Part IV · Quantum Mechanics — Chapter 4.2

The Linear Algebra of Quantum States

A table of renamings, seven postulates, and nothing else.

Tools you'll need  — Chapter 0.5, all of it, and this chapter spends nothing else of comparable weight. Section 1 has the inner-product axioms and the choice of linear slot. Section 1.4 has Cauchy–Schwarz, used in §8 here and cashed as the uncertainty principle in Chapter 4.9. Section 2 has orthonormal bases, Parseval and the resolution of the identity. Section 3 has orthogonal projection and $\hat P^{2}=\hat P$. Section 4 has the adjoint and the three species of operator. Section 6 has the spectral theorem in all three of its parts. Section 7 has functions of operators and $\ee^{\ii A}$. Section 8 has commuting operators, simultaneous diagonalisation and its figure fig-joint. And its Worked example 1 has a $2\times2$ matrix that §10 here uses three times. Chapter 0.4 §2 for basis and dimension, which §9 needs to build the tensor product. Its §6.1 for the cyclic property of the trace, which §8 turns into a proof that this subject cannot be finite-dimensional. Its §7 for the one theorem Part 0 imported rather than proved. Chapter 1.3 §6 for Poisson brackets and the fundamental brackets $\{q^{i},p_{j}\}=\delta^{i}_{j}$, and §7 for the statement that a conserved quantity is the generator of its symmetry. Sections 7 and 8 here repeat both with operators. Chapter 4.1 §6.5 for $\hbar$ and $p^{\mu}=\hbar k^{\mu}$, and §6.6 for the tension this chapter is here to resolve. Chapter 2.5 §4.3 for $E^{2}=p^{2}c^{2}+m^{2}c^{4}$, which §10.3 expands to get the neutrino oscillation length. Chapter 0.3 for Euler's formula, used in §10 without comment.

1 · The claim, stated before it is earned

Announce the destination. This section does no mathematics. It states exactly what the chapter will assert and exactly what it will not. It gives the total number of assumptions quantum mechanics requires, and it sets out the two rules by which the rest of Part IV should be read. Doing this first costs two pages. What it buys is the ability to check, at every later line, whether something was derived or put in by hand.

1.1 · What Chapter 0.5 claimed

Chapter 0.5 proved a sequence of theorems about operators on a complex inner-product space, with no physics anywhere in the argument. Then, in an insight box at the end of its §6, it read three of them back with two words changed:

"(a) Eigenvalues are real becomes: measurement outcomes are real numbers. … (b) Eigenvectors are orthogonal becomes: distinct outcomes are perfectly distinguishable. … (c) The eigenvectors form a complete orthonormal basis becomes: any state whatsoever can be written as a superposition of outcomes… These are three of the postulates of quantum mechanics as they are usually taught, and they are not postulates. They are theorems about Hermitian matrices, proved above, in a chapter containing no physics. Chapter 4.2 will do exactly one thing that this chapter did not: assert that a physical state is a unit vector and a physical observable is a Hermitian operator. That is one postulate, not four. Everything else is renaming."

That is the claim. This chapter either honours it, or the design of the book fails retrospectively. So the chapter is organised to make the claim checkable. Section 2 lays out the renamings as a table, and every later section either points at a row of that table or announces an assumption in a box. If you find a place where something has been slipped in without one or the other, the claim has failed there.

1.2 · The complete bill, said out loud

Here is the number, because a subject that has to assume things should be willing to say how many. Quantum mechanics as this book develops it rests on eight postulates and one experimental measurement. Seven of the eight arrive in this chapter. The eighth is the symmetrisation of identical-particle states, which arrives in Chapter 4.18. The measurement is that the electron carries half-integer angular momentum, which is Stern–Gerlach's, quoted in Chapter 4.12. Nothing else in Part IV is asserted without being derived. Chapter 4.20 §9 puts the whole list back on one page, with an honest statement of what each item does and does not settle.

	What is asserted	Where
P1	A pure state is a unit vector, defined only up to an overall phase	§3
P2	An observable is a Hermitian operator	§4
P3	The Born rule. The probability of outcome $\lambda_{k}$ is $\norm{\hat P_{k}\ket\psi}^{2}$	§5
P4	The state update. After outcome $\lambda_{k}$ the state is $\hat P_{k}\ket\psi/\norm{\hat P_{k}\ket\psi}$	§6
P5	The generator of time evolution is the energy	§7
P6	Canonical quantisation. $[\hat x_{i},\hat p_{j}]=\ii\hbar\delta_{ij}$	§8
P7	A composite system's space is the tensor product of the parts'	§9
P8	Identical-particle states are totally symmetric or totally antisymmetric	Chapter 4.18
E1	Measured, not postulated: the electron carries $j=\half$	Chapter 4.12

Now set that against the list of things which are not on it. Every one of them is a theorem of Chapter 0.5 that you have already proved:

• that measured values are real numbers (0.5 §6.1).
• that distinct measured values belong to exactly perpendicular states (0.5 §6.2).
• that every state can be resolved into the possible outcomes of any observable (0.5 §6.3).
• that the probabilities of the outcomes add to one (0.5 §2, Parseval).
• that two quantities can be sharp together exactly when their operators commute (0.5 §8).
• that a norm-preserving evolution preserves every amplitude, so that conserving total probability and being unitary are the same condition (0.5 §4.3).
• that a Hermitian generator produces exactly such an evolution, and every such evolution has one (0.5 §7.1 and its grind box).

Seven theorems against seven assertions. The seven theorems are the ones usually presented as the mysterious part. That contrast is the payoff for nine chapters of mathematics, and it is the thing this chapter exists to make visible.

1.3 · Two rules for reading the rest of Part IV

Every postulate is announced in a box, before it is used, and carries a ⚑. Part III never had to do this, because general relativity was cornered rather than posited. There you had the equivalence principle plus one identity, and the field equations follow. Quantum mechanics has no such route. The response is not to hide that fact but to timestamp it. A box means this is being asserted. The ⚑ means the book uses this and does not derive it. Both, every time, in place.

Say each time whether a step is a derivation or an identification. These are different things, and Part IV moves between them from sentence to sentence. That $\ee^{-\ii\hat Ht/\hbar}$ is unitary when $\hat H$ is Hermitian is algebra, proved in Chapter 0.5 §7.1. That the operator in that exponent is the energy is a physical identification with experiments behind it. Chapter 4.1 already ran this distinction once. The form of the cavity spectrum was derived and its normalisation was matched, and the chapter said so rather than letting the two blur. You have been trained for twenty-nine chapters to ask which is which, and this is the part where the answer changes most often.

2 · The table of renamings

Announce the destination. We set out, in one place, every result of Chapter 0.5 that this part of the book uses, beside the physical statement it becomes. Nothing in this section is assumed and nothing is proved. The proofs are all in Chapter 0.5, and each one is cited by equation. The purpose is to make the size of what has to be added visible before any of it is added.

2.1 · One convention, restated because everything rests on it

Chapter 0.5 §1.1 defined the inner product to be linear in its second argument and conjugate-linear in its first, and gave the reason in advance: "in Chapter 4.2 the object $\avg{\phi,\psi}$ becomes a probability amplitude, read right-to-left as 'amplitude to find $\psi$ in state $\phi$', and the linear slot had better be the one holding the state that is actually evolving." That reason can now be stated properly rather than promised.

The number $\avg{\phi|\psi}$ is going to be an amplitude, meaning a complex number whose squared modulus is a probability. The state that is evolving, being prepared, being superposed, is $\ket\psi$, and superposition is a linear operation on it. Split a beam into two paths and recombine it, and what comes out is $\ket{\psi_{1}}+\ket{\psi_{2}}$. The amplitude to find that in $\ket\phi$ must be the sum of the two amplitudes, or interference could not be computed at all. So the slot holding $\ket\psi$ has to be the linear one.

The conjugation then lands on $\ket\phi$, and that is exactly what makes $\avg{\phi|\psi}=\overline{\avg{\psi|\phi}}$, and hence $\abs{\avg{\phi|\psi}}^{2}=\abs{\avg{\psi|\phi}}^{2}$. Read that last equality in words. The probability of finding $\psi$ in $\phi$ equals the probability of finding $\phi$ in $\psi$, which is a symmetry a probability had better have.

Mathematics texts make the other choice, and then every formula below acquires its conjugates on the other side. Nothing physical depends on which convention you pick. What does depend on it is arithmetic. Mixing the two conventions inside one calculation is the single most common way to produce a wrong sign.

2.2 · The table

Read it once from left to right and once from right to left. Left to right it is a dictionary. Right to left it is a list of physical facts, with the place each one was proved.

Chapter 0.5 proved	Where	Which says, about a physical system
A complex inner-product space: vectors add and scale, and $\avg{u,v}$ is linear in $v$	§1	Superposition. If two states are possible, so is any combination of them, and $\avg{\phi|\psi}$ is the amplitude to find $\psi$ in $\phi$
$\norm{v}^{2}=\avg{v,v}\gt0$ for $v\neq0$	§1.1, axiom (iii)	Two different states can always be told apart: no non-zero state has zero length
Cauchy–Schwarz, $\abs{\avg{u,v}}^{2}\le\avg{u,u}\avg{v,v}$	§1.4	No probability exceeds one — and, in Chapter 4.9, the uncertainty relation, with nothing added but the meaning of the letters
Orthonormal basis $\{e_{i}\}$, $\avg{e_{i},e_{j}}=\delta_{ij}$	§2	A complete set of mutually exclusive outcomes of one measurement
$v=\sum_{i}\avg{e_{i},v}\,e_{i}$	§2	Every state is a superposition of the outcomes, with the coefficients read off as single overlaps
Parseval, $\norm{v}^{2}=\sum_{i}\abs{\avg{e_{i},v}}^{2}$	§2	The probabilities add to one, and they add to one automatically rather than by a normalisation imposed afterwards
Resolution of the identity, $\sum_{i}\ket{e_{i}}\bra{e_{i}}=\hat I$	§2.3	The set of outcomes has missed nothing: insert it anywhere and a hard amplitude becomes a sum of easy ones
Orthogonal projection $\hat P$, with $\hat P^{2}=\hat P$ and $\hat P^{\dagger}=\hat P$	§3	An outcome. Idempotence becomes the statement that repeating a measurement immediately returns the same value
The adjoint, $\avg{\hat Au,v}=\avg{u,\hat A^{\dagger}v}$	§4	The rule for moving an operator across an amplitude — used on every page of Part IV
Hermitian, $\hat A^{\dagger}=\hat A$	§4.2	An observable
Unitary, $\hat U^{\dagger}\hat U=\hat I$, preserves every inner product	§4.3	Evolution, and every symmetry: exactly the maps under which total probability is conserved
Eigenvalue $\lambda$, eigenvector $v$, $\hat Av=\lambda v$	§5	A measured value, and the state that gives it with certainty
Over $\C$ every operator has an eigenvalue	§5, from 0.4 §7	Every observable has possible measured values. Over $\R$ this is false, and that is why the space is complex
Hermitian $\Rightarrow$ every eigenvalue is real	§6.1	Measured values are real numbers. A dial cannot read $3+2\ii$
Hermitian $\Rightarrow$ eigenvectors with different eigenvalues are orthogonal	§6.2	Distinct outcomes are perfectly distinguishable. The chance of confusing one for another is zero, not small
The spectral theorem: an orthonormal eigenbasis exists	§6.3	Any state at all is a superposition of the outcomes of any observable
$\hat A=\sum_{k}\lambda_{k}\hat P_{k}$, with $\hat P_{k}\hat P_{l}=\delta_{kl}\hat P_{k}$ and $\sum_{k}\hat P_{k}=\hat I$	§6.4	An observable is its list of possible values together with the outcomes they belong to, and nothing else
Degeneracy: the eigenspace is unique, a basis inside it is not	§6.4	Two states can share a measured value, and then that measurement does not determine the state
$f(\hat A)=\sum_{k}f(\lambda_{k})\hat P_{k}$	§7	Functions of observables, including the one that generates motion
$\hat A$ Hermitian $\Rightarrow\ \ee^{\ii\hat A}$ unitary, and conversely	§7.1	Observables generate symmetries and evolution, and every symmetry has an observable behind it
$[\hat A,\hat B]=0\iff$ a common orthonormal eigenbasis exists	§8	Compatible measurements, and the quantum numbers that label a state
$[\hat A,\hat B]\neq0\Rightarrow$ no common eigenbasis	§8	There is no state in which both quantities are sharp — the qualitative half of Chapter 4.9

2.3 · What has been added so far, exactly

Nothing. Not one line of the table is an assumption about the world. Every left-hand entry is a theorem with a proof you have read, and every right-hand entry is that theorem with two or three words changed. Chapter 0.5 §6.5 put the count in its own words: "That is one postulate, not four. Everything else is renaming". The table is that sentence, itemised.

The physics enters at exactly seven places, and the sections that follow take them one at a time.

• Two of the seven, §3 and §4, are the pair Chapter 0.5 named: a state is a unit vector, an observable is a Hermitian operator.
• Two more, §5 and §6, are the ones Chapter 0.5 never had to mention, because a chapter of linear algebra has no reason to talk about probability.
• The last three, §§7–9, are dynamics, the relation between position and momentum, and how two systems combine.

That is the whole of it.

3 · States are unit rays

Announce the destination. The first assertion. A state is a unit vector, and two unit vectors differing by an overall complex factor of modulus one describe the same state. The first clause is bookkeeping, and we show why. The second clause is physics, and we show what it forbids and what it permits. The section ends with a parameter count that turns the abstraction into something you can hold: a two-state system has two real parameters, not four.

3.1 · Why the length is fixed at one, and why that is not a restriction

The reason to normalise is that the numbers we are about to call probabilities are squared overlaps, and Parseval says their sum is $\norm\psi^{2}$. So expand any state in an orthonormal basis $\{\ket{e_{i}}\}$, which Chapter 0.5 §2 permits:

Setting $\norm\psi=1$ makes the right-hand sum equal to one, which is what a complete list of probabilities has to do. Notice the direction of the logic. Normalisation is chosen so that Parseval's identity reads as a statement about probability. Probability is not being normalised afterwards, by hand.

Scaling $\ket\psi$ by any non-zero complex $c$ scales every $c_{i}$ by $c$, and every $\abs{c_{i}}^{2}$ by $\abs{c}^{2}$. So it changes no ratio and describes nothing new. A state is therefore a direction, and the unit-length convention picks one representative from each direction.

3.2 · The overall phase, and why it is invisible

Fixing the length leaves one freedom untouched. If $\norm\psi=1$ then $\norm{\ee^{\ii\alpha}\psi}=1$ as well, since $\abs{\ee^{\ii\alpha}}=1$. So unit length does not pick a unique vector. P1 asserts that this leftover freedom is physically empty. The reason can be given completely only once §5 has supplied the Born rule. But the computation itself is one line, and it does not depend on anything §5 adds beyond the form of the expression, so we can do it now. For any projection $\hat P_{k}$,

because the first slot conjugates (Chapter 0.5 §1.1) and $\ee^{-\ii\alpha}\ee^{\ii\alpha}=1$. Every probability the theory produces is of this form, so every one of them is untouched. The same one-line argument kills the phase in every expectation value, since $\bra{\ee^{\ii\alpha}\psi}\hat A\ket{\ee^{\ii\alpha}\psi}=\bra\psi\hat A\ket\psi$ by the identical cancellation. There is no measurement in the theory that can see $\alpha$.

Verified. Taking the Hermitian matrix $\hat A=\operatorname{diag}(2)\oplus\big(\begin{smallmatrix}1&1/2\\ 1/2&1\end{smallmatrix}\big)$ and the state $\ket\psi=\tfrac13(1,2,2)$, symbolic algebra gives the three probabilities $0$, $\tfrac89$, $\tfrac19$ both with and without a factor $\ee^{\ii\alpha}$, identically in $\alpha$, and they sum to $1$.

3.3 · The relative phase, which is everything

Now the contrast that makes §3.2 content rather than triviality. Multiply one term of a superposition by a phase and nothing cancels. Take a two-state system, the two states being the two paths of an interferometer, and compare

Both are unit vectors. Now we want the number a detector actually reports, so project onto the recombined state $\ket{+}=\tfrac{1}{\sqrt2}(\ket1+\ket2)$, which is what a detector at one output port of an interferometer responds to. The amplitude is $\avg{+|\psi_{\beta}}=\tfrac12(1+\ee^{\ii\beta})$, and squaring its modulus gives

which runs from $1$ at $\beta=0$ to $0$ at $\beta=\pi$. That is a fringe, and it is what an interferometer measures. So a phase applied to every component is unobservable, and a phase applied to one component is the difference between full transmission and complete extinction.

This is not a subtlety to be filed away. It is the whole reason the theory needs complex numbers, and the whole content of the two-slit experiment. Chapter 4.1 §6.6 insisted that light's wave behaviour was untouched by the evidence for parcels of energy, and (4.2.4) is where the wave behaviour now lives: in the relative phase between the terms of a superposition.

3.4 · How many parameters a state actually has

Count, because the count is the cleanest way to feel the difference between a vector and a ray. A unit vector in $\C^{n}$ has $2n$ real components, subject to one real constraint $\norm\psi=1$, so the unit sphere has $2n-1$ real dimensions. Identifying vectors that differ by a phase removes one more. So

For $n=2$ that is two, not four. Two real parameters is a sphere, and we can see the sphere directly by writing the general unit vector as

which uses the phase freedom to make the first coefficient real and non-negative, and that exhausts the freedom. Every state of a two-level system is (4.2.6) for exactly one $(\theta,\varphi)$, and $(\theta,\varphi)$ are polar coordinates on a sphere. That sphere is the Bloch sphere. It is not a picture drawn for intuition. It is the state space, exactly, and §12's Problem 1 shows that the three expectation values $\avg{\sigma_{x}},\avg{\sigma_{y}},\avg{\sigma_{z}}$ are the Cartesian coordinates of the corresponding point. The sphere is therefore something you can measure.

For $n=3$ the count gives four real parameters, and there is no comparably nice picture. The two-level case is special, and its picture does not generalise. Better to say that out loud than to let the sphere become a mental model of quantum mechanics in general.

a natural place to stop  ·  the dictionary is laid out; what follows is one postulate at a time

4 · Observables are Hermitian operators

Announce the destination. The second assertion, and the one Chapter 0.5 named in advance as "the one postulate that has to be made". We state it, then collect the three theorems it immediately buys without re-proving any of them. Then we read the spectral decomposition as a physical statement about what an observable is. Then we use Chapter 0.5 §8 to define a term the rest of Part IV lives on: a quantum number. One warning is attached to the postulate itself and cannot wait, because the word "Hermitian" is not strong enough for the spaces Chapter 4.3 builds.

4.1 · Three theorems, collected rather than re-proved

With P2 in place, the following are facts about nature that were established in a chapter containing no physics. Each is cited at the section that states it, and none is proved again here. Proving them again would be a misreading of what this chapter is.

Measured values are real. Chapter 0.5 §6.1 showed that a Hermitian operator's eigenvalues satisfy $\bar\lambda=\lambda$. A dial reads a real number, and now it has to.

Distinct measured values belong to orthogonal states. Chapter 0.5 §6.2 showed that eigenvectors with $\lambda\neq\mu$ satisfy $\avg{v,w}=0$. Section 5 will make $\abs{\avg{v,w}}^{2}$ the probability of finding one where the other is, so that probability is exactly zero. Not small, and not zero to within experimental error. Two states with different definite values of the same observable are perfectly distinguishable in a single measurement.

Every state is a superposition of outcomes. Chapter 0.5 §6.3 produced an orthonormal basis of eigenvectors. So for any state $\ket\psi$ and any observable $\hat A$ whatever, $\ket\psi=\sum_{i}c_{i}\ket{e_{i}}$ with $\sum_{i}\abs{c_{i}}^{2}=1$ by Parseval. The probabilities of the outcomes add to one automatically, and no separate assumption is needed to make them.

4.2 · What an observable is, read off the spectral decomposition

We want the form of the decomposition that will still be true when the space becomes infinite dimensional, so group Chapter 0.5 §6.4's decomposition by distinct eigenvalue rather than by basis vector. That is the form Part IV uses throughout:

with the $\lambda_{k}$ real and distinct. Let's read (4.2.7) as a definition of the object rather than as a factorisation of it. An observable is a list of real numbers, together with a set of mutually orthogonal outcomes that exhaust the space. It is a labelled partition of the state space into perpendicular pieces. The operator is not something extra that the list has. The list is the operator. Two observables are the same observable exactly when they have the same list, which is why in practice one specifies a measurement by saying what its possible answers are and which states give each of them with certainty.

The degeneracy clause carries physics. If $\hat P_{k}$ has rank greater than one, several independent states share the value $\lambda_{k}$, and Chapter 0.5 §6.4 was careful that the eigenspace is unique but a basis inside it is not, and no theorem prefers one. Physically: that measurement, on its own, does not determine the state. Something else has to be measured, and the next subsection says exactly what "something else" is allowed to be.

4.3 · Compatible observables, and the definition of a quantum number

Chapter 0.5 §8 proved that two Hermitian operators admit a common orthonormal eigenbasis if and only if they commute. Rename both halves.

If $[\hat A,\hat B]=0$ there are states in which both quantities have definite values at once, and enough of them to span the space. Call two such observables compatible. Chapter 0.5 §8.2's Step 4 is where the content sits. Inside a degenerate eigenspace of $\hat A$, where $\hat A$ has said all it can say, the restriction of $\hat B$ is still Hermitian. So the spectral theorem applies inside that eigenspace, and $\hat B$ chooses the basis $\hat A$ could not.

If $[\hat A,\hat B]\neq0$ there is no common eigenbasis, so there is no state at all in which both quantities are sharp. That is the qualitative content of the uncertainty principle. Chapter 4.9 supplies the quantitative version by applying Cauchy–Schwarz to two particular vectors. That is a calculation Chapter 0.5 §1.4 already set up, and it adds nothing but the meaning of the symbols.

Now the definition this chapter owes the rest of the part. A set $\{\hat A_{1},\dots,\hat A_{m}\}$ of mutually commuting observables is a complete set of commuting observables when their common eigenspaces are all one-dimensional. Said the other way round, the list of eigenvalues $(a_{1},\dots,a_{m})$ determines the state uniquely up to phase. That list is called the quantum numbers of the state. Writing a hydrogen state as $\ket{n,\ell,m_{\ell}}$ is naming the eigenvalues of three commuting operators and nothing more. Chapter 0.5's insight box promised the term to Chapter 4.11. The term is defined here instead, where the machinery arrives. Chapter 4.9 §3 asks how one knows a set is complete, and Chapters 4.11 and 4.13 spend it on angular momentum and on hydrogen.

One caution, and Chapter 0.5's figure fig-joint is where to see it rather than have it described again. The theorem promises that a common eigenbasis exists. It does not promise that the basis a calculation happens to produce is that one. Drive the commutator of two operators to exactly zero, and a standard eigensolver will still hand back states in which the second observable has a spread. Inside a degenerate eigenspace it returned an eigenbasis of the first, rather than the one the second prefers. Go and move the sliders. The point is worth the minute, and it is exactly what Chapter 4.13's degeneracies will need.

4.4 · Which operators are observables, and which are not

P2 says every observable is Hermitian. It does not say every Hermitian operator corresponds to something an apparatus can measure, and that converse is genuinely open. For a system with superselection rules, or for gauge-dependent quantities, some Hermitian operators correspond to nothing measurable. This book does not need the converse and does not assert it. What it does need is the direction P2 states, plus one immediate negative consequence worth recording now.

A unitary operator is not, in general, an observable. Chapter 0.5's Problem 2 showed that its eigenvalues have modulus one, so they are complex unless they are $\pm1$, and P2 requires real outcomes. Unitary operators are how the state changes. Hermitian operators are what can be read. Chapter 0.5 §7.1's correspondence $\hat U=\ee^{\ii\hat A}$ says the two classes carry the same information, and §7 below is the section that spends it.

5 · The Born rule

Announce the destination. This is the section the book's honesty rests on. We state the rule connecting a state to the probabilities of outcomes. We say without hedging that it is not derivable, and that nothing in this book or outside it has derived it. We deal properly with the theorem that comes closest. Then we derive from the rule everything that is derivable from it: that probabilities are non-negative and sum to one, that overall phase is invisible, and that the average and the spread of a measurement are the two short expressions everyone uses.

5.1 · Why nothing so far has produced a probability

Look at what P1 and P2 have supplied. There is a state, which is a unit ray. There is an observable, which by (4.2.7) is a set of real values with orthogonal outcomes summing to the identity. The state resolves into those outcomes, $\ket\psi=\sum_{k}\hat P_{k}\ket\psi$, since $\sum_{k}\hat P_{k}=\hat I$. The pieces are mutually orthogonal, so Chapter 0.5 §1.3's Pythagoras applies with no cross terms, and their squared lengths add to one:

the cross terms vanishing because $\avg{\hat P_{k}\psi,\hat P_{l}\psi}=\avg{\psi,\hat P_{k}\hat P_{l}\psi}=\delta_{kl}\norm{\hat P_{k}\psi}^{2}$ by Hermiticity and orthogonality of the projections. So there is a list of non-negative numbers, one per possible outcome, adding to one, sitting in the formalism already.

And that is as far as mathematics goes. A list of non-negative numbers summing to one is consistent with being a probability distribution. It is not a demonstration that it is one. Nothing in Chapter 0.5, and nothing in P1 or P2, connects any number in the theory to the relative frequency of an outcome in a long run of experiments. That connection has to be asserted, and here it is.

5.2 · What follows immediately, and what does not

The probabilities are legitimate. Each is a squared norm, hence real and non-negative, and (4.2.8) says they add to one. Neither fact was assumed in P3. Both are Chapter 0.5's, and P3 was written in the form $\norm{\hat P_{k}\psi}^{2}$ precisely so that they would be automatic.

The overall phase is invisible. Proved in (4.2.2), and it is worth noticing that §3.2's computation was written before P3 arrived and needed only the form $\norm{\hat P_{k}\psi}^{2}$. P1 and P3 fit together exactly. Had the rule been anything not quadratic in $\ket\psi$, P1's identification of $\ket\psi$ with $\ee^{\ii\alpha}\ket\psi$ would have been inconsistent.

What does not follow: which outcome occurs. P3 gives the distribution and says nothing about the individual run. This is not a gap the chapter will close.

5.3 · The expectation value and the variance, derived

These are the two quantities actually reported by an experiment, and both come out of P3 in two lines each. We want the mean of the measured value over many repetitions. By P3 that mean is the weighted average $\sum_{k}\lambda_{k}\Pr(\lambda_{k})$. So substitute P3 into it, and then run (4.2.7) backwards:

The middle step used $\norm{\hat P_{k}\psi}^{2}=\avg{\hat P_{k}\psi,\hat P_{k}\psi} =\avg{\psi,\hat P_{k}^{\dagger}\hat P_{k}\psi}=\avg{\psi,\hat P_{k}\psi}$, which is $\hat P_{k}^{\dagger}=\hat P_{k}$ and $\hat P_{k}^{2}=\hat P_{k}$ from Chapter 0.5 §3. The last step is the spectral decomposition read from right to left. So the ubiquitous formula $\avg{\hat A}=\bra\psi\hat A\ket\psi$ is derived from P3. It is not a separate definition of what "average" means.

The spread comes the same way. The variance of the measured value is $\sum_{k}(\lambda_{k}-\avg{\hat A})^{2}\Pr(\lambda_{k})$. We want that in operator form, so expand the square, then use $\sum_{k}\lambda_{k}^{2}\hat P_{k}=\hat A^{2}$ (which is Chapter 0.5 §7's function-of-an-operator with $f(\lambda)=\lambda^{2}$) together with $\sum_{k}\hat P_{k}=\hat I$:

The second form is the one Chapter 4.9 needs, because it exhibits $\Delta A$ as the length of a vector, and Cauchy–Schwarz is a statement about lengths. That is the whole reason the uncertainty principle is one line there rather than a new idea.

Verified. With $\hat A=\big(\begin{smallmatrix}2&\half\\ \half&1\end{smallmatrix}\big)$ and $\ket\psi=(\tfrac35,\tfrac45)$, computing the two projections and summing gives $\sum_{k}\Pr=1$, $\sum_{k}\lambda_{k}\Pr(\lambda_{k})=\tfrac{46}{25}$ against $\bra\psi\hat A\ket\psi=\tfrac{46}{25}$, and $\sum_{k}(\lambda_{k}-\avg{\hat A})^{2}\Pr(\lambda_{k})=\tfrac{961}{2500}$ against $\avg{\hat A^{2}}-\avg{\hat A}^{2}=\tfrac{961}{2500}$. Exact rationals both ways.

5.4 · Amplitudes are not probabilities

6 · Measurement, and the state after it

Announce the destination. A fourth assertion is needed, and most books fold it into the third. We keep them apart, because they are logically independent, and because the separation is what lets Chapter 4.20 say precisely which of the two decoherence addresses. We then derive the one consequence that makes the postulate testable, which is that an immediately repeated measurement returns the same value with certainty. Next we set the algebra beside the conditioning you do every day, and say exactly where the two part company. We end by naming the one imported theorem that all four postulates rest on.

6.1 · Why P3 is not enough

P3 says what the odds are. It says nothing whatever about the system afterwards, and something has to be said, because measurements are performed in sequence. An apparatus is calibrated by measuring the same thing twice. A Stern–Gerlach beam is sent through a second magnet. A qubit is read out and then used. Without a rule for the post-measurement state, the theory cannot predict the second measurement at all.

6.2 · Repeatability, derived

The postulate has one immediate consequence that is a genuine prediction, and it costs three lines. Measure $\hat A$ again straight away, on the updated state $\ket{\psi'}$. By P3 the probability of getting $\lambda_{l}$ is $\norm{\hat P_{l}\ket{\psi'}}^{2}$, and

using $\hat P_{l}\hat P_{k}=\delta_{lk}\hat P_{k}$ from (4.2.7). So the probability is $1$ for $l=k$ and $0$ otherwise: an immediately repeated measurement returns the same value with certainty. That is a fact about laboratories, and it is what makes the word "measurement" mean anything. It comes from $\hat P^{2}=\hat P$, which is Chapter 0.5 §3's idempotence, proved there as the algebraic statement that projecting twice is projecting once.

Notice also which vector $\ket{\psi'}$ is when $\lambda_{k}$ is non-degenerate. Then $\hat P_{k}$ has rank one, so $\hat P_{k}\ket\psi=\ket{e_{k}}\avg{e_{k}|\psi}$, and the update gives $\ket{e_{k}}$ up to a phase, which by P1 is $\ket{e_{k}}$. The state after the measurement is the eigenstate, and every memory of $\ket\psi$ except the fact of the outcome is gone. When $\lambda_{k}$ is degenerate the update keeps more. It keeps the component of $\ket\psi$ inside the eigenspace, direction and all. That distinction is not decorative. It is what makes a measurement of a degenerate observable weaker than a measurement of a complete set, and it is the reason Chapter 4.13's hydrogen states need three labels.

6.3 · What is not settled, named

P3 and P4 together let you predict every laboratory result in this book. They do not explain anything about the process they describe. Three questions are left completely open, and it is better to list them than to let them accumulate as unease.

What counts as a measurement? P4 refers to "a measurement" as a primitive. The apparatus is itself made of atoms and should be describable by P1 and P5 like anything else, in which case nothing in the joint evolution ever produces a single outcome. This is the measurement problem and this book does not solve it.

Why one outcome rather than another? P3 supplies the distribution and P4 the conditional state. Neither says what selects the individual result.

When does the update happen? P4 says "immediately afterwards", which is not a time. Chapter 4.20 §9 states which of these decoherence answers, essentially the first and only in part, and which it leaves untouched.

6.4 · What all four postulates rest on

One structural remark, and it is owed. Every measurement postulate above is a statement about the projections $\hat P_{k}$ of an observable, and those exist because of Chapter 0.5's spectral theorem. Look at Step 1 of that theorem's proof: "Because $V$ is complex, [Chapter 0.4] supplies an eigenvalue." Chapter 0.4 §7 supplied it from the fundamental theorem of algebra, which says that every non-constant complex polynomial has a root. That chapter flagged the theorem as the one result Part 0 imported rather than proved ⚑.

So the whole apparatus of measurement in this book descends from one imported theorem. That is a small debt, and it is worth knowing exactly where it sits, because it is not a hidden one. Without it, an observable might have no eigenvalues at all, and there would be no possible measured values to assign probabilities to. Chapter 0.4 §7 already made the point in those words. The debt is paid in Chapter 5.4, where complex analysis is built and the fundamental theorem of algebra becomes a three-line corollary of Liouville's theorem. Until then it is quoted, and it is the only thing in this chapter that is quoted rather than either derived or postulated.

a natural place to stop  ·  states, observables and measurement are in place; what follows is motion

7 · Evolution has a Hermitian generator

Announce the destination. Here almost everything is derived, and the assertion is a single clause. In four steps we show that time evolution must be unitary, that a unitary flow must be the exponential of a Hermitian operator, and that the operator therefore exists and has the dimensions of an energy. Then we state the one thing that does not follow, which is that it is the energy. Differentiating the result gives the Schrödinger equation, which Chapter 4.6 takes seriously. The section closes by collecting Chapter 1.3's promise that observables generate symmetries, with operators in place of functions and every word unchanged.

7.1 · Step one: evolution preserves the norm

Write $\hat U(t)$ for the map taking the state at time $0$ to the state at time $t$, and require two things of it.

The first is linearity. If a system can be prepared in $\ket{\psi_{1}}$ or $\ket{\psi_{2}}$, P1 says it can be prepared in any superposition, and evolution must take that superposition to the corresponding superposition of the evolved states. Otherwise the two beams of an interferometer, prepared together and travelling together, could not be treated as they are.

The second is conservation of total probability. At every time, the outcomes of any measurement must have probabilities summing to one, which by §5.2 is the statement $\norm{\psi(t)}=1$. Put the two together, and $\hat U(t)$ is a linear map preserving the norm of every state.

7.2 · Step two: norm-preserving and linear means unitary

Preserving lengths sounds weaker than preserving inner products. It is not. The missing inner-product information can be extracted from lengths alone, and that extraction is worth doing on the page, because it is three lines and it is the hinge of the section. Apply Chapter 0.5 §1.3's expansion $\norm{u+v}^{2}=\norm{u}^{2}+\norm{v}^{2}+2\operatorname{Re}\avg{u,v}$ to $\hat Uu+\hat Uv$, which is $\hat U(u+v)$ by linearity, and use $\norm{\hat Uw}=\norm{w}$ three times:

so $\operatorname{Re}\avg{\hat Uu,\hat Uv}=\operatorname{Re}\avg{u,v}$. That is the real part. Now we want the imaginary part, so run the same line with $v$ replaced by $\ii v$. The second slot is the linear one, so $\avg{\hat Uu,\hat U(\ii v)}=\ii\avg{\hat Uu,\hat Uv}$, and $\operatorname{Re}(\ii z)=-\operatorname{Im}z$, which gives $\operatorname{Im}\avg{\hat Uu,\hat Uv}=\operatorname{Im}\avg{u,v}$. Both parts agree, so

The equivalence on the right is Chapter 0.5 §4.3 read backwards. We have $\avg{\hat Uu,\hat Uv} =\avg{u,\hat U^{\dagger}\hat Uv}$ by the definition of the adjoint, and an operator whose inner products with everything agree with the identity's is the identity. So time evolution is unitary, and this is a consequence rather than an assumption. Chapter 0.5 §4.3 said the same thing forward, in the words "$U(t)$ must be unitary because $\norm{\psi}^{2}=1$ is a total probability". Here is the argument it was pointing at.

7.3 · Step three: a unitary flow has a Hermitian generator

Now bring in time. Take a system whose physical arrangement is not itself changing, with no field being switched on and no apparatus being moved. For such a system, evolving for $s$ and then for $t$ must be the same as evolving for $s+t$, since there is nothing to distinguish the intermediate instant. So

We want a differential equation rather than a functional one, so differentiate (4.2.14) with respect to $s$ at $s=0$. The left side gives $\hat U'(t)$ and the right gives $\hat U(t)\hat U'(0)$, so writing $\hat K\equiv\hat U'(0)$,

the solution being the matrix exponential $\sum_{m}t^{m}\hat K^{m}/m!$, which converges for every operator and is the unique solution with $\hat U(0)=\hat I$. Once $\hat K$ is identified below as $-\ii\hat G/\hbar$ with $\hat G$ Hermitian, that same exponential is Chapter 0.5 §7's function of an operator with $f(\lambda)=\ee^{t\lambda}$.

What kind of operator is $\hat K$? Unitarity is the condition we have not yet spent, so differentiate $\hat U^{\dagger}(t)\hat U(t)=\hat I$ at $t=0$, using $(\hat U')^{\dagger}=(\hat U^{\dagger})'$:

so $\hat K$ is anti-Hermitian. Then $\ii\hat K$ is Hermitian, because $(\ii\hat K)^{\dagger}=-\ii\hat K^{\dagger}=\ii\hat K$ using $(\alpha\hat A)^{\dagger}=\bar\alpha\hat A^{\dagger}$ from Chapter 0.5 §4.1. Every anti-Hermitian operator is $-\ii$ times a Hermitian one and conversely, which is the same correspondence as Chapter 0.5 §7.1's $\hat U=\ee^{\ii\hat A}$, seen at the level of generators rather than group elements.

Now the dimensions. $\hat K$ has the dimensions of one over time, since $t\hat K$ has to be dimensionless. Multiply and divide by the constant with the dimensions of action that Chapter 4.1 produced, writing $\hat K=-\ii\hat G/\hbar$. Then $\hat G$ is Hermitian, and it has the dimensions of $\hbar/\text{time}$, which is energy. Nothing has been assumed to get here. The existence of a Hermitian operator with the dimensions of energy generating the evolution is forced by unitarity, the group law and dimensional analysis.

7.4 · Step four: the one thing that does not follow

Differentiating (4.2.17) with $\hat G=\hat H$ and multiplying by $\ii\hbar$ gives the equation the next chapter is named after:

Chapter 0.5's closing insight box named this exact equation as where its $\ee^{\ii A}$ would be spent, and said the sentence "energy is observable, therefore probability is conserved" is not a physical argument but a line of algebra. That is now literally true. (4.2.13) is the algebra, and P5 is the only physics in the paragraph.

7.5 · Observables generate symmetries: Chapter 1.3, with operators

Nothing in §7.3 used the fact that the parameter was time. Any Hermitian $\hat G$ generates a one-parameter family of unitaries $\hat U_{\epsilon}=\ee^{-\ii\epsilon\hat G/\hbar}$, and the statement that this family is a symmetry of the system is the statement that it leaves $\hat H$ alone. So let's work out what such a family does to an arbitrary observable, to first order in $\epsilon$. Expand the exponential, keeping terms to first order:

the two cross terms combining into the commutator because $\hat G\hat f-\hat f\hat G=[\hat G,\hat f]$ by definition. Subtracting $\hat f$ leaves the change itself, which is the object we are after:

Set that beside Chapter 1.3 §7, which proved for classical mechanics that the change in $f$ under the flow generated by $G$ is $\delta f=\epsilon\{f,G\}$. The two are the same statement under the single replacement

which is precisely the substitution Chapter 1.3 §6.4 announced in advance. Taking $\hat G=\hat H$ and $\epsilon=t$ turns (4.2.20) into the equation of motion for an observable, $\dv{\hat f}{t}=\frac{1}{\ii\hbar}[\hat f,\hat H]$, which is Chapter 1.3's $\dv{f}{t}=\{f,H\}$ with the same replacement.

Now read the same line the other way. The condition $[\hat f,\hat H]=0$ says two things at once: that $\hat f$ is conserved, and that $\hat H$ is unchanged by the symmetry $\hat f$ generates. A conserved quantity is the generator of its symmetry, in operator form, which is what Chapter 1.3 said this chapter would repeat word for word.

Two things are being collected there, and it is worth separating them. The algebraic statement (4.2.20) is derived, from §7.3 and nothing else. The claim that (4.2.21) is the right correspondence for every pair of classical observables is a different animal. It is not derived, it is not assumed here either, and it is false. Section 8 postulates it for the single pair it needs, and Chapter 4.10 §8 proves that it cannot be extended consistently to all of them.

8 · Canonical quantisation

Announce the destination. Position and momentum have to be related to each other somehow, and the classical theory already supplies the relation. Chapter 1.3 computed the Poisson bracket of a coordinate with its conjugate momentum and got one. We postulate the operator version, under §7.5's substitution, for that pair alone. Then we spend three lines proving something that decides the shape of the next two chapters. No finite-dimensional space can carry that relation. Quantum mechanics is infinite-dimensional before any physics is done, and you can prove it here with Chapter 0.4's trace.

8.1 · The classical relation, and the operator version

Chapter 1.3 §6 built the Poisson bracket and computed the fundamental brackets

which say that momentum generates translation of position, and that neither generates any motion of its own kind. Section 7.5 showed that the operator analogue of a Poisson bracket is $\frac{1}{\ii\hbar}[\hat f,\hat g]$, derived for a generator acting on an observable. Applying it to (4.2.22) gives the relation this part runs on. It is an assumption, and here is its box.

8.2 · Three lines that decide the next two chapters

Suppose the state space were finite-dimensional, of dimension $n$, so that $\hat x$ and $\hat p$ are $n\times n$ matrices. Take the trace of both sides of P6's first relation. Chapter 0.4 §6.1 proved that the trace is cyclic, $\operatorname{tr}(AB)=\operatorname{tr}(BA)$, for any two square matrices of matching size. Hence the trace of any commutator vanishes:

Now take the trace of the right-hand side. The identity matrix in dimension $n$ has $n$ ones on its diagonal, so

Two numbers that must be equal are $0$ and $\ii\hbar n$. There are no $n\times n$ matrices satisfying P6, for any $n$ whatever.

Stop and look at what that argument is. It uses one fact about the trace, proved in Chapter 0.4 in a chapter about determinants and change of basis, with no analysis, no limits and no physics. It is complete. And it says that the state space of a particle with a position cannot be $\C^{n}$ for any $n$. The reason is not that $\C^{n}$ is too coarse an approximation, and not that a continuum is more elegant. It is that the arithmetic is impossible. The infinite-dimensionality of quantum mechanics is forced by one postulate and one line of linear algebra, before any physics has been done at all.

8.3 · How badly finite matrices fail, quantified

"Impossible" invites the response that some large matrix must come close, and the question deserves a number rather than a reassurance. It does not come close, and the bound is exact.

Use the inner product on matrices from Chapter 0.5 §1.2's table, $\avg{A,B}=\operatorname{tr}(A^{\dagger}B)$, whose norm is $\norm{A}_{F}=\big(\sum_{ij}\abs{A_{ij}}^{2}\big)^{1/2}$. Apply Chapter 0.5 §1.4's Cauchy–Schwarz inequality in that space, to $\hat I$ and any matrix $M$:

Now we want to apply that to the miss itself, so set $M=[\hat x,\hat p]-\ii\hbar\hat I$, the amount by which a candidate pair fails. By (4.2.23) its trace is $0-\ii\hbar n$, so $\abs{\operatorname{tr}M}=\hbar n$, and (4.2.25) gives $\hbar^{2}n^{2}\le n\norm{M}_{F}^{2}$, that is

for every pair of $n\times n$ matrices and every $n$. Read the right-hand equality. The error is at least as large as the thing being reproduced. In relative terms the miss is at least $100\%$, at every size, so a finite matrix pair cannot get the canonical commutator even half right.

Verified numerically, and the bound is attained. Minimising $\norm{[\hat x,\hat p]-\ii\hbar\hat I}_{F}/\norm{\ii\hbar\hat I}_{F}$ directly over all complex $n\times n$ pairs by Nelder–Mead from twelve random starts bottoms out at $1.000000$ for $n=2,3,4,5$, so the bound is right and it is attained. Problem 3 identifies the equality case, and it is worth having in advance. Equality needs $M$ parallel to $\hat I$, which forces $[\hat x,\hat p]=0$. So the best a finite-dimensional model can do is to commute, that is to reproduce none of the relation at all, and every pair that makes the commutator non-zero does strictly worse. For the natural truncation $\hat x=\operatorname{diag}(0,1,\dots,n-1)$ with a nearest-neighbour $\hat p$, the trace of the commutator is $0$ to machine precision at $n=4,16,64$ and $256$, exactly as (4.2.23) requires.

8.4 · Why spin is not a counterexample, and what has to happen next

Here is the objection to raise. Chapter 4.11 will describe electron spin with $2\times2$ matrices, and that is finite-dimensional. It is, and no contradiction arises, because a spin has no position operator. The three spin components satisfy $[\hat S_{i},\hat S_{j}]=\ii\hbar\epsilon_{ijk}\hat S_{k}$, whose right-hand side is not a multiple of the identity. Its trace therefore can vanish, and does, since each $\hat S_{k}$ is traceless. Section 8.2 rules out finite dimensions only for a system carrying a canonically conjugate pair. That it rules them out so cleanly while leaving spin alone is a check on the argument rather than a limitation of it.

What §8.2 forces, then, is this. The space of states for a particle has to be infinite-dimensional, and every theorem quoted in this chapter was proved in Chapter 0.5 in finite dimensions. Chapter 0.5 was explicit about the four places its proofs used that hypothesis, naming them as "in the induction, in rank–nullity, in the interchange of sums, in the claim that an injective map is surjective", and it said the bill would come due. It comes due in two instalments, and the division is worth stating precisely rather than gesturing at:

• Chapter 4.3 builds the space. The Riemann integral is discarded and rebuilt as the Lebesgue integral. That is what makes $L^{2}$ complete, so that a limit of states is a state, and what makes the Fourier modes genuinely an orthonormal basis rather than an assumption. Everything in §§1–3 of this chapter that used "an orthonormal basis exists" is on credit until then.
• Chapters 4.4 and 4.5 build the operators on it. Unbounded operators, domains, the difference between symmetric and self-adjoint that P2's box already warned about, spectra with no eigenvectors, the spectral theorem in the form that survives, and the meaning of $\ket x$ and $\ket p$. Everything in §§4–7 that used "the spectral theorem" is on credit until then.

That shape is not new. Chapter 0.4 built the space and Chapter 0.5 built the operators on it. Chapter 4.3 builds the space and Chapters 4.4 and 4.5 build the operators on it, in the same order and for the same reason.

9 · Two systems: the tensor product

Announce the destination. Everything so far has described one system. Two systems require a rule for combining their state spaces, and there are only two candidates: the dimensions add, or the dimensions multiply. We build the object in which they multiply out of Chapter 0.4's basis and dimension, and it turns out to be a basis of pairs and nothing more. We postulate that it is the right one, and then we count. The counting is what makes entanglement a fact about arithmetic rather than a mystery, and it is the whole of what Chapter 4.19 needs from this chapter.

9.1 · A basis of pairs

Let system $A$ have state space $V$ with orthonormal basis $\{\ket{a_{1}},\dots,\ket{a_{m}}\}$ and system $B$ have $W$ with $\{\ket{b_{1}},\dots,\ket{b_{n}}\}$. Whatever the joint space is, it must contain a state for each way of specifying both systems separately: $A$ in $\ket{a_{i}}$ and $B$ in $\ket{b_{j}}$. Write that state $\ket{a_{i}}\otimes\ket{b_{j}}$, abbreviated $\ket{a_{i}b_{j}}$, and there are $mn$ such pairs.

Now use P1. If those $mn$ states are all possible states of the pair, then by the superposition principle so is every complex combination of them, and the joint space contains their span. Declare them to be an orthonormal basis of that span. That declaration is the definition of the tensor product $V\otimes W$, and by Chapter 0.4 §2.2's definition of dimension it gives

The inner product comes with it, and nothing new has to be chosen. Declare $\avg{a_{i}b_{j}\,|\,a_{k}b_{l}}=\delta_{ik}\delta_{jl}$ on the basis and extend by conjugate-linearity in the first slot and linearity in the second, which is Chapter 0.5 §1's construction unchanged. For a general pair of states $\ket\phi=\sum_{i}\phi_{i}\ket{a_{i}}$ and $\ket\chi=\sum_{j}\chi_{j}\ket{b_{j}}$ this gives

the second identity following by expanding both sides on the basis and collapsing with the deltas. Amplitudes for independent systems multiply, which is what they had better do.

9.2 · Entanglement, as a dimension count

Ask how many of the $mn$ dimensions are reached by states of the form $\ket\phi\otimes\ket\chi$. Count parameters. Choosing $\ket\phi$ takes $m$ complex numbers and $\ket\chi$ takes $n$, but rescaling $\ket\phi$ by $c$ and $\ket\chi$ by $1/c$ gives the same product, so one complex parameter is shared. The set of product vectors therefore has complex dimension $m+n-1$, inside a space of complex dimension $mn$. For two two-level systems that is $3$ inside $4$. For two ten-level systems it is $19$ inside $100$.

So almost every state of a composite system is not a pair of states of its parts. Such states are called entangled, and the word names a counting fact rather than an influence: the joint space has more directions in it than the product construction reaches.

One explicit case, checkable by hand in three lines. Take two two-level systems and the state $\tfrac{1}{\sqrt2}\big(\ket{00}+\ket{11}\big)$. If it were $\big(a\ket0+b\ket1\big)\otimes\big(c\ket0+d\ket1\big)=ac\ket{00}+ad\ket{01}+bc\ket{10}+bd\ket{11}$, then matching coefficients requires

From $ad=0$ either $a=0$, which contradicts $ac=1/\sqrt2$, or $d=0$, which contradicts $bd=1/\sqrt2$. No solution exists. The state assigns no state at all to either half on its own, and the question "what is the first system's state?" has no answer of the kind P1 provides. That is why Chapter 4.19 has to build a different object, the density operator, to answer it.

9.3 · The size of the space, with a number

Iterating (4.2.27) over $N$ two-level systems gives a state space of dimension $2^{N}$, so specifying a general state takes $2^{N}$ complex amplitudes, while specifying a state of each system separately takes $2N$ real angles by (4.2.5). At $N=300$:

The observable universe contains of order $10^{80}$ atoms. Three hundred two-level systems means three hundred atoms, which is a small molecule. They have more amplitudes in their joint description than there are atoms available to record them, and the excess is entirely the gap between $mn$ and $m+n-1$, compounded three hundred times. That gap is what makes simulating quantum systems on ordinary computers hard, what makes building quantum ones interesting, and what Chapter 4.20 measures with Bell's inequality. It follows from P7 and nothing else.

a natural place to stop  ·  the framework is complete; what follows is three systems worked in full

10 · Three two-state systems, in full

Announce the destination. Chapter 0.5's Worked example 1 diagonalised one $2\times2$ Hermitian matrix completely and then said what it was: "It is the ammonia molecule… it is neutrino oscillation… it is the qubit. Chapter 4.2 will do all three, and the linear algebra will already be finished." Here it is done. We solve the matrix once with §7's evolution operator, and obtain a single formula. Then we read that formula three times, in three sets of units: picoseconds, kilometres, nanoseconds. The linear algebra takes half a page because it was finished nine chapters ago. The rest is identification and arithmetic.

10.1 · One matrix, solved once

Two states, called $\ket1$ and $\ket2$, whatever they turn out to be physically. The most general Hermitian operator on their span has four real parameters. One of them, the phase of the off-diagonal entry, can be absorbed into the definition of $\ket2$, so take it real. Write the remaining three as a mean, a difference and a coupling:

Chapter 0.5's Worked example 1 is the case $\delta=0$. The eigenvalues come from the characteristic polynomial, $(\bar E+\delta-\lambda)(\bar E-\delta-\lambda)-\Delta^{2}=0$, giving

Real, as Chapter 0.5 §6.1 requires, and the level splitting is $\Delta E=2R$. Notice the shape. The coupling $\Delta$ pushes the two levels apart whatever the diagonal difference is, and the splitting is never smaller than $2\Delta$. Two levels that would cross as $\delta$ passes through zero instead approach to $2\Delta$ and separate again. That is an avoided crossing, and the phenomenon is entirely (4.2.32).

Define the mixing angle $\theta$ by

which is legitimate because $\cos^{2}2\theta+\sin^{2}2\theta=(\delta^{2}+\Delta^{2})/R^{2}=1$. Then the normalised eigenvectors are

orthogonal by inspection, as Chapter 0.5 §6.2 guarantees without being consulted. The grind box verifies them and runs §7's evolution operator on the state that starts as $\ket1$. The result is one formula, and it is the whole of this section:

with $P_{1\to1}=1-P_{1\to2}$ exactly. Four features are worth naming before the physics starts, because each is one of this chapter's postulates doing visible work.

The mean energy $\bar E$ does not appear. It cancelled, because it contributes an overall factor $\ee^{-\ii\bar Et/\hbar}$ to the state. That is a global phase, which P1 says is not there. Only energy differences are observable, and that is P1 rather than a separate principle.

The oscillation never stops and never damps. By §7's familiar box, the eigenvalues of the generator are purely imaginary, so nothing decays. A classical two-state rate process would relax to fifty-fifty and stay there.

Depth and speed move in opposite directions. Increasing the detuning $\delta$ at fixed coupling makes $\sin^{2}2\theta$ smaller, which is a shallower oscillation, and $R$ larger, which is a faster one.

And at short times the two effects cancel exactly. Expanding (4.2.35) for small $t$, the $\sin^{2}$ contributes $R^{2}t^{2}/\hbar^{2}$ and the prefactor contributes $\Delta^{2}/R^{2}$, so

and the leading term does not contain $\delta$ at all. However far off resonance the system is driven, the transition probability leaves zero along the same parabola. Detuning changes when the curve turns over, not how fast it starts. That is the seed of the transition-rate formula Chapter 4.17 derives, and the figure below makes it visible.

Grind box — the eigenvectors, the evolution, and $P_{1\to2}$

Step 1 · check the eigenvectors. Subtract $\bar E\hat I$, which shifts both eigenvalues and moves no eigenvector. Using (4.2.33),

$$\big(\hat H-\bar E\hat I\big)\ket{+} = \begin{pmatrix}\delta&\Delta\\ \Delta&-\delta\end{pmatrix}\begin{pmatrix}\cos\theta\\ \sin\theta\end{pmatrix} = \begin{pmatrix}\delta\cos\theta+\Delta\sin\theta\\ \Delta\cos\theta-\delta\sin\theta\end{pmatrix} = R\begin{pmatrix}\cos\theta\\ \sin\theta\end{pmatrix},$$

the last equality being the two identities $\cos2\theta\cos\theta+\sin2\theta\sin\theta =\cos\theta$ and $\sin2\theta\cos\theta-\cos2\theta\sin\theta=\sin\theta$, both instances of $\cos(2\theta-\theta)$ and $\sin(2\theta-\theta)$. The same computation with $\ket-$ gives $-R$.

Step 2 · invert. (4.2.34) is a rotation by $\theta$, so its inverse is a rotation by $-\theta$:

$$\ket1 = \cos\theta\,\ket+ \;-\; \sin\theta\,\ket-, \qquad \ket2 = \sin\theta\,\ket+ \;+\; \cos\theta\,\ket-.$$

Step 3 · evolve. By §7, each eigenstate picks up its own phase and nothing else:

$$\ket{\psi(t)} = \ee^{-\ii\bar Et/\hbar}\Big(\cos\theta\,\ee^{-\ii Rt/\hbar}\ket+ \;-\; \sin\theta\,\ee^{+\ii Rt/\hbar}\ket-\Big).$$

Step 4 · project. By P3 the amplitude to find $\ket2$ is $\avg{2|\psi(t)}$, and Step 2 gives $\avg{2|+}=\sin\theta$, $\avg{2|-}=\cos\theta$:

$$\avg{2|\psi(t)} = \ee^{-\ii\bar Et/\hbar}\sin\theta\cos\theta\Big(\ee^{-\ii Rt/\hbar}-\ee^{+\ii Rt/\hbar}\Big) = -\ii\,\ee^{-\ii\bar Et/\hbar}\,\sin2\theta\,\sin\!\frac{Rt}{\hbar},$$

using $\ee^{-\ii u}-\ee^{\ii u}=-2\ii\sin u$ from Euler's formula and $2\sin\theta\cos\theta=\sin2\theta$. Squaring the modulus kills both the global phase and the $-\ii$, leaving (4.2.35). $\blacksquare$

Verified symbolically. Forming $\ee^{-\ii\hat Ht/\hbar}$ directly from (4.2.31) and taking the modulus squared of its entries returns $\Delta^{2}\sin^{2}(Rt/\hbar)/R^{2}$ for the off-diagonal transition, and gives $P_{1\to1}+P_{1\to2}-1=0$ identically in $\delta,\Delta,t$. Not to some precision. Identically. The series expansion of the same expression returns (4.2.36) term by term.

10.2 · The ammonia molecule

Ammonia is a nitrogen atom and three hydrogens arranged as a shallow pyramid. The nitrogen can sit above the plane of the hydrogens or below it, and these two arrangements are mirror images with identical energy. Call them $\ket L$ and $\ket R$. Two facts fix (4.2.31) completely.

The detuning is zero, by symmetry. Reflecting the molecule exchanges $\ket L$ and $\ket R$ and changes no energy, so the two diagonal entries of $\hat H$ are equal and $\delta=0$. This is not an approximation or a convenient choice. It is a symmetry of the Coulomb interaction between the same four nuclei and the same ten electrons. Hence by (4.2.33), $\theta=\pi/4$ and $\sin^{2}2\theta=1$, and the oscillation goes all the way from one configuration to the other and back.

The coupling is not zero. Classically the nitrogen cannot get from one side to the other, because passing through the plane of the hydrogens costs about $2020\ \mathrm{cm^{-1}}$, which is $0.2504\ \mathrm{eV}$, far more than the molecule has. The off-diagonal entry $\Delta$ of a Hermitian operator connecting $\ket L$ and $\ket R$ is under no such prohibition. Nothing in P1 to P5 requires a state to get from one place to another through the intervening places, and Chapter 4.7 computes $\Delta$ for a barrier. Here it is read off a measurement.

So the energy eigenstates are the symmetric and antisymmetric combinations, $\ket\pm=(\ket L\pm\ket R)/\sqrt2$, split by $\Delta E=2\Delta$, and

We want a number out of that rather than a shape, so put the measured $\nu_{0}=23.8701\ \mathrm{GHz}$ into it. The splitting is

and the time for the molecule to turn itself inside out and back is $1/\nu_{0}=41.89\ \mathrm{ps}$, with the halfway point, nitrogen fully on the other side, at $20.95\ \mathrm{ps}$. Three consequences follow, and each of them is a postulate of this chapter made visible.

A molecule in its ground state has no shape. The stationary states are $\ket\pm$, not $\ket L$ or $\ket R$. A state of definite shape is a superposition of the two energy levels, and therefore not stationary. This is §4.3 in the most concrete form available. Shape and energy are represented by operators that do not commute, so there is no state with both sharp, and the molecule in its lowest energy level is not in either configuration.

The splitting is tiny because the barrier is large. The ratio of barrier to splitting is $0.2504\ \mathrm{eV}/98.72\ \mu\mathrm{eV}=2536$. That the coupling is suppressed by three orders of magnitude rather than being zero is the quantitative content of tunnelling, and Chapter 4.10's WKB approximation computes the suppression as an exponential in the barrier's width and height.

The frequency lands in the microwave band, which is why the maser came first. The wavelength is $c/\nu_{0}=1.2559\ \mathrm{cm}$. Chapter 4.1's Worked example 3 derived the ratio of stimulated to spontaneous emission for an atom in thermal radiation as $1/(\ee^{h\nu/k_{B}T}-1)$. At $23.8701\ \mathrm{GHz}$ and $300\ \mathrm{K}$, $h\nu/k_{B}T=3.819\times10^{-3}$, and the ratio is

so stimulated emission beats spontaneous emission by a factor of $260$ at room temperature at this frequency, against $10^{-33}$ for an optical transition. Now sort the two energy eigenstates with an inhomogeneous electric field, which works because $\ket\pm$ have opposite parity and so respond oppositely. Feed the upper one into a cavity tuned to $1.26\ \mathrm{cm}$, and the third process Einstein was forced to invent in Chapter 4.1 §4.5 amplifies. That is the ammonia maser, built in 1954, and every number in its design is in this paragraph.

10.3 · Neutrino oscillation

The same matrix, and the identification is where all the work is. A neutrino is produced by a weak interaction in a definite flavour, with a muon or with an electron, and it is detected the same way. But flavour is not what propagation cares about. Propagation is generated by $\hat H$, whose eigenstates are the states of definite mass.

Call the flavour states $\ket{\nu_{\mu}}$ and $\ket{\nu_{\tau}}$, and the mass states $\ket{\nu_{1}}$ and $\ket{\nu_{2}}$. The assertion tested by every oscillation experiment is that these are two different orthonormal bases of the same two-dimensional space, related by (4.2.34) for some angle $\theta$. Nothing about that is unusual. It is Chapter 0.4's change of basis, and §4.3 of this chapter says the two observables do not commute.

Now the energies, and this is the one step needing Part II. A neutrino of definite mass $m_{i}$ and momentum $p$ has, by Chapter 2.5 §4.3, energy $E_{i}=\sqrt{p^{2}c^{2}+m_{i}^{2}c^{4}}$. Neutrinos in these experiments carry energies in the MeV to GeV range, while their masses are below an electronvolt, so $m_{i}c^{2}\ll pc$ and the square root can be expanded. We want the difference of two nearly equal energies, and the leading terms will cancel, which is exactly why expanding is the right move:

using $\sqrt{1+u}=1+u/2+O(u^{2})$ from Chapter 0.3. The common $pc$ cancels in the difference, and with $E\approx pc$ for the beam energy,

That is the level splitting in (4.2.35). One more identification is needed. A neutrino travels at essentially $c$, so the proper substitution for the elapsed time is $t=L/c$, where $L$ is the distance from source to detector. That distance is the quantity an experiment actually controls. Substituting both into (4.2.35):

which is the formula every oscillation experiment is analysed with, obtained from (4.2.35) by two substitutions and no new physics. Now convert the phase to the units experiments use, with $\Delta m^{2}$ in $\mathrm{eV^{2}}$, $L$ in kilometres and $E$ in GeV. Using $\hbar c=1.973269804\times10^{-7}\ \mathrm{eV\,m}$,

the familiar coefficient, derived rather than quoted. Next we want the distance over which the pattern repeats. That is the oscillation length, the analogue of ammonia's $41.89\ \mathrm{ps}$, and it is the $L$ making the phase advance by $\pi$:

Numbers, for the T2K experiment. A muon-neutrino beam is made at Tokai and detected at Kamioka, $L=295\ \mathrm{km}$ away, with the beam tuned to $E\approx0.600\ \mathrm{GeV}$. With $\Delta m^{2}_{32}=2.45\times10^{-3}\ \mathrm{eV^{2}}$ the phase is $1.26693\times2.45\times10^{-3}\times295/0.600=1.5261\ \mathrm{rad}$, which is $0.486\pi$, within three per cent of $\pi/2$, the first oscillation maximum. So

for maximal mixing: essentially every muon neutrino in the beam has become something else by the time it arrives. The oscillation length is $L_{\text{osc}}=2.4797\times0.600/2.45\times10^{-3}=607.3\ \mathrm{km}$, so the first minimum sits at $303.6\ \mathrm{km}$ against a baseline of $295\ \mathrm{km}$. Equivalently, $295\ \mathrm{km}$ is exactly the first minimum for a beam energy of $0.583\ \mathrm{GeV}$. The beam energy was chosen to put the detector at the minimum, and (4.2.42) is how it was chosen.

The same formula with a small mixing angle gives the other kind of experiment. At the Daya Bay reactor, electron antineutrinos of about $4\ \mathrm{MeV}$ are counted at $L=1.65\ \mathrm{km}$. The phase is $1.26693\times2.5\times10^{-3}\times1.65/0.004=1.307\ \mathrm{rad}$, so with $\sin^{2}2\theta_{13}=0.085$ the predicted disappearance is $0.085\times\sin^{2}(1.307)=0.079$. That is a deficit of about eight per cent, which is what is measured, and it is how $\theta_{13}$ is known.

One honesty note about the derivation. Treating the two mass states as plane waves with a common momentum, and setting $t=L/c$, is a shortcut. A neutrino is produced as a localised wave packet, the two mass components travel at slightly different speeds, and the careful treatment follows the packets and asks when they still overlap at the detector. It gives (4.2.42) in every regime these experiments operate in, and the corrections are suppressed by the ratio of the packet width to $L_{\text{osc}}$. The shortcut is standard, and it is a shortcut. The machinery to do it properly is Chapter 4.6's wave packets.

10.4 · The qubit

The third reading inverts the relationship between the formula and the apparatus. In ammonia the parameters are whatever chemistry supplies. In a neutrino they are whatever the mass matrix supplies. In an engineered two-level system both are set by the designer, and (4.2.35) becomes a specification rather than a prediction.

Take a superconducting circuit with two levels split by $\hbar\omega_{0}$, with $\omega_{0}/2\pi=5\ \mathrm{GHz}$. That is a splitting of $20.68\ \mu\mathrm{eV}$, which is twelve times $k_{B}T=1.72\ \mu\mathrm{eV}$ at the $20\ \mathrm{mK}$ such circuits run at. So Chapter 0.6's Boltzmann factor $\ee^{-\Delta E/k_{B}T}=6\times10^{-6}$ leaves the system in its lower level rather than thermally stirred.

Now drive it with a microwave field at exactly $\omega_{0}$. Chapter 4.17 shows that in a frame rotating with the drive the problem becomes (4.2.31) with $\delta=0$ and $\Delta=\hbar\Omega_{R}/2$, where $\Omega_{R}$ is proportional to the drive amplitude. That reduction is Chapter 4.17's and is not derived here. With it granted, (4.2.35) reads

the Rabi formula. With $\Omega_{R}/2\pi=25\ \mathrm{MHz}$ the full period is $40\ \mathrm{ns}$, so a pulse lasting

takes $\ket0$ to $\ket1$ with probability $1$. That is a NOT gate, performed by leaving the drive on for exactly half an oscillation. Half that, $10\ \mathrm{ns}$, gives probability $\half$ and produces the operator whose square is NOT. Worked example 3 builds it and shows that no classical stochastic process has such a square root. Detuning the drive off resonance is $\delta\neq0$, and (4.2.35) says the gate then fails to reach probability $1$ no matter how long it is left on, with the shortfall $1-\sin^{2}2\theta=\delta^{2}/(\delta^{2}+\Delta^{2})$. That is where the tolerance on a control line comes from.

Set the three side by side. Same matrix, same formula, three sets of units:

	$\ket1,\ket2$ are	$\delta$	Splitting $\Delta E$	Full period of $P_{1\to2}$
Ammonia	nitrogen above / below the hydrogens	$0$, by mirror symmetry	$h\times23.8701\ \mathrm{GHz}=98.72\ \mu\mathrm{eV}$	$41.89\ \mathrm{ps}$
Neutrino	the two detectable flavours	set by the mass matrix	$\Delta m^{2}c^{4}/2E=2.04\ \mathrm{peV}$	$607.3\ \mathrm{km}$ of flight
Qubit	the two engineered levels	drive detuning, chosen	$\hbar\Omega_{R}=0.1034\ \mu\mathrm{eV}$	$40.0\ \mathrm{ns}$

The splittings span a factor of $4.8\times10^{7}$, from $98.72\ \mu\mathrm{eV}$ down to $2.04\ \mathrm{peV}$, and the periods span exactly the same factor the other way, because (4.2.35) makes period and splitting reciprocal and nothing else enters. The linear algebra is one $2\times2$ matrix that Chapter 0.5 diagonalised before any of this was mentioned.

10.5 · One computed evolution, read in three sets of units

The figure below integrates (4.2.18) numerically for (4.2.31) and plots what comes out. Nothing in it uses (4.2.35). The closed form appears only in the readout, as the thing being checked against. Two of the readouts are the numerical confirmation this chapter owes.

detuning δ / Δ

0.00

cycles shown

2.00

global phase α

0.000 pi

ammonia & maximal mixing (δ = 0)

reactor θ₁₃: sin²2θ = 0.085

badly detuned drive

normalisation: max | |c1|^2 + |c2|^2 - 1 | = 5.22e-15 integration against the closed form: max difference = 8.88e-15 (nothing plotted used the closed form)

measured from the curve: period = 1.0000000 pi, amplitude = 1.0000000 formula: pi/sqrt(1+d^2) = 1.0000000 pi, sin^2 2theta = 1/(1+d^2) = 1.0000000

the same curve, three sets of units: ammonia 41.893 ps neutrinos 607.27 km of flight qubit 40.00 ns (half of each is a complete transfer)

global phase alpha = 0.000 pi: the plotted curve differs from the alpha = 0 run by at most 0.00e+0 -- P1, measured

11 · Worked examples

12 · Your turn