spectrum from below. These are a† and a — the first operator algebra in the book, and the rehearsal for ch08's angular momentum.	—	operators; in the number basis, the off-diagonal matrices with √(n+1) and √n on their first super/sub-diagonal	ch. 6
$ω_{m, n}$	beat frequency between two levels	\|Eₘ − Eₙ\|/ħ — the rate at which the interference between two energy eigenfunctions oscillates. For the harmonic oscillator every one of these is an integer multiple of ω, which is why every superposition is exactly periodic; for the box they are incommensurate, which is why it has revivals instead.	$rad s^{- 1}$	real scalar, one per PAIR of levels	ch. 6
$A, α$	classical amplitude and phase	The two free constants of classical simple harmonic motion, x = A cos(ωt + α), fixed by the initial conditions. Both are absent from quantum mechanics: energy is quantized rather than set by A, and a state of definite energy has no phase to observe. CAUTION: α is the FOURTH meaning of that glyph in this book.	$m, radians$	real scalars; properties of the motion, not of the oscillator	ch. 6
$x_{m, n}$	position matrix element	∫ψ*ₘ x ψₙ dx. Zero unless \|m−n\| = 1, which is why ⟨x⟩ oscillates at ω alone however many levels are mixed in, and why vibrational transitions are between ADJACENT levels. This is a selection rule, and as a matrix it is a sparsity pattern.	$m$	a matrix indexed by two quantum numbers; tridiagonal, in fact	ch. 6
$\overset{n}{ˉ}$	mean excitation	The mean of the Poisson distribution \|cₙ\|² = n̄ⁿe^(−n̄)/n! defining a quasi-classical state. Large n̄ means ΔE/⟨E⟩ → 1/√n̄, so the motion approaches classical simple harmonic motion.	—	dimensionless real scalar	ch. 6
$δ_{m, n}$	Kronecker delta	1 if m = n and 0 otherwise. States the orthonormality of an eigenfunction set in one symbol: ∫ψ*ₘψₙ dx = δₘₙ, i.e. the basis is the identity matrix in disguise.	—	the identity matrix, written elementwise	ch. 6
$n_{x}, n_{y}, n_{z}$	three-dimensional quantum numbers	One per Cartesian direction, for any separable 3-D problem. For the BOX (ch04) they start at 1 and E ∝ nₓ²+n_y²+n_z², giving irregular degeneracies; for the OSCILLATOR (ch06) they start at 0 and E = (nₓ+n_y+n_z+3/2)ħω, so only the sum enters and the degeneracies are the triangular numbers 1, 3, 6, 10. Same glyphs, two conventions — check which system you are in.	—	integers; from 1 for the box, from 0 for the oscillator	ch. 4
$μ$	reduced mass	The effective mass of a two-body relative motion, μ = m₁m₂/(m₁+m₂) — so μ = m_p/2 for two protons. It turns a two-particle problem into a one-particle problem in the separation r.	$kg$	real scalar	ch. 5
$l, m_{l}$	orbital angular momentum quantum numbers	l fixes the magnitude, L = √(l(l+1))ħ; m_l fixes the z component, L_z = m_lħ, and takes the 2l+1 values −l…+l. Both are integers — ch08 problem 5(c) shows m_l must be, because a wave function has to be single-valued after going round by 2π.	—	integers; l ≥ 0 and \|m_l\| ≤ l	ch. 8
$s, m_{s}$	spin quantum numbers	The same structure as l and m_l, for intrinsic angular momentum: S = √(s(s+1))ħ and S_z = m_sħ. The difference that matters is that s MAY BE HALF-INTEGER — the electron has s = ½ and therefore only two states.	—	integer or half-integer; \|m_s\| ≤ s	ch. 8
$j, m_{j}$	total angular momentum quantum numbers	For orbital and spin combined, and also for a general angular momentum. Eq. (8.6) gives j = l+s, l+s−1, … \|l−s\|, and the state count survives: (2l+1)(2s+1) = Σ(2j+1), because coupling is a change of basis.	—	integer or half-integer; \|m_j\| ≤ j	ch. 8
$\hat{L}, \hat{L}_{x}, \hat{L}_{y}, \hat{L}_{z}$	orbital angular momentum operators	L̂ = r̂ × p̂ = −iħ r × ∇, a VECTOR of three separate operators. No single operator has "the direction of L" as its eigenvalue, which is precisely why the direction has no value. In spherical polars L̂_z = −iħ ∂/∂φ, the generator of rotations about z.	$J s$	three first-order differential operators	ch. 8
$\hat{L}^{2}$	squared magnitude of orbital angular momentum	L̂ₓ² + L̂ᵧ² + L̂_z² — a single SCALAR operator with eigenvalues l(l+1)ħ². Unlike the components, it commutes with each of them (ch07 problem 3), which is what lets a state carry both l and m_l as labels.	$J^{2} s^{2}$	second-order differential operator	ch. 8
$\hat{S}$	spin operator	Acts on a quantum state that includes the particle's spin, and is usually represented by MATRICES rather than derivatives — because spin has no spatial coordinate to differentiate. This book mentions it in a footnote and never writes the matrices down.	$J s$	three matrices (2×2 for spin-½)	ch. 8
$μ, μ_{z}$	magnetic moment and its z component	μ = (q/2m)L classically, from I·A for a current loop. In quantum physics it inherits the fuzzy-vector structure, so only μ_z has a definite value. NOTE: μ alone is the reduced mass elsewhere in this book — see the ch05 entry.	$J T^{- 1}$	vector; only one component measurable at a time	ch. 8
$μ_{B}, μ_{N}$	Bohr and nuclear magneton	μ_B = eħ/2m_e = 9.274×10⁻²⁴ J T⁻¹ and μ_N = eħ/2m_p = 5.05×10⁻²⁷ J T⁻¹. Same formula, different mass — so their ratio is exactly m_p/m_e = 1836, which is why nuclei are such weak magnets and why MRI needs tesla-scale fields.	$J T^{- 1}$	physical constants	ch. 8
$g$	Landé g-factor	g = 1 + [j(j+1) − l(l+1) + s(s+1)]/2j(j+1). Exactly 2 when the magnetism is all spin, exactly 1 when all orbital. Because it depends on the state, a Zeeman pattern identifies WHICH state you are looking at, not merely how strong the field is.	—	dimensionless, state-dependent	ch. 8
$Y_{l, m_{l}} (θ, ϕ)$	spherical harmonic	The simultaneous eigenfunctions of L̂² and L̂_z — the complete set of angular shapes on a sphere, orthonormal under ∫…dΩ. \|Y\|² has NO φ dependence, so definite L_z means a wave function perfectly smeared around z.	—	complex function on the sphere, indexed by two integers	ch. 8
$Z_{m_{l}} (ϕ), F_{l, m_{l}} (θ)$	azimuthal and polar factors of a spherical harmonic	Y = F(θ)e^{im_lφ}, and Z_{m_l} = e^{im_lφ}/√(2π) is the normalized azimuthal basis — the Fourier basis on a circle. CAUTION: most other texts call these Φ_m and Θ_lm; this book uses Z and F.	—	complex function of one angle each	ch. 8
$d Ω$	solid angle element	sin θ dθ dφ, integrated over θ from 0 to π and φ from 0 to 2π, giving 4π for the whole sphere. The sin θ is the reason bands near the equator carry more area — and more probability — than bands near the poles.	$sr$	measure on the sphere	ch. 8
$(r, θ, ϕ)$	spherical polar coordinates	x = r sinθ cosφ, y = r sinθ sinφ, z = r cosθ. PHYSICS CONVENTION: θ is the polar angle from the z axis and φ the azimuthal angle — the opposite of most engineering and mathematics texts, and the one place prior familiarity actively misleads.	$m, rad, rad$	coordinates; θ ∈ [0,π], φ ∈ [0,2π)	ch. 8
$V_{e} (r)$	effective potential	l(l+1)ħ²/2mr² + V(r) — the true potential plus the centrifugal barrier. Radial motion behaves exactly like a one-dimensional problem in this potential, classically (Eq. 9.3) and quantum mechanically (Eq. 9.10) alike, which is what lets ch00's grid/eigh recipe solve the hydrogen atom.	$J$	real function of r, one per value of l	ch. 9
$u (r) = r R (r)$	radial function	The function that actually obeys a 1-D Schrödinger equation. NOT the radial part R itself: the factor of r removes the first-derivative term, and it is u — not R — that is normalized by ∫\|u\|²dr = 1 and that carries the boundary condition u(0) = 0.	$m^{- 1/2}$	real function on the half-line r > 0	ch. 9
$n_{r}$	radial quantum number	The number of nodes of u(r) strictly between r = 0 and infinity. More radial nodes means more curvature, hence more radial kinetic energy, hence higher energy. Related to the principal quantum number by n = n_r + l + 1 — and the book switches between the two labellings without warning.	—	integer ≥ 0	ch. 9
$p_{r}, p_{t}$	radial and transverse momentum	p_r = m dr/dt and p_t = L/r for planar motion in a central potential; together they carry the whole kinetic energy, p²/2m = p_r²/2m + L²/2mr². NOTE: the book prints p_t = L/m on p. 180, which has the wrong dimensions — Eq. (9.2) on the next line confirms it must be L/r.	$kg m s^{- 1}$	real scalars	ch. 9
$p_{n_{r}, l} (r)$	radial polynomial	The polynomial factor in Eq. (9.24), with exactly n_r zeros — it supplies the radial nodes. These are the associated Laguerre polynomials, though the book never names them; §9.7 derives them as a power series that must terminate, and it is that termination which quantizes the energy.	—	polynomial of degree n_r in r	ch. 9
$\hat{d} = - e \hat{r}$	electric dipole operator	The operator whose matrix element between two states governs radiative transitions, via the interaction Ĥ_I = −d̂·E (Eq. 9.26). What matters for the selection rule is only that it is ODD under r → −r: its magnitude sets the rate, its parity sets whether the rate is zero at all.	$C m$	vector operator	ch. 9
$Δ l$	change in orbital quantum number	l_f − l_i during a transition. Electric dipole transitions require Δl = ±1 (Eq. 9.28) — parity alone forbids every EVEN change, and the spherical-harmonic algebra narrows the survivors to exactly ±1. A transition violating it is forbidden, not merely improbable.	—	integer, restricted to ±1 for electric dipole radiation	ch. 9
$ρ = r / a_{0}, γ^{2} = - E / E_{R}$	the dimensionless variables of §9.7	Scaling used to strip every physical constant out of the radial equation (Eq. 9.36). The eigenvalue condition then reads simply γ = 1/(n_r + l + 1), so the energy quantization appears as a statement about a pure number rather than about e, ε₀ and m.	—	dimensionless; γ = 1/n at the eigenvalues	ch. 9
$n l_{j}$	spectroscopic term symbol	A one-electron state written as principal quantum number, orbital letter and total-angular-momentum subscript — 2p_{3/2} means n = 2, l = 1, j = 3/2. Needed once spin–orbit coupling makes j rather than m_l and m_s the good quantum numbers. ch11 extends the notation to many electrons with capital letters.	—	a label, not a quantity	ch. 9
$Ψ (r_{p}, r_{q}, t)$	two-particle wave function	One function of BOTH positions, not two functions of one position — so it lives on a six-dimensional configuration space, and for two particles in one dimension on a plane. Eq. (10.1) makes \|Ψ\|² the joint probability density. The labels p and q are scaffolding: footnote 1 concedes that we name the particles and then forbid the names from mattering, and that quantum field theory avoids the contrivance entirely.	$m^{- 3} p er p a r t i c l e, so m^{- 6} f or t h e p ai r in t h r ee d im e n s i o n s$	complex; symmetric or antisymmetric under swapping its two position arguments	ch. 10
$Ψ^{(S)}, Ψ^{(A)}, Ψ^{(D)}$	symmetric, antisymmetric and distinguishable two-particle states	The parenthesised superscript records behaviour under exchange: (S) unchanged, (A) sign-reversed, (D) neither — available only to particles you can tell apart. Eq. (10.4) proves (S) and (A) are the only options for identical particles. BEWARE: a bare S elsewhere in this chapter is the combined spin quantum number, and in ch08 it was the magnitude of a spin angular momentum — three meanings, one letter. The superscript is always parenthesised and always sits on a Ψ or a χ.	$s am e a s Ψ$	a symmetry label, not a quantity	ch. 10
$x = x_{p} - x_{q}, X = (x_{p} + x_{q}) /2$	separation and centre of mass	§10.2's change of variables, and the reason Fig. 10.1 exists. In these coordinates the two-particle states of Eqs. (10.12)–(10.13) differ by a single factor — the symmetric one carries X, the antisymmetric one carries x — which is why only the antisymmetric state is forced to vanish at zero separation. Integrating \|Ψ\|² over X leaves the separation distributions of Eqs. (10.14)–(10.16), which are normalized over the MAGNITUDE \|x\|, from 0 to infinity.	$m$	real; x may be either sign, but the distributions are written in \|x\|	ch. 10
$χ_{s, m_{s}} (p)$	spin eigenvector of particle p	The spin analogue of a spherical harmonic: a particle of spin s has 2s+1 of them, exactly as orbital angular momentum l has 2l+1 harmonics Y_{l,m_l}. Footnote 2 gives the spin-half pair as the two-component columns (1,0) and (0,1) and then declines to use them — but with them the singlet/triplet split is just the decomposition of a 4-dimensional tensor product, which is the site's Bridge at §10.3.	$d im e n s i o n l ess$	a (2s+1)-component column vector, not a function of position	ch. 10
$Φ (p) = ψ (r_{p}) χ (p)$	complete single-particle state, space × spin	Capital phi, distinct from the spatial wave function psi. Writing a state this way is what lets exchange symmetry be shared between the two factors: the symmetry of the product is the PRODUCT of the two symmetries, which is the whole content of Eqs. (10.22)–(10.23). Valid only when the spatial and spin properties are independent.	$s am e a s p s i$	complex; the product factorizes only when space and spin are uncoupled	ch. 10
$S, M_{S}$	combined spin quantum numbers of two or more particles	Magnitude sqrt(S(S+1)) hbar and z component M_S hbar for the TOTAL spin. Two spin-halves give S = 1 (three symmetric states, the triplet) or S = 0 (one antisymmetric state, the singlet). Footnote 3 concedes that capital S already meant the magnitude of a spin angular momentum in ch08, and the superscript (S) on a wave function means 'symmetric' — three meanings for one letter, in one chapter.	$d im e n s i o n l ess q u an t u mn u mb er s$	S is a non-negative integer or half-integer; M_S runs from -S to +S	ch. 10
$z (r), Z^{*}$	effective charge at radius r; effective point charge	Eq. (11.5) gives z(r) = (Z-1)e^{-r/a} + 1, which equals Z at the nucleus and 1 far away — the whole central field approximation compressed into one function. Z* is the constant point charge that would produce the binding an electron actually feels: for carbon it is 4.49 in 1s but only 1.24 in 2p, because higher l penetrates the screening cloud less.	$m u l t i pl eso f t h ee l e m e n t a r y c ha r g ee$	real; z(r) decreases monotonically from Z to 1	ch. 11
$V_{A}, V_{B}, V_{C}$	the screened potential and its two limits	The three curves of Fig. 11.1. V_C is the bare nucleus seen at small r (Eq. 11.4); V_B is the fully screened potential seen at large r (Eq. 11.3); V_A interpolates between them (Eq. 11.5). Every chemical property lives in the crossover region between the two.	$J or E_{R}$	real functions of r, all negative and all monotonically increasing	ch. 11
$E_{n l}$	orbital energy, depending on n AND l	The break from hydrogen. In a 1/r potential the energy depends only on n and states of different l are accidentally degenerate; in any OTHER central potential that degeneracy is lifted, and E rises with l because the centrifugal barrier keeps high-l electrons out where the nucleus is screened. Without this splitting there would be no periodic table.	$eV or E_{R}$	real, negative for a bound orbital; increases with both n and l	ch. 11
$L, M_{L}, S, M_{S}, J, M_{J}$	many-electron angular momentum quantum numbers	CAPITALS for two or more electrons, lower case (l, s, j) for one — the convention §11.1b states as a bullet because there is nothing to derive. L combines all the orbital angular momenta, S all the spins, and J combines L with S. In j–j coupling the order is reversed: each electron's own l and s pair into a j first, and the j's then combine into J.	$d im e n s i o n l ess q u an t u mn u mb er s$	non-negative integers or half-integers; M runs from -X to +X in integer steps	ch. 11
$^{2 S + 1} L_{J}$	term symbol	A many-electron energy level: multiplicity 2S+1 as a superscript, a CAPITAL letter S, P, D, F for L = 0, 1, 2, 3, and J as a subscript. Carbon's ground state is 3P_0. TRAP: the letter and the superscript describe different quantities, so 1S means L = 0 AND S = 0, and 3S means L = 0 with S = 1. The rescue is position — the middle letter is always L, the superscript always 2S+1, the subscript always J.	—	a label, not a quantity	ch. 11
$f$	electron–electron avoidance parameter	Section 11.3 only. Defined by R_ee = fR: the typical electron–electron separation as a multiple of the atom size. Expected to be about 1, larger if the electrons are good at keeping apart, and fitted ONCE to helium at 1.67. It is the single empirical input to the whole "what if there were no Pauli principle" calculation — a stand-in for correlation, not a derived quantity.	—	real scalar, dimensionless, of order 1	ch. 11
$A_{Z}, B_{Z}$	the coefficients of §11.3’s variational estimate	E = A_Z/R^2 - B_Z/R, with A_Z the cost of localizing Z electrons in a region of size R and B_Z the net Coulomb attraction after pair repulsion is subtracted. The minimum is at R = 2A/B with E = -B^2/4A, which for Z = 1 gives the Bohr radius and the Rydberg energy exactly.	$A in J \cdot m^{2}, B in J \cdot m$	real and positive; A_Z = Z·A_1 and B_Z = [Z² - Z(Z-1)/2f]·B_1	ch. 11
$σ$	standard deviation	The statistical spread of a distribution, and the same quantity Δx and Δp measure — σ² is the variance ⟨x²⟩ − ⟨x⟩². The book mostly writes Δ, but σ appears wherever a Gaussian's width is meant, and ch03's problems page warns that σ, a and α all look alike in running text while meaning a spread, a length and a decay constant respectively.	$w ha t e v er t h e q u an t i t y ha s$	real and non-negative	ch. 3
$a, a^{†}$	lowering and raising (annihilation and creation) operators	Section 6.6's factorization of the oscillator Hamiltonian. a lowers the energy by one quantum and a† raises it, so the whole spectrum follows from the ground state without solving a differential equation. The dagger is the HERMITIAN CONJUGATE — for a matrix, the conjugate transpose, which is why ch07 can say A† = A defines an observable. They do not commute: [a, a†] = 1, and that single relation generates E = (n + ½)ħω.	$d im e n s i o n l ess in t h esc a l e d v a r iab l e q$	operators, i.e. matrices; NOT Hermitian, and not observables — only their combinations are	ch. 6

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1.A page shows a dashed slate-grey curve next to a solid violet one. What is the dashed curve?
2.Which of these is the correct statement about $∣Ψ (x, t) ∣^{2}$ in one dimension?
3.In the linear-algebra dictionary, what corresponds to "a value you can actually measure"?
4.The bridge box lists four places where "quantum mechanics is just Hermitian matrices" breaks down. Which of these is one of them?
1 eV = $1.602 \times 1 0^{- 19}$ J. Do it in your head.
5.A visible photon at $λ = 663 nm$ carries about $3 \times 1 0^{- 19} J$ . Roughly how many electronvolts is that?
6.Why does the site distinguish capital $Ψ$ from lower-case $ψ$ so carefully?