States and Probability Amplitudes

States and probability amplitudes¶

Learning objectives¶

After this chapter, you should be able to:

• represent a spin-$\tfrac12$ state using kets and column matrices;

• expand a state in any orthonormal basis;

• normalize a state and apply the Born rule;

• calculate selected routes through sequential analyzers;

• distinguish relative phase from overall phase;

• translate among the $x$, $y$, and $z$ spin bases;

• distinguish a coherent pure state from a classical mixture;

• in the optional extension, construct spinors for an arbitrary analyzer direction; and

• add amplitudes for coherent alternatives and probabilities for exclusive recorded routes.

The subsections labeled “optional extension” provide compact matrix machinery for readers ready to go further; the core conceptual path does not depend on them.

2.1 States are vectors¶

Chapter 1 ended with a wish list: whatever mathematical object represents a spin state, it needs two mutually exclusive outcomes for every analyzer direction, a way to represent a preparation that is certain for one of them, continuous families of other preparations, and phase information that can produce interference. An ordinary table of probabilities cannot do all of this at once, because it has no room for phase. Complex vectors can, and this chapter makes that idea precise.

The notation $|\cdot\rangle$, read “ket,” simply names a vector in that complex vector space. There is no more mystery in the symbol than in writing $\vec v$ for an arrow in ordinary space—it is just a label, chosen to remind us that we are working with a state rather than an ordinary number.

The preparation called $z+$ is represented by the ket $|+z\rangle$, and the preparation $z-$ by $|-z\rangle$. These two state vectors form an orthonormal basis for the spin state space: every other spin-$\tfrac12$ preparation can be built from them by superposition, in exact analogy to how two perpendicular unit vectors span a plane. Any pure spin state can therefore be written

where $a$ and $b$ are complex numbers. Think of $a$ and $b$ as coordinates: they fix how much of $|+z\rangle$ and how much of $|-z\rangle$ go into building $|\psi\rangle$, in the same way that the numbers 3 and 4 fix an arrow once we agree on $x$- and $y$-axes. They are not themselves probabilities—Section 2.3 will show exactly how probabilities are built from them.

It is often convenient to write $|\psi\rangle$ as a column of numbers. Choosing the representation

gives $|\psi\rangle\doteq(a,b)^{\mathsf T}$. The symbol $\doteq$ means “is represented, in the chosen basis, by.” A ket is an abstract vector; a column matrix is only its representation once a basis has been chosen.

This distinction is the same one made for an ordinary geometric vector. The arrow in space is not identical to the list of three numbers used to represent it: rotating the coordinate axes changes the numbers but not the arrow itself. In exactly the same way, a spin ket does not change when we rewrite it in the $x$ basis instead of the $z$ basis—only its column of numbers changes.

Superposition is a physical principle¶

Because states are vectors, any nonzero linear combination of two allowed states can be normalized to represent an allowed state. This is the superposition principle, and it says something stronger than “the atom is secretly in one state or the other.” In

the complex coefficients preserve phase information that can affect a later measurement. Compare this with a classical mixture, in which a fraction $|a|^2$ of atoms is separately prepared in $z+$ and the rest in $z-$. That mixture and the superposition above generally make different predictions, even though both give the same probabilities for a $z$ measurement. The section on pure states and mixtures makes this comparison explicit.

2.2 Bras and inner products¶

A ket by itself cannot yet produce a number we could compare to laboratory data—it is a direction in an abstract space, not a prediction. To extract predictions, we pair every ket $|\psi\rangle$ with a companion object called a bra, written $\langle\psi|$. A bra combines with a ket to its right to produce a single complex number, called the inner product. This pairing is what lets us ask “how much of outcome $\phi$ is contained in state $\psi$?” and get back a definite amplitude. The bra corresponding to $|\psi\rangle$ is its Hermitian conjugate,

The $z$ basis is orthonormal, meaning each basis ket has unit length and the two are perpendicular in the sense of the inner product:

Using these two facts, pairing $\langle\psi|$ with $|\psi\rangle$ gives

A physical pure state is normalized, meaning $|a|^2+|b|^2=1$. If a nonzero vector $|v\rangle$ is not already normalized, we can always rescale it into a valid physical state by dividing by its length:

Example 2.1: normalization¶

Normalize $|v\rangle=2|+z\rangle+(1+i)|-z\rangle$. Applying equation (9) term by term, the first coefficient contributes $|2|^2=4$ and the second contributes $|1+i|^2=(1+i)^*(1+i)=(1-i)(1+i)=1+1=2$, so the squared norm is $\langle v|v\rangle=4+2=6$, hence

Algebra check¶

If $|\chi\rangle=(|+z\rangle-i|-z\rangle)/\sqrt2$, what are $\langle\chi|$ and $\langle\chi|\chi\rangle$?

Inner-product habits¶

Three small algebraic rules prevent most early errors. First, swapping the order of an inner product conjugates it:

Second, a bra acts linearly on a ket to its right:

Third, the inner product is conjugate-linear in its first slot. If a ket is combined first and then converted into a bra, its coefficients are conjugated:

For example, if $|\phi\rangle=(|+z\rangle+i|-z\rangle)/\sqrt2$, then $\langle\phi|=(\langle+z|-i\langle-z|)/\sqrt2$. Once a bra has already been written as a row, ordinary row–column multiplication is linear in the entries of that row. Keeping these two statements separate prevents the common error of forgetting conjugation when forming the bra.

Complex numbers as amplitude geometry¶

Complex numbers enter quantum mechanics because an amplitude needs to carry both a magnitude and a phase at once, and a single real number cannot do that. In polar form,

Multiplying two amplitudes multiplies their magnitudes and adds their phases. Adding two amplitudes is ordinary geometric addition in the complex plane—and this is the key fact that makes interference possible. Because vectors in a plane can point in opposite directions, two nonzero amplitudes can add to zero. For example,

even though the squared magnitude of each term separately is $1/4$. This possibility—two nonzero contributions canceling exactly—is the algebraic core of destructive interference.

For arbitrary amplitudes $A_1$ and $A_2$, expanding $|A_1+A_2|^2$ directly gives

The final term is the interference term. It depends on the relative phase between $A_1$ and $A_2$, and it can be positive, negative, or zero. If the two alternatives are instead recorded separately, or their relative phase is randomized by the apparatus, this cross term averages to zero and only $|A_1|^2+|A_2|^2$ remains—the ordinary rule for adding probabilities of mutually exclusive events.

2.3 The Born rule¶

We now have the machinery to turn a state into an actual prediction. The amplitude for obtaining $z+$ from $|\psi\rangle$ is $\langle+z|\psi\rangle=a$—simply the coefficient of $|+z\rangle$ in the expansion of $|\psi\rangle$. The probability of that outcome is the squared magnitude of that amplitude:

This rule for turning an amplitude into a probability is the Born rule, and it is one of the central postulates of quantum mechanics. Notice that the order matters in the amplitude: the bra labels the possible measurement outcome, and the ket labels the prepared state.

Because the $z$ basis spans the state space, the two overlaps recover the whole ket:

Comparing this with equation (4) shows that $a$ and $b$ are precisely the amplitudes $\langle+z|\psi\rangle$ and $\langle-z|\psi\rangle$. Chapter 3 will package this basis expansion into an identity operator after operators have been introduced.

A probability calculation in four steps¶

For any ideal spin measurement, the same workflow applies:

1. Write the prepared ket and the outcome ket in a common basis.

2. Hermitian-conjugate the outcome ket to make the bra.

3. Form the amplitude $\langle\text{outcome}|\text{prepared}\rangle$.

4. Take its squared magnitude, and check that all exclusive outcome probabilities sum to one.

The amplitude computed in step 3 is generally complex, and there is no requirement that it lie between zero and one—only its squared magnitude in step 4 is a probability. Keeping the amplitude, rather than squaring it immediately, is essential whenever the alternative will later be recombined with another one.

Concept check 2.1¶

After computing $\langle\phi|\psi\rangle=-i/2$, a student reports a probability of $-i/2$. What should the student report instead?

2.4 Changing the question means changing the basis¶

Every measurement so far has used the $z$ basis, but nothing forces that choice—an analyzer built along $x$ or $y$ asks a different physical question, and it needs its own basis kets. The 50–50 result alone fixes the magnitudes of the two $z$-basis coefficients but not their relative phases. The formulas below therefore contain two ingredients: equal magnitudes demanded by the experiment, and a standard phase convention that ties the labels $x$ and $y$ to perpendicular directions in physical space. Chapter 3 will recover the same kets as eigenvectors of the spin-component matrices. With that convention, the $x$ basis is

And the $y$ basis is

Example 2.2: a relative phase becomes a definite result¶

Prepare

and ask for the probability of $y+$. From equation (21),

Therefore

so $P(y+)=1$ and $P(y-)=0$. The state is exactly $|+y\rangle$, written in the $z$ basis. This deliberately simple example exposes the important algebra: the $i$ in the ket becomes $-i$ in the bra, and $(-i)i=1$. A missed complex conjugation would destroy the certainty that defines the $y+$ state.

Let’s check that this reproduces the known result. Applying the Born rule to a $z+$ input,

exactly as observed in Chapter 1. To express any state $|\psi\rangle$ in the $x$ basis instead of the $z$ basis, project it onto the two $x$-basis kets:

It is important to see what has and has not changed here. No physical state has changed—the atom’s preparation is exactly what it was. Only the coordinates used to describe that state have changed, in the same way that rewriting an arrow’s components in a rotated coordinate frame does not move the arrow.

Optional extension: basis-change matrices¶

The inner-product method above is the core skill. Readers who want a compact matrix version can organize all the overlaps into one basis-change matrix.

The same calculation can be organized as a single matrix multiplication, which is often more convenient for bookkeeping. If $[\psi]_z=(a,b)^{\mathsf T}$ is the column representing $|\psi\rangle$ in the $z$ basis, its $x$-basis column is

Each row of this matrix is one of the $x$-basis bras, written out in $z$ coordinates. A matrix $U$ is called unitary when $U^\dagger U=UU^\dagger=I$; this condition guarantees that lengths and inner products are preserved. The change-of-basis matrix is unitary, so a properly normalized state stays normalized after the change of basis. Chapter 3 will develop operators and unitary matrices systematically. Reading the entries here as inner products also makes clear that a basis change is not a physical Stern–Gerlach measurement—no outcome has been selected, and the abstract ket itself has not been disturbed in any way.

Quick calculation: carrying out the matrix multiplication¶

It is worth performing this multiplication by hand once, so that equation (27) reads as an ordinary matrix–vector product—row dotted into column, exactly as in the vector review of Section 2.1—rather than as an unfamiliar piece of bra–ket bookkeeping. Take $|\psi\rangle=(|+z\rangle+i|-z\rangle)/\sqrt2$, so $[\psi]_z=\frac{1}{\sqrt2}\binom1i$. Multiplying,

The top entry comes from dotting the row $(1,1)$ into the column $(1,i)$, giving $1+i$; the bottom entry comes from dotting $(1,-1)$ into $(1,i)$, giving $1-i$; each is then scaled by the $\frac1{\sqrt2}\cdot\frac1{\sqrt2} =\frac12$ collected from the two prefactors. As a check, normalization must survive the change of basis:

confirming directly, by arithmetic rather than by appeal to unitarity alone, that the $x$-basis column is a properly normalized state.

Example 2.3: probabilities in another basis¶

For

the $x+$ amplitude is

Therefore $P(x+)=\tfrac12$ and $P(x-)=\tfrac12$. The same state has unequal $z$ probabilities, $3/4$ and $1/4$. Probabilities belong to a state–measurement pair, not to the state alone.

2.5 Sequential measurements with amplitudes¶

We can now redo the analyzer chains of Chapter 1 using the vector formalism, and see that it reproduces the same answers by a more systematic route. If $+a$ is prepared, $+b$ is selected, and $+c$ is finally detected, the probability for that selected route is

The two factors multiply because the first measurement prepares $|+b\rangle$, and it is that state, not the original $|+a\rangle$, which enters the second stage. It would be incorrect to replace this product with $|\langle+c|+a\rangle|^2$: the intervening selected measurement genuinely changes the state, and the calculation must reflect that.

Example 2.4: a three-stage analyzer problem¶

A $z+$ beam enters $\mathrm{SG}_x$. The $x+$ output is retained and sent into $\mathrm{SG}_y$. What fraction of the original beam emerges as $y-$?

The route probability is

Using the basis kets,

Both squared magnitudes are $1/2$, so one quarter of the original $z+$ beam reaches the $y-$ detector. Notice that the second amplitude is complex even though its probability is simple.

2.6 Relative and overall phase¶

Not every phase in a ket matters physically, and it is important to know which ones do. Multiplying an entire ket by an overall phase does not change any probability at all:

Since $|e^{i\gamma}|^2=1$, the kets $|\psi\rangle$ and $e^{i\gamma}|\psi\rangle$ represent exactly the same physical state, for any choice of $\gamma$.

A relative phase—one that appears on only one term of a superposition, not on the whole ket—is entirely different, and it does change interference. Compare

Concept check 2.2¶

Do $|+y\rangle$ and $|-y\rangle$ differ only by an overall phase?

Both $|+x\rangle$ and $|-x\rangle$ give 50–50 probabilities in the $z$ basis—the two states look identical if all we ever do is measure $S_z$. Yet a relative sign is all that separates them, and an $S_x$ measurement distinguishes them perfectly. This is the essential fact about relative phase: it can be completely invisible to one choice of measurement and yet fully visible to another.

More generally, consider the family of states

parameterized by a single relative phase $\phi$. Its $x+$ probability is

Concept check 2.3¶

A normalized ket has $P(z+)=P(z-)=1/2$. Is the state determined uniquely?

Every member of this family has exactly the same $z$ statistics—a 50–50 split, regardless of $\phi$. Rotating the final analyzer to $x$ is what converts the otherwise hidden relative phase into a visible count rate. Interference, seen this way, is not some extra phenomenon bolted onto the vector formalism; it is simply what ordinary basis projection does to complex coefficients once more than one term is present.

2.7 A complete coherent path calculation¶

We can now close the experimental loop opened in Chapter 1. Consider an ideal device that splits $z+$ into coherent $x+$ and $x-$ paths and then implements the inverse routing operation, returning both paths to the same spatial and motional state. With no uncontrolled path-dependent phase, the amplitude for each final outcome is the sum of the two indistinguishable route amplitudes:

The two route amplitudes are each $1/2$, so

For the other exit,

The mathematics describes the ideal apparatus specified above; merely writing a sum over intermediate states would not prove that an arbitrary physical beam splitter preserves coherence or recombines every other degree of freedom.

Concept check 2.4¶

Both route amplitudes in equation (41) have squared magnitude $1/4$. Why is the final probability zero rather than $1/2$?

The cancellation is valid only because the intermediate $x$ result was never recorded. If it is recorded, the two complete routes become exclusive, and we add their probabilities:

The same route magnitudes produce certainty or a 50–50 split depending on whether the intermediate alternatives remain coherent.

Concept check 2.5¶

Does changing the phase convention of $|-x\rangle$ change the interference prediction?

A controllable phase¶

Suppose one path adds a phase $e^{i\delta}$ before recombination, perhaps because it travels a slightly longer distance. Repeating the calculation,

so that

At $\delta=0$, the original $z+$ preparation is recovered. At $\delta=\pi$, the two exits exchange roles and $z-$ occurs with certainty. In between, a continuously adjustable phase controls discrete detector outcomes through continuously changing ensemble frequencies.

The calculation supplies a reusable rule:

1. identify each complete, indistinguishable route to one final record;

2. multiply amplitudes along each route;

3. include every transformation phase;

4. add the route amplitudes; and

5. square the magnitude only after the coherent sum is complete.

If a record distinguishes routes, group together only routes that remain indistinguishable to that record, and add probabilities between the resulting exclusive groups.

2.8 The Bloch-sphere coordinates¶

After normalizing a state and discarding its irrelevant overall phase, every pure spin-$\tfrac12$ state can be written in the standard form

with $0\leq\theta\leq\pi$ and $0\leq\phi<2\pi$. Because only two real parameters remain once normalization and overall phase are fixed, the angles $(\theta,\phi)$ locate a point on an ordinary sphere—the Bloch sphere—and every possible pure spin state corresponds to exactly one point on it. The half-angle is essential here, not optional. The angles $(\theta,\phi)$ label a direction on the Bloch sphere; they should not be confused with a history of physical rotation undergone by the atom. Chapter 4 will show that when a representative spinor is physically rotated through $2\pi$, its ket acquires a minus sign. The corresponding physical ray is unchanged because the sign is an overall phase, though it can be observed relative to another coherent amplitude.

For a measurement along a unit vector $\mathbf n$ making angle $\theta$ with $+z$, equation (45) gives

This spin-$\tfrac12$ rule resembles Malus’s law for polarized light, but with one important difference: it uses half the geometric angle on the Bloch sphere, not the angle itself.

For two arbitrary pure states, let their Bloch angles be $(\theta_a,\phi_a)$ and $(\theta_b,\phi_b)$. Substituting equation (45) for both kets gives the amplitude

Taking its squared magnitude and using the half-angle identities produces

The expression in square brackets after the 1 is exactly $\mathbf a\cdot\mathbf b=\cos\alpha$, where $\alpha$ is the ordinary angle between the two analyzer directions. Therefore

This completes the derivation, promised in Chapter 1, of the empirical angle rule from state vectors. Notice a geometric subtlety worth pausing on: orthogonal directions in ordinary space ($\alpha=90^\circ$) do not correspond to orthogonal kets. They give an overlap probability of $1/2$, not zero. The state that is orthogonal to $|+\mathbf a\rangle$ is $|-\mathbf a\rangle$, represented by the antipodal, oppositely pointing point on the Bloch sphere.

Example 2.5: recover Bloch angles from a ket¶

Consider

Comparison with equation (45) gives $\cos(\theta/2)=\sqrt3/2$, so $\theta=\pi/3$. The relative phase is $e^{i\phi}=-i$, so one convenient choice is $\phi=3\pi/2$. The state therefore lies $60^\circ$ from $+z$ and points toward negative $y$.

2.9 Pure states and mixtures¶

Section 2.1 promised that a coherent superposition and a classical mixture are not the same thing, even when they agree on some measurements. We can now make that comparison precise. A pure state assigns one ket to every member of an ideally prepared ensemble. Compare the coherent state $|+x\rangle$ with a different preparation: flip a fair classical coin, and produce $z+$ on heads and $z-$ on tails. Both preparations give identical $z$ statistics,

An $x$ measurement, however, tells the two apart immediately. The coherent $|+x\rangle$ state gives $x+$ with certainty, by construction. For the coin-toss ensemble, half of the atoms began in $z+$ and half in $z-$; each of those subensembles independently gives 50–50 $x$ results, so the combined statistics also remain 50–50.

No single ket can represent that classical mixture—there is simply no linear combination of $|+z\rangle$ and $|-z\rangle$ that reproduces its statistics in every basis. The core operational lesson is therefore: identical probabilities for one measurement do not prove that two preparations are the same quantum state.

2.10 Optional extension: spinors for an arbitrary direction¶

The core chapter is complete. This optional extension packages an arbitrary analyzer orientation into a pair of spinors and a basis-change matrix.

The Bloch-sphere coordinates do more than just label a state—they also give us the eigenstates of an analyzer built along any direction we like. For an analyzer oriented along

a convenient phase convention gives its two outcome states as

A direct calculation shows that each ket has unit norm and that $\langle+\mathbf n|-\mathbf n\rangle=0$, so together they form a valid orthonormal basis, exactly as $|\pm z\rangle$, $|\pm x\rangle$, and $|\pm y\rangle$ did. The overall phase of either ket is conventional: multiplying $|+\mathbf n\rangle$ or $|-\mathbf n\rangle$ by its own overall phase changes its coordinates but no outcome probability. The relative phase written inside each ket, by contrast, is not a matter of convention once the $z$-basis convention has already been fixed.

Checking special cases first¶

Before applying equation (55) to an unfamiliar direction, it is a good habit to check that it reproduces the three directions already known, since a mistake in a general formula often shows up as soon as you specialize it. Setting $\theta=0$ (pointing along $+z$) gives $|+\mathbf n\rangle=|+z\rangle$, exactly as it should, since the formula’s $\phi$-dependent term is multiplied by $\sin(\theta/2)=0$. Setting $\theta=\pi/2,\ \phi=0$ (pointing along $+x$) gives $|+\mathbf n\rangle=(|+z\rangle+|-z\rangle)/\sqrt2=|+x\rangle$, matching equation (20). Setting $\theta=\pi/2,\ \phi=\pi/2$ (pointing along $+y$) gives $|+\mathbf n\rangle=(|+z\rangle+i|-z\rangle)/\sqrt2=|+y\rangle$, matching equation (21). With these three checks passed, the formula can be trusted for a direction that is not simply one of the six coordinate poles.

Example 2.6: an analyzer not in a coordinate plane¶

Let $\theta=2\pi/3$ and $\phi=\pi/2$. Then

A $z+$ input gives

The same result follows from the real-space angle rule: $\cos^2(\theta/2)=\cos^2(\pi/3)=1/4$. Notice that the azimuth $\phi$ does not affect a $z+$ input, because rotating the analyzer around $z$ leaves its angle from the preparation axis unchanged. It does affect inputs that carry their own transverse phase information.

The column matrix whose two columns are the two spinors above,

is unitary, and it converts coordinates between the analyzer basis and the $z$ basis. Its columns describe how the new basis vectors look in old coordinates, while $V_{\mathbf n}^\dagger$ performs the reverse change, from old coordinates to new. Keeping those two directions of conversion straight prevents a common basis-change error.

Summary¶

• A ket is an abstract state vector; its column depends on the chosen basis.

• Inner products produce amplitudes, and squared magnitudes produce probabilities.

• An orthonormal basis supplies expansion amplitudes through inner products.

• Overall phase is unobservable, while relative phase controls interference.

• Every pure spin-$\tfrac12$ state corresponds to a point on the Bloch sphere.

• A selected intermediate result resets the state used in subsequent amplitudes, while unobserved coherent alternatives must be added at the amplitude level.

• A coherent superposition and a classical mixture can agree in one basis and disagree in another.

• In the optional extension, arbitrary analyzer directions correspond to orthonormal spinors whose half-angle coordinates form a unitary basis-change matrix.

• Coherent routes are multiplied along each route and added across indistinguishable alternatives before the Born rule is applied.

Exercises¶

Exercises 1–7 establish the core notation and probability skills, 8–14 consolidate basis changes, phases, and mixtures, and 15–21 are extensions on arbitrary directions and coherent paths. Complete the first group before moving to the more algebraic problems.

1. Normalize $(3,4i)^{\mathsf T}$ and find the probabilities of $z+$ and $z-$.

2. Verify directly that the two states in equation (20) are normalized and orthogonal.

3. Write $|+z\rangle$ and $|-z\rangle$ in the $x$ basis.

4. Calculate all four quantities $|\langle\pm y|\pm x\rangle|^2$. Interpret the result as a sequence of analyzers.

5. For the state in Example 2.3, calculate $P(y+)$ and $P(y-)$. Explain why the relative phase matters here although it did not affect the $x$ probabilities.

6. Show that equation (45) is normalized. Identify the Bloch-sphere angles for the six states $|\pm x\rangle$, $|\pm y\rangle$, and $|\pm z\rangle$.

7. Give two normalized kets that have identical $z$ probabilities but are orthogonal to one another.

8. Prove that replacing both basis kets by phase-shifted basis kets changes the column coordinates but not any measurement probability.

9. Use equation (27) to express $(|+z\rangle+i|-z\rangle)/\sqrt2$ in the $x$ basis. Check the norm before and after the transformation.

10. A $y+$ beam enters $\mathrm{SG}_z$; the $z-$ output is retained and sent to $\mathrm{SG}_x$. Calculate the probability for every complete detected route from the original beam.

11. Derive equation (38) and find $P(x-)$ without doing a second inner-product calculation.

12. Find the Bloch angles $(\theta,\phi)$ for the normalized ket $(1,\sqrt3 i)^{\mathsf T}/2$. What spin direction does it represent?

13. Without using density operators, show that a coherent $|+x\rangle$ preparation gives $x+$ with certainty while a 50–50 classical mixture of $z+$ and $z-$ gives $P(x+)=1/2$. State where probabilities, rather than amplitudes, must be added in the mixture calculation.

14. Construct two physically different ensembles that give identical $z$ statistics. Specify one additional analyzer orientation that distinguishes them, and calculate the predicted probabilities.

15. Verify the normalization and orthogonality of the two spinors in equation (55).

16. Write $|-\mathbf n\rangle$ for $\theta=\pi/2$ and $\phi=\pi/2$. Identify it as one of the six coordinate-axis states, up to overall phase.

17. Verify by direct matrix multiplication that $V_{\mathbf n}^\dagger V_{\mathbf n}=\hat I$ for equation (59).

18. Starting from equation (17), let $A_1=a$ and $A_2=be^{i\delta}$ for real nonnegative $a$ and $b$. Find the largest and smallest possible total probabilities as $\delta$ varies.

19. Reproduce equations (40) and (41) with the $y$ basis inserted instead of the $x$ basis. Track every factor of $i$.

20. A phase shifter adds $e^{i\delta}$ to the $x-$ route of the recombiner. Predict both final probabilities for $\delta=\pi/2$, $2\pi/3$, and $\pi$.

21. A path marker leaves the $x+$ route unchanged but correlates the $x-$ route with a distinguishable marker state. Explain, without yet using composite state notation, why equations (44) no longer describe full-visibility fringes.

Selected exercise guidance¶

Use these answers only after making a complete attempt.