Time Evolution

Time evolution¶

Learning objectives¶

After this chapter, you should be able to:

• use the Hamiltonian to evolve a quantum state in time;

• construct and interpret the unitary time-evolution operator;

• distinguish stationary states from general superpositions;

• predict spin precession in a uniform magnetic field;

• connect relative phase evolution to changing measurement probabilities;

• compose piecewise evolution and measurement steps in a complete protocol;

• translate controlled field pulses into Bloch-sphere rotations; and

• predict a Ramsey interference signal.

Sections 4.1–4.7 form the core path through unitary evolution and spin precession. Sections 4.8–4.9 are mathematical extensions, while Sections 4.10–4.11 form an applications track on laboratory pulses and Ramsey interference. Those tracks may be assigned independently after the core.

4.1 The dynamical postulate¶

Every chapter so far has treated the spin state as something that only changes at the instant of a measurement. Between measurements, we have simply left the state alone. That gap needs to be filled: real experiments involve waiting time, traveling atoms, and applied fields, and the state must be doing something during all of that. This chapter supplies the missing piece. Between measurements, the state of an isolated system evolves according to the time-dependent Schrödinger equation,

The Hamiltonian $\hat H$ is the observable associated with energy, and it plays a second role here as the generator of time evolution: the object that tells the state how to change from one instant to the next. Equation (1) is a first-order differential equation for a vector, which means specifying the initial state $|\psi(0)\rangle$ and the Hamiltonian $\hat H(t)$ is enough to determine the state at every later time.

When the Hamiltonian does not depend on time, this differential equation has an explicit solution:

Check directly that this solves equation (1), rather than accepting it purely on authority. Differentiating the defining power series of $\hat U(t)$ term by term with respect to $t$,

because $\hat H$ commutes with every power of itself, so each term of the series differentiates exactly as an ordinary power of $t$ would. Multiplying both sides by $i\hbar$ and applying the result to $|\psi(0)\rangle$ reproduces equation (1) exactly. This calculation used only that $\hat H$ does not depend on time; nothing about spin specifically entered.

Because $\hat H$ is Hermitian, the operator $\hat U$ built from it is automatically unitary: $\hat U^\dagger\hat U=\hat I$. This single fact has an important consequence. It guarantees that a state’s normalization never drifts as it evolves:

So unitary evolution preserves normalization and, more generally, every inner product between two evolving states. Something here can feel paradoxical at first: within the ideal isolated-system model, this evolution is completely deterministic, even though the outcomes of a later projective measurement are probabilistic. The Schrödinger equation itself introduces no random step; the measurement postulate supplies probabilities when a measurement is made.

What an evolution problem is asking¶

Most two-state dynamics problems, however different they look on the surface, can be organized into the same five-step sequence:

1. Write the Hamiltonian and choose a basis in which to represent it.

2. Construct $\hat U(t)$ by using energy eigenstates, diagonalizing the matrix, or exploiting Pauli-matrix identities.

3. Apply $\hat U(t)$ to the initial ket.

4. Project the evolved ket onto the states of the measurement performed at time $t$.

5. Check normalization, the $t=0$ limit, and any conserved probabilities.

The distinction between steps 3 and 4 is the single most common source of error in this material, so it is worth stating plainly: the ket evolves continuously under $\hat U(t)$, but a probability is only ever defined once a measurement question has been specified. Two different choices of final analyzer can reveal very different time dependence from the very same evolving state, so step 4 cannot be skipped or guessed.

For a time-independent Hamiltonian, the exponential in step 2 can always be understood through its power series, which is most useful once $\hat H$ has been diagonalized. If $\hat H=\hat V\hat D\hat V^\dagger$, with $\hat D$ diagonal, then

In words: diagonalize the Hamiltonian, let each diagonal component evolve by its own simple phase, and transform back to the original basis. This is a completely general method for solving the Schrödinger equation, not a trick that happens to work only for spin.

4.2 Energy eigenstates and phases¶

Let $|E_n\rangle$ satisfy the time-independent eigenvalue equation

Because the evolution operator is built from the exponential of $\hat H$, it acts especially simply on one of $\hat H$’s own eigenstates:

Only an overall phase changes here, and Chapter 2 established that an overall phase never affects any measurement probability. For this reason, an energy eigenstate is called a stationary state: every time-independent measurement probability computed from it stays exactly constant, even though the ket itself is visibly changing.

Concept check 4.1¶

Can a stationary state have a time-dependent ket?

A general initial state need not be a single energy eigenstate; it can be any superposition of them,

and by linearity of the Schrödinger equation, each term simply evolves on its own:

Here is the key qualitative difference from a single stationary state: the energy probabilities $|c_n|^2$ stay constant, but the relative phases between different terms evolve, at angular frequencies $(E_n-E_m)/\hbar$ set by the energy differences. Any observable that mixes different energy eigenstates can therefore vary in time, even though no individual energy probability ever does.

Example 4.1: a two-level superposition¶

Suppose $E_+>E_-$ and

Removing the physically irrelevant common phase $e^{-iE_-t/\hbar}$ gives the equivalent representative

The energy probabilities remain one half at all times. But in a basis built from coherent sums of $|E_+\rangle$ and $|E_-\rangle$, the corresponding probabilities oscillate with frequency $\omega$. If these two energy states are chosen as the $z$ basis of a spin-$\tfrac12$ system, the $x$ and $y$ bases are familiar examples of such mixing bases.

Only energy differences drive observable phase changes¶

For two energy components, the cross term in any measurement probability contains a phase of the form

This predicts an experimentally visible angular frequency $|E_n-E_m|/\hbar$, a useful fact to remember, since it also gives a quick dimensional check: energy divided by $\hbar$ has units of inverse time, as an angular frequency should. A common phase built from, say, the average energy of two levels drops out of every isolated-system probability, exactly like the overall phase discussed in Section 4.1.

Example 4.2: evolution from a nondiagonal matrix¶

In the $z$ basis, consider

The energy eigenstates are $|+x\rangle$ and $|-x\rangle$, with energies $E_0\pm\hbar\Omega/2$. Because the identity commutes with $\sigma_x$,

Applied to $|+z\rangle$, and suppressing the common phase,

Therefore

Notice what is happening here: the initial state was definite along $z$, but the Hamiltonian was not diagonal in the $z$ basis. It is exactly this mismatch—the off-diagonal entries connecting $|+z\rangle$ and $|-z\rangle$—that drives coherent transfer between the two $z$ outcomes over time.

4.3 Spin in a uniform magnetic field¶

We now specialize the general machinery above to the physical situation that will occupy the rest of this chapter: a spin in a magnetic field. A magnetic moment in a uniform field has Hamiltonian

the same energy expression introduced in Chapter 1, now promoted to an operator. For a spin-$\tfrac12$ particle whose magnetic moment is proportional to its spin, write $\hat{\boldsymbol\mu}=\gamma\hat{\mathbf S}$, where $\gamma$ is the gyromagnetic ratio. For $\mathbf B=B_0\hat{\mathbf z}$, this becomes

The sign of $\omega_0$ depends on the sign convention chosen for $\gamma$. Keeping track of that sign carefully, rather than dropping it, avoids silently reversing the direction of precession later on.

Because $\sigma_z^2=\hat I$, its exponential can be evaluated directly from the power series, without any diagonalization step. Every even power of $\sigma_z$ collapses back to $\hat I$, and every odd power collapses back to $\sigma_z$ itself: $\sigma_z^3=\sigma_z(\sigma_z^2)=\sigma_z$, $\sigma_z^4=(\sigma_z^2)^2=\hat I$, and so on. Writing $\theta=\omega_0t/2$ for brevity and separating the exponential’s power series into its even and odd terms,

using $\sigma_z^n=\hat I$ for even $n$ and $\sigma_z^n=\sigma_z$ for odd $n$ to pull $\sigma_z$ out of every odd term. The two bracketed sums are exactly the ordinary Taylor series whose sum is $\cos\theta$ and $i\sin\theta$ respectively: the same series reviewed above for $e^{i\phi}$, now split into its even and odd parts instead of read off directly. This gives

The identical argument applies to $e^{i\alpha\,\mathbf n\cdot\boldsymbol\sigma/2}$ for any unit vector $\mathbf n$, since $(\mathbf n\cdot\boldsymbol\sigma)^2=\hat I$ as well; this is exactly the fact reused in Section 4.5 for a field pointing in an arbitrary direction and in Section 4.10 for a general control pulse.

In the $z$ basis, where $\hat H$ is already diagonal, this is simply

Concept check 4.2¶

A spin begins in $|+z\rangle$ in a uniform $z$-directed field. Does it precess into $|-z\rangle$?

4.4 Larmor precession¶

The matrix above shows that a $z+$ or $z-$ state, being an eigenstate of $\hat H$, only picks up a phase and never changes. To see the field actually do something visible, we need a state with some component transverse to $z$. Prepare $|+x\rangle=(|+z\rangle+|-z\rangle)/\sqrt2$ and let it evolve in the $z$-directed field. Then

After removing an overall phase, the relative phase between the two terms is $e^{-i\omega_0t}$, and Chapter 2 already showed exactly what a relative phase like this does to measurement probabilities in a different basis. The probabilities for a later $S_x$ measurement are

Meanwhile $P(z+)=P(z-)=1/2$ at every time $t$: the $z$-basis populations never change at all. Geometrically, the Bloch vector simply rotates around the $z$ axis at the Larmor angular frequency $|\omega_0|$. This is the general pattern to remember: dynamics changes a relative phase, and it is the choice of final analyzer that converts an otherwise hidden phase into a measurable probability.

Concept check 4.3¶

A spin in a $z$-directed field always has $P(z+)=P(z-)=1/2$. Must its state be constant in time?

This experiment is a two-state interferometer, in the same spirit as the recombination experiments of Chapters 1 and 2. Preparing $|+x\rangle$ creates two equal $z$-basis amplitudes, the field gives them different phases as time passes, and the final $x$ analyzer recombines those amplitudes into an interference pattern. The observed oscillation is the spin-space analog of moving a phase plate through one arm of an ordinary path interferometer.

The classical system with the same equation of motion is a gyroscope: a torque perpendicular to an angular momentum does not tip it over but walks it around a cone at fixed polar angle, exactly as a transverse field walks the Bloch vector around the equator without changing $P(z\pm)$.

Example 4.3: a spin-flip time¶

Starting from $|+x\rangle$, the first time at which an $S_x$ measurement gives $x-$ with certainty satisfies $|\omega_0|t=\pi$. Thus

Be precise about what this is not: it is not a transition between the $S_z$ energy levels, since their populations remain fixed throughout. It is a rotation of the state relative to the $x$ measurement basis—the energy eigenstates are untouched; only the relative phase between them has advanced by $\pi$.

Example 4.4: infer a field from counts¶

A $+x$ state evolves for $2.0\,\mu\text{s}$ in a $z$-directed field. A later $S_x$ measurement gives $P(x+)=3/4$. Equation (26) implies

The smallest positive phase is $|\omega_0|t=\pi/3$, giving

Two ambiguities remain, and neither should be overlooked. The probability alone does not determine the sign of $\omega_0$, since $\cos^2$ cannot distinguish clockwise from counterclockwise rotation. And because the cosine is periodic, this same data is also consistent with larger frequencies differing by whole cycles. Additional measurements at other times, or a $y$ analyzer sensitive to the direction of rotation, remove these ambiguities.

4.5 A field in an arbitrary direction¶

Everything so far assumed the field points along $z$. For a constant field along an arbitrary unit vector $\mathbf n$,

Since $(\mathbf n\cdot\boldsymbol\sigma)^2=\hat I$—the same identity used in Chapter 3—the same power-series trick used for $\sigma_z$ generalizes immediately:

This operator rotates the Bloch vector around $\mathbf n$, and it also reveals why the exponent naturally involves a half-angle, echoing the half-angle rule from Chapter 1. A $2\pi$ spatial rotation gives $\hat U=-\hat I$, and only a $4\pi$ rotation returns the ket to exactly itself. That minus sign after $2\pi$ is an overall phase for one isolated evolving path, so it has no effect by itself, but it becomes observable through interference with a second, unrotated path.

Three geometric cases deserve separate attention. If the initial Bloch vector is parallel to $\mathbf n$, the state is an energy eigenstate and gains only an overall phase, never changing direction. If it starts perpendicular to $\mathbf n$, it sweeps out a full circle around $\mathbf n$. At any intermediate angle, it traces a cone around $\mathbf n$, with its component along $\mathbf n$ staying fixed throughout.

Example 4.5: rotation about $x$¶

Take the Hamiltonian $\hat H=-(\hbar\omega/2)\sigma_x$ and initial state $|+z\rangle$. Equation (31) gives

The probability of a $z-$ result is $\sin^2(\omega t/2)$. At $|\omega|t=\pi/2$, the state has equal $z$ probabilities but a definite transverse direction; at $|\omega|t=\pi$, it is $|-z\rangle$ up to phase. Equal $z$ populations halfway through the rotation do not mean coherence has been lost, however: the state is still a perfectly definite pure state, just one pointing along a different axis.

4.6 Measurement interrupts unitary evolution¶

So far we have evolved a state continuously and only asked about measurement at the very end. Real experimental protocols often measure, select an outcome, and then let the resulting state evolve further, and it is important to know exactly how to handle that. Suppose the spin evolves until time $t_1$, when $S_x$ is measured. Just before the measurement,

The outcome probabilities follow from projecting this state onto $|\pm x\rangle$, exactly as in Chapter 3. If the result is $x+$, the state used for all subsequent evolution is $|+x\rangle$ itself, not the pre-measurement superposition:

Unitary evolution and ideal projective measurement are two genuinely distinct parts of the model, governed by different rules, and confusing them leads to a common mistake: continuing to evolve the pre-measurement superposition after a definite outcome has already been selected, as if the measurement had never happened.

Example 4.6: a complete prepare–evolve–measure protocol¶

A spin is prepared in $|+x\rangle$, evolves in a $z$-directed field for a time $t_1=\pi/(2|\omega_0|)$, and is measured along $x$. Suppose the $x-$ outcome is selected. The same field then acts for another time $t_2=\pi/(2|\omega_0|)$ before a final $S_x$ measurement.

Just before the intermediate measurement, $P(x+)=P(x-)=1/2$. Selecting $x-$ prepares $|-x\rangle$; it does not leave the earlier phase-evolved ket in place, per the rule just stated. Starting fresh from $|-x\rangle$, another quarter-period of evolution produces equal final $x$ probabilities. So the probability for either final result, conditional on having selected the $x-$ branch, is $1/2$. The probability for the complete route starting from the original preparation must also include the probability of the intermediate selection itself, giving $1/4$ overall.

4.7 Constants in the Hamiltonian¶

Adding a constant energy $E_0$ to every state changes the Hamiltonian to $\hat H'=\hat H+E_0\hat I$. Because $E_0\hat I$ commutes with anything, the corresponding evolution operator factors cleanly:

Every state acquires exactly the same overall phase from this shift, so isolated-system probabilities are completely unchanged. This confirms a pattern seen throughout the chapter: it is energy differences, not an arbitrary choice for the zero of energy, that control observable relative phases.

4.8 Optional mathematical extension: time-dependent Hamiltonians¶

Every result so far assumed a time-independent Hamiltonian, which is what let us write $\hat U(t)=e^{-i\hat Ht/\hbar}$ so simply. If $\hat H$ instead changes with time, that simple exponential formula no longer applies, and the evolution operator must instead be defined through its own differential equation:

If the Hamiltonians at different times all commute with each other, the solution still takes a simple exponential form, just with an integral in place of a product:

But if $[\hat H(t_1),\hat H(t_2)]\ne0$ for some pair of times, this shortcut fails, and the ordering of successive infinitesimal rotations genuinely matters; a time-ordered exponential or a piecewise product must be used instead. For example, a pulse about $x$ followed by a pulse about $y$ generally does not produce the same final state as the reversed sequence. This is the dynamical counterpart of the noncommuting analyzer questions from Chapter 3.

Example 4.7: a piecewise-constant Hamiltonian¶

Suppose a control system applies

Within each interval the Hamiltonian is constant. The complete evolution is therefore the ordered product

The earliest operation stands nearest the initial ket. Starting from $|+z\rangle$, the first interval gives $\hat R_x(\pi/2)|+z\rangle=|-y\rangle$. Applying the $z$ rotation then gives

The displayed phase is overall, so the final Bloch direction is $+x$. If the two intervals are reversed, the $z$ pulse first changes $|+z\rangle$ only by an overall phase, after which the $x$ pulse leaves the Bloch vector at $-y$. The two schedules use the same Hamiltonians for the same durations but end in different physical states because the interval propagators do not commute. This piecewise construction is the practical finite-step version of time ordering.

4.9 Optional mathematical extension: motion of expectation values¶

The Schrödinger equation for the ket also implies a direct equation of motion for the expectation value of any observable, which is often a faster route to physical insight than tracking the full ket. For an operator with no explicit time dependence of its own,

Specializing to $\hat H=-\gamma\mathbf B\cdot\hat{\mathbf S}$ and using the spin commutators from Chapter 3, this becomes

Equation (42) has exactly the form of classical magnetic precession, even though each individual component measurement remains fully quantized, with only two discrete outcomes. It offers a useful bridge between two pictures developed so far: the geometric Bloch-sphere picture of Chapter 2, and the explicit matrix evolution of a ket developed in this chapter. The expectation vector traces out a smooth, classical-looking trajectory, even though the individual Stern–Gerlach outcomes behind it remain discrete.

4.10 Applications track: controlled rotations and pulse language¶

Laboratory control of a spin is usually described not in terms of a continuously running field, but in terms of discrete pulses—a field switched on for a controlled duration to produce a specific rotation. A constant control Hamiltonian applied for a chosen duration implements exactly such a rotation. Define

For $\hat H=(\hbar\Omega/2)\mathbf n\cdot\boldsymbol\sigma$ applied for time $\tau$, the rotation angle is $\alpha=\Omega\tau$. This control Hamiltonian is defined with the opposite sign from the field Hamiltonian $\hat H=-(\hbar\omega/2)\mathbf n\cdot\boldsymbol\sigma$ of Sections 4.3–4.5; matching the two conventions, $\Omega=-\omega$, recovers $\hat R_{\mathbf n}(\Omega\tau)=\hat U(\tau)$ from equation (31). Two particular pulses come up often enough to deserve names: a $\pi$ pulse carries a Bloch vector to the opposite side of its rotation circle, while a $\pi/2$ pulse can turn a definite $z$ state into an equal-amplitude superposition. For example,

The factor $-i$ may be dropped here because it multiplies the entire state; subsequent unitary operations cannot turn a truly global phase into a relative one. In multibranch calculations, however, a phase belonging to only one branch is not global and must be retained until the branches are recombined.

If an $x$ pulse acts first and a $y$ pulse second, the combined operator is

Read this expression from right to left: the earliest operation in time stands closest to the initial ket. Rotations about different axes generally do not commute, so reversing the order of these two matrices describes a genuinely different laboratory sequence, not merely a different way of writing the same one.

Example 4.8: pulse order on a prepared spin¶

Begin in $|+z\rangle$. An $x$-axis $\pi/2$ pulse produces

A subsequent $y$-axis pulse changes this $S_y$ eigenstate only by an overall phase, since $|-y\rangle$ is already an eigenstate of the rotation axis. In the reverse order, the first $y$ pulse produces $|+x\rangle$, which the $x$ pulse then changes only by an overall phase, for the same reason. The two final Bloch vectors, $-y$ and $+x$, are clearly different states. This geometric shortcut reaches the same conclusion as direct matrix multiplication, with considerably less arithmetic.

4.11 Applications track: Ramsey interference¶

We can now assemble everything in this chapter into one of the most important protocols in atomic physics: a Ramsey sequence, which is really an interferometer built entirely in a two-state space, using pulses instead of beam splitters. The protocol has five steps:

1. prepare $|+z\rangle$;

2. apply $\hat R_y(\pi/2)$ to create $|+x\rangle$;

3. allow free phase evolution for time $T$;

4. apply $\hat R_y(-\pi/2)$; and

5. measure $S_z$.

Represent the free evolution in step 3 by

where $\delta$ is the relative angular frequency in the chosen reference frame. Combining all five steps, the final state is

Projecting onto $|\pm z\rangle$ gives the observed fringes

It is worth naming what each step contributes: the first pulse creates two coherent amplitudes out of a single definite state, free evolution changes their relative phase by an amount set by $\delta$ and $T$, and the second pulse recombines those two amplitudes so that the final measurement can read out the accumulated phase as a probability. This is precisely the coherent path logic first introduced in Chapter 2, now implemented with controlled rotations rather than physically separated paths.

Example 4.9: the first fringe minimum¶

If $|\delta|=2\pi(25\,\mathrm{kHz})$, the first $z+$ minimum occurs at $|\delta|T=\pi$:

Scanning $T$ at fixed detuning reveals these oscillations directly. Scanning the control frequency instead, at fixed $T$, also produces fringes, now as a function of detuning rather than time. This sensitivity of a simple probability to a small phase is the operating principle behind atomic clocks and many modern quantum sensors.

If the accumulated phase varies unpredictably between different trials of the experiment, averaging over those trials reduces the fringe contrast. Each individual run may still be perfectly unitary; it is the pooled ensemble that becomes mixed, because its relative phase is no longer controlled from run to run. Density operators, introduced in Chapter 3, are exactly the tool needed to distinguish this loss of ensemble coherence from ordinary, coherent precession.

Concept check 4.4¶

Why not measure $S_z$ after the first Ramsey pulse to verify both alternatives?

Summary¶

• The Hamiltonian generates deterministic, unitary evolution between measurements.

• Energy eigenstates acquire only overall phases and are stationary.

• Superpositions acquire changing relative phases set by energy differences.

• A uniform magnetic field rotates a spin state; a field gradient can analyze it.

• Larmor precession converts phase evolution into oscillating measurement probabilities.

• Adding a constant to every energy changes only an unobservable overall phase.

• A complete dynamics calculation specifies a preparation, an evolution operator, and a final measurement basis.

• Time-dependent Hamiltonians require attention to the order of rotations when the Hamiltonians at different times do not commute.

• Spin expectation values precess smoothly even though individual component measurements have only two discrete outcomes.

• Controlled pulses implement rotations whose matrix order follows the laboratory sequence from right to left.

• Ramsey sequences turn accumulated relative phase into measurable population fringes.

Exercises¶

Exercises 1–9 establish the core evolution and measurement workflow. Exercises 10–13 consolidate spin precession and interrupted evolution. Exercises 14–20 support the optional mathematical and applications tracks; a focused first assignment need not include all three groups.

1. Starting from equation (1), show directly that the norm of a state is constant when $\hat H$ is Hermitian.

2. A two-level system has energies 0 and $\epsilon$. Evolve the initial state $(|E_0\rangle+i|E_1\rangle)/\sqrt2$ and find the probability of returning to the initial state at time $t$.

3. Derive equation (23) by separating the exponential power series into even and odd powers.

4. Derive equation (26) from equation (25) using inner products.

5. For the same evolving state, calculate $P(y+,t)$ and identify the direction of precession for positive $\omega_0$ under the conventions used here.

6. Find the earliest time at which a spin initially in $|+z\rangle$ becomes $|-z\rangle$ under a uniform field in the $+x$ direction.

7. Prove that equation (31) is unitary.

8. An $S_x$ measurement is made halfway through a full Larmor period and its $x+$ result is selected. Describe the state immediately before and after the measurement, then find $P(x+)$ one quarter-period later.

9. Show that adding $E_0\hat I$ to a Hamiltonian cannot affect the expectation value of any time-independent observable in an isolated system.

10. For the Hamiltonian in Example 4.2, calculate $P(y+,t)$ for an initial $|+z\rangle$ state. At what earliest time is the state $|+y\rangle$ up to overall phase?

11. A $+x$ state precesses in a $z$-directed field. Find both $P(x+,t)$ and $P(y+,t)$, and explain how their relative timing reveals the sign of $\omega_0$.

12. Derive the result of Example 4.5 by expanding $|+z\rangle$ in the energy basis $|\pm x\rangle$, evolving the two components, and transforming back.

13. Work through every branch of Example 4.6 rather than selecting only $x-$. Show that pooling the intermediate outcomes gives final statistics different from a protocol with no intermediate measurement.

14. A Hamiltonian equals $\hbar\Omega\sigma_x/2$ for a duration $\tau$ and then $\hbar\Omega\sigma_y/2$ for the same duration. Write the total evolution operator in the correct order. Repeat for the reversed pulse sequence and show that the results differ unless $\Omega\tau$ takes a special value.

15. Derive equation (41) from the Schrödinger equation and its Hermitian conjugate.

16. Use equation (42) to identify a conserved component of $\langle\mathbf S\rangle$ in a constant magnetic field and connect it to energy conservation.

17. Verify both pulse actions following equation (43) using explicit Pauli matrices.

18. Calculate both pulse orders in Example 4.8 by matrix multiplication.

19. Derive equation (49) from equation (48) and check $\delta T=0,\pi,2\pi$.

20. Insert an unread $S_z$ measurement after the first Ramsey pulse and use density operators to show that both final outcomes have probability $1/2$.

Selected exercise guidance¶

Use these answers only after making a complete attempt.