Chapter 7: Time Evolution and the Schrödinger vs. Heisenberg Pictures

How quantum states change in time — and the surprising fact that you get to choose whether it is the states or the operators that evolve

Opening: The Question of Becoming

Everything we have done so far has been, in a sense, frozen. We solved the time-independent Schrödinger equation in Chapters 3 through 5, finding energy eigenstates that sit there, perfectly still, like notes written on a page. In Chapter 6, we built the operator formalism — a powerful grammar for the language of quantum mechanics — but we used it primarily to describe what you can measure, not how things change.

Yet the universe is emphatically not frozen. Atoms absorb light and jump to excited states. Wave packets spread as they propagate. Radioactive nuclei decay. Quantum computers process information by evolving qubit states through carefully designed sequences of operations. If quantum mechanics is to describe the real world, it must describe dynamics — the passage of time.

This chapter is about time. We will construct the mathematical machinery that governs how quantum systems evolve, discover that there are multiple equivalent ways to think about this evolution (each with its own power and elegance), and encounter our first genuinely time-dependent quantum system: a two-level atom driven by an oscillating field.

The payoff is not merely mathematical. By the end of this chapter, you will understand why classical mechanics emerges from quantum mechanics in the appropriate limit, why wave packets spread and sometimes miraculously reassemble, and how the same mathematical structure that describes a spinning magnetic moment in a laboratory also describes the qubits inside a quantum computer.

Let us begin with the most fundamental question: given a quantum state at time $t = 0$, what is the state at a later time $t$?

7.1 The Time-Evolution Operator

From the Schrödinger Equation to the Propagator

The time-dependent Schrödinger equation is the master equation of nonrelativistic quantum mechanics:

$$i\hbar \frac{\partial}{\partial t} \Psi(x, t) = \hat{H} \Psi(x, t)$$

This is a first-order differential equation in time. That means if you know $\Psi(x, 0)$, the state at $t = 0$, then the Schrödinger equation completely determines $\Psi(x, t)$ for all future (and past) times. Quantum mechanics is deterministic between measurements — a point that often surprises students who have heard only about quantum randomness.

We want to formalize this. Define the time-evolution operator $\hat{U}(t, t_0)$ as the operator that takes the state at time $t_0$ to the state at time $t$:

$$\Psi(x, t) = \hat{U}(t, t_0)\, \Psi(x, t_0)$$

What properties must $\hat{U}$ have?

Property 1: Identity at equal times. If no time has elapsed, nothing has changed:

$$\hat{U}(t_0, t_0) = \hat{I}$$

Property 2: Composition. Evolving from $t_0$ to $t_1$ and then from $t_1$ to $t_2$ must be the same as evolving directly from $t_0$ to $t_2$:

$$\hat{U}(t_2, t_1)\, \hat{U}(t_1, t_0) = \hat{U}(t_2, t_0)$$

Property 3: Unitarity. Probabilities must be conserved. Since $\int |\Psi|^2 dx = 1$ must hold at all times, $\hat{U}$ must be unitary:

$$\hat{U}^\dagger(t, t_0)\, \hat{U}(t, t_0) = \hat{I}$$

This is not optional — it is the mathematical expression of the fact that probability is never created or destroyed in quantum mechanics. A universe where probability leaks away would be a universe where the Born rule (Chapter 2) cannot be trusted.

Deriving the Explicit Form

Substituting $\Psi(x, t) = \hat{U}(t, t_0)\, \Psi(x, t_0)$ into the Schrödinger equation:

$$i\hbar \frac{\partial}{\partial t} \hat{U}(t, t_0)\, \Psi(x, t_0) = \hat{H}\, \hat{U}(t, t_0)\, \Psi(x, t_0)$$

Since this must hold for any initial state $\Psi(x, t_0)$, we obtain an operator equation:

$$i\hbar \frac{\partial}{\partial t} \hat{U}(t, t_0) = \hat{H}\, \hat{U}(t, t_0)$$

This is the Schrödinger equation for the time-evolution operator itself.

Case 1: Time-independent Hamiltonian. When $\hat{H}$ does not depend on time (the most common situation in introductory quantum mechanics), this ordinary differential equation has a simple solution. Setting $t_0 = 0$:

$$\boxed{\hat{U}(t) = e^{-i\hat{H}t/\hbar}}$$

This is the single most important equation in this chapter. Let us unpack it carefully.

The exponential of an operator is defined by its Taylor series:

$$e^{-i\hat{H}t/\hbar} = \hat{I} - \frac{i\hat{H}t}{\hbar} + \frac{1}{2!}\left(\frac{-i\hat{H}t}{\hbar}\right)^2 + \cdots = \sum_{n=0}^{\infty} \frac{1}{n!}\left(\frac{-i\hat{H}t}{\hbar}\right)^n$$

You can verify directly that this satisfies the differential equation, the initial condition $\hat{U}(0) = \hat{I}$, and the unitarity condition $\hat{U}^\dagger \hat{U} = e^{+i\hat{H}t/\hbar} e^{-i\hat{H}t/\hbar} = \hat{I}$ (since $\hat{H}$ is Hermitian, $\hat{H}^\dagger = \hat{H}$).

💡 Key Insight: The Hamiltonian is the generator of time translations, just as momentum is the generator of spatial translations (a connection we will explore deeply in Chapter 10). The structure $e^{-i\hat{G}\alpha/\hbar}$, where $\hat{G}$ is a Hermitian operator and $\alpha$ is a continuous parameter, is the universal form of a unitary transformation in quantum mechanics.

Case 2: Time-dependent Hamiltonian. When $\hat{H} = \hat{H}(t)$ depends explicitly on time — for example, an atom in a time-varying electromagnetic field — the situation is far more subtle. The naive expression $e^{-i\int_0^t \hat{H}(t')dt'/\hbar}$ does not work in general because $\hat{H}(t_1)$ and $\hat{H}(t_2)$ may not commute at different times.

The correct expression involves the time-ordered exponential (or Dyson series):

$$\hat{U}(t, 0) = \mathcal{T} \exp\left(-\frac{i}{\hbar}\int_0^t \hat{H}(t')\, dt'\right) = \sum_{n=0}^{\infty} \left(\frac{-i}{\hbar}\right)^n \int_0^t dt_1 \int_0^{t_1} dt_2 \cdots \int_0^{t_{n-1}} dt_n\, \hat{H}(t_1)\hat{H}(t_2)\cdots\hat{H}(t_n)$$

where $\mathcal{T}$ is the time-ordering operator that arranges the product so that later times stand to the left. We will not need this machinery until Chapter 21 (time-dependent perturbation theory), but it is important to know it exists — and to understand why it is necessary. The non-commutativity of operators at different times is a genuinely quantum phenomenon with no classical analogue.

🔗 Connection: The time-ordering problem is closely related to the Baker-Campbell-Hausdorff formula: $e^{\hat{A}}e^{\hat{B}} \neq e^{\hat{A}+\hat{B}}$ unless $[\hat{A}, \hat{B}] = 0$. You explored commutator algebra in Chapter 6. That algebra is now driving real physics.

The Photon in a Mach-Zehnder Interferometer

Let us see the time-evolution operator in action with our anchor example. Consider a single photon entering a Mach-Zehnder interferometer — two beam splitters separated by two mirrors, with path lengths that can be adjusted.

The photon has two possible paths: the upper arm ($|U\rangle$) and the lower arm ($|L\rangle$). After the first beam splitter, a photon that entered in state $|L\rangle$ is in the superposition:

$$|\psi(0)\rangle = \frac{1}{\sqrt{2}}\left(|U\rangle + i|L\rangle\right)$$

The time evolution through the two arms introduces phase factors. If the upper arm has length $\ell_U$ and the lower arm has length $\ell_L$, the Hamiltonian for free propagation gives:

$$|\psi(t)\rangle = \frac{1}{\sqrt{2}}\left(e^{-i\phi_U}|U\rangle + i\, e^{-i\phi_L}|L\rangle\right)$$

where $\phi_{U,L} = k \ell_{U,L}$ are the accumulated phases. The second beam splitter then recombines the paths, and the detection probability depends on the relative phase $\Delta\phi = \phi_U - \phi_L$:

$$P(\text{detector 1}) = \cos^2\!\left(\frac{\Delta\phi}{2}\right), \qquad P(\text{detector 2}) = \sin^2\!\left(\frac{\Delta\phi}{2}\right)$$

This is single-photon interference — the quantum phenomenon that launched a thousand thought experiments (recall Chapter 1). What we have now that we lacked in Chapter 1 is the mathematical machinery to describe why it happens: the time-evolution operator produces phase factors, and these phase factors interfere.

⚠️ Common Misconception: Students sometimes think the photon "splits in half" at the beam splitter. It does not. The photon is always detected as a single, indivisible quantum at one detector or the other. What splits is the probability amplitude — and it is the interference of these amplitudes, governed by the time-evolution operator, that produces the characteristic pattern.

Worked Example 7.1: The Beam Splitter Matrix

The action of a 50/50 beam splitter on the two-path basis $\{|U\rangle, |L\rangle\}$ can be represented as a $2 \times 2$ unitary matrix:

$$\hat{B} = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & i \\ i & 1 \end{pmatrix}$$

The factor of $i$ represents the 90-degree phase shift acquired upon reflection (a consequence of energy conservation at the beam splitter interface — you can verify that this matrix is indeed unitary: $\hat{B}^\dagger \hat{B} = \hat{I}$).

For a photon entering in the lower port, $|L\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$:

$$\hat{B}|L\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} i \\ 1 \end{pmatrix} = \frac{1}{\sqrt{2}}(i|U\rangle + |L\rangle)$$

Now the photon propagates through the interferometer. The phase evolution in each arm is described by:

$$\hat{P} = \begin{pmatrix} e^{i\phi_U} & 0 \\ 0 & e^{i\phi_L} \end{pmatrix}$$

The second beam splitter applies $\hat{B}$ again. The full interferometer transformation is:

$$\hat{U}_{\text{MZ}} = \hat{B}\, \hat{P}\, \hat{B}$$

Working this out (a good exercise in matrix multiplication):

$$\hat{U}_{\text{MZ}} = e^{i(\phi_U + \phi_L)/2}\begin{pmatrix} -\sin(\Delta\phi/2) & i\cos(\Delta\phi/2) \\ i\cos(\Delta\phi/2) & -\sin(\Delta\phi/2) \end{pmatrix}$$

where $\Delta\phi = \phi_U - \phi_L$. For a photon entering in the lower port:

$$P(\text{upper output}) = \cos^2(\Delta\phi/2), \qquad P(\text{lower output}) = \sin^2(\Delta\phi/2)$$

When $\Delta\phi = 0$ (equal arm lengths), all photons exit through the upper port. When $\Delta\phi = \pi$, all exit through the lower port. At intermediate values, the photon has a probability of exiting through either port — the signature of quantum interference.

This is a concrete example of the general principle: time evolution is a unitary transformation, and unitary transformations preserve total probability while redistributing it among the possible outcomes.

✅ Check Your Understanding (CYU-7.1): 1. Verify that $\hat{U}(t) = e^{-i\hat{H}t/\hbar}$ satisfies the composition property: $\hat{U}(t_2)\hat{U}(t_1) = \hat{U}(t_1 + t_2)$. 2. Why does unitarity of $\hat{U}$ require $\hat{H}$ to be Hermitian? Start from $\hat{U}^\dagger \hat{U} = \hat{I}$ and show that this demands $\hat{H}^\dagger = \hat{H}$. 3. In the Mach-Zehnder interferometer, what happens to the detection probabilities when $\Delta\phi = 0$? When $\Delta\phi = \pi$?

7.2 Stationary States Revisited

Why "Stationary"?

In Chapter 3, we found the energy eigenstates of the infinite square well: $\psi_n(x) = \sqrt{2/a}\sin(n\pi x/a)$ with energies $E_n = n^2\pi^2\hbar^2/(2ma^2)$. In Chapter 4, we found the QHO energy eigenstates with $E_n = (n + 1/2)\hbar\omega$. But we solved the time-independent Schrödinger equation, which means we found the spatial parts only.

Now we can write the full time-dependent solution. If $\hat{H}\psi_n = E_n \psi_n$, then:

$$\Psi_n(x, t) = \hat{U}(t)\, \psi_n(x) = e^{-iE_n t/\hbar}\, \psi_n(x)$$

The time dependence is nothing but a phase factor $e^{-iE_n t/\hbar}$. This is why energy eigenstates are called stationary states: the probability density is

$$|\Psi_n(x, t)|^2 = |e^{-iE_n t/\hbar}|^2 |\psi_n(x)|^2 = |\psi_n(x)|^2$$

which is independent of time. The state "oscillates" in phase, but nothing measurable changes.

More precisely, for any observable $\hat{A}$ that does not explicitly depend on time:

$$\langle \hat{A} \rangle_n(t) = \int \Psi_n^*(x,t)\, \hat{A}\, \Psi_n(x,t)\, dx = \int \psi_n^*(x)\, e^{+iE_n t/\hbar}\, \hat{A}\, e^{-iE_n t/\hbar}\, \psi_n(x)\, dx = \langle \hat{A} \rangle_n(0)$$

All expectation values are constant. The energy eigenstate is, in every measurable sense, unchanging.

🔵 Historical Note: The term "stationary state" was coined by Bohr in his 1913 model of the hydrogen atom, where he postulated (without derivation) that electrons in certain orbits did not radiate. The Schrödinger equation finally explained why: an energy eigenstate has no time-varying charge distribution, so it produces no time-varying electromagnetic field, and therefore does not radiate.

The Energy-Time Phase

The phase $e^{-iE_n t/\hbar}$ is often written as $e^{-i\omega_n t}$ where $\omega_n = E_n/\hbar$. This is the angular frequency associated with energy level $n$. It is the quantum mechanical version of the relation $E = \hbar\omega$ that Planck introduced for photons (Chapter 1) — but now it applies to any quantum system.

🔗 Spaced Review (Ch 4): For the QHO, the energy levels are equally spaced: $E_n = (n + 1/2)\hbar\omega$. This means the phase frequencies are also equally spaced: $\omega_n = (n + 1/2)\omega$. This equal spacing will turn out to be profoundly important for the dynamics of coherent states (Chapter 27) and is already lurking behind the quantum revival phenomenon we discuss in Section 7.4.

Worked Example 7.2: Two-State Superposition in the Infinite Well

Consider a particle in the infinite square well in the state:

$$\Psi(x, 0) = \frac{1}{\sqrt{2}}\left[\psi_1(x) + \psi_2(x)\right]$$

This is a superposition of the ground state ($n = 1$) and the first excited state ($n = 2$). Using $\hat{U}(t)$:

$$\Psi(x, t) = \frac{1}{\sqrt{2}}\left[e^{-iE_1 t/\hbar}\,\psi_1(x) + e^{-iE_2 t/\hbar}\,\psi_2(x)\right]$$

The probability density is:

$$|\Psi(x,t)|^2 = \frac{1}{2}\left[|\psi_1|^2 + |\psi_2|^2 + 2\text{Re}\left(\psi_1^* \psi_2\, e^{-i(E_2 - E_1)t/\hbar}\right)\right]$$

The cross term oscillates at the beat frequency $\omega_{21} = (E_2 - E_1)/\hbar = 3E_1/\hbar$ (since $E_2 = 4E_1$ and $E_1 = \pi^2\hbar^2/(2ma^2)$). This is the simplest example of quantum dynamics: the probability density sloshes back and forth inside the well at a frequency determined by the energy difference.

At $t = 0$, the two wave functions add constructively on the left side of the well (where both $\psi_1$ and $\psi_2$ are positive). At $t = \pi\hbar/(E_2 - E_1)$, the relative phase has flipped, and the probability is concentrated on the right side. The particle oscillates between left and right with a well-defined period:

$$T_{21} = \frac{2\pi\hbar}{E_2 - E_1} = \frac{2\pi\hbar}{3E_1}$$

This is the classical period for a particle bouncing back and forth in the well with an energy midway between $E_1$ and $E_2$.

The lesson is fundamental: a single energy eigenstate is static; a superposition of two or more eigenstates oscillates at the beat frequency. The richer the superposition (more energy eigenstates involved), the more complex the dynamics — until, in the infinite well, we get the remarkable revival phenomenon of Section 7.4.

7.3 Time-Dependent Phenomena: Wave Packets

Superposition and Dynamics

The magic happens when the system is not in a single energy eigenstate. Consider a general state expanded in the energy eigenbasis:

$$\Psi(x, 0) = \sum_n c_n\, \psi_n(x)$$

where $c_n = \int \psi_n^*(x)\, \Psi(x, 0)\, dx$ are the expansion coefficients (with $\sum_n |c_n|^2 = 1$). Time evolution acts on each component independently:

$$\Psi(x, t) = \sum_n c_n\, e^{-iE_n t/\hbar}\, \psi_n(x)$$

Each energy eigenstate picks up its own phase factor, oscillating at its own frequency. Because these frequencies are (in general) different, the relative phases between the components change with time. The probability density $|\Psi(x,t)|^2$ is no longer time-independent — it moves, spreads, and reshapes.

This is the fundamental mechanism of all quantum dynamics: time evolution is the dephasing and rephasing of energy eigenstate superpositions.

Free-Particle Wave Packets

The simplest and most important example is a free particle ($V = 0$). The energy eigenstates are plane waves $\psi_k(x) = e^{ikx}/\sqrt{2\pi}$ with $E_k = \hbar^2 k^2/(2m)$, and a general wave packet is:

$$\Psi(x, t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \phi(k)\, e^{i(kx - \omega_k t)}\, dk$$

where $\omega_k = \hbar k^2/(2m)$ and $\phi(k)$ is the momentum-space wave function.

Gaussian wave packet. Let us take a Gaussian initial state centered at $x_0 = 0$ with initial width $\sigma_0$ and average momentum $\hbar k_0$:

$$\Psi(x, 0) = \left(\frac{1}{2\pi\sigma_0^2}\right)^{1/4} \exp\left(-\frac{x^2}{4\sigma_0^2} + ik_0 x\right)$$

The momentum-space representation is also Gaussian:

$$\phi(k) = \left(\frac{2\sigma_0^2}{\pi}\right)^{1/4} \exp\left(-\sigma_0^2(k - k_0)^2\right)$$

Performing the integral (a Gaussian integral — see Appendix A), the time-evolved wave packet is:

$$\Psi(x, t) = \left(\frac{1}{2\pi\sigma(t)^2}\right)^{1/4} \exp\left(-\frac{(x - v_g t)^2}{4\sigma(t)^2} + i\Phi(x,t)\right)$$

where the group velocity is

$$v_g = \frac{\hbar k_0}{m} = \frac{p_0}{m} = \frac{\partial \omega}{\partial k}\bigg|_{k_0}$$

and the spreading width is

$$\sigma(t) = \sigma_0 \sqrt{1 + \left(\frac{\hbar t}{2m\sigma_0^2}\right)^2}$$

Group Velocity and Dispersion

Two crucial phenomena are packed into these formulas:

1. The center moves at the group velocity $v_g = p_0/m$. This is exactly the classical velocity of a particle with momentum $p_0$. Quantum mechanics recovers classical motion for the center of the wave packet.

2. The wave packet spreads. The width $\sigma(t)$ grows monotonically. At short times ($t \ll 2m\sigma_0^2/\hbar$), the spreading is negligible. At long times ($t \gg 2m\sigma_0^2/\hbar$), $\sigma(t) \approx \hbar t/(2m\sigma_0)$ — the packet spreads linearly in time.

The spreading occurs because the free-particle dispersion relation $\omega = \hbar k^2/(2m)$ is nonlinear in $k$. Different momentum components travel at different speeds (the phase velocity $v_p = \omega/k = \hbar k/(2m)$ depends on $k$), and the packet disperses.

📊 By the Numbers: How fast does spreading matter? For an electron ($m = 9.11 \times 10^{-31}$ kg) with $\sigma_0 = 1$ nm, the characteristic spreading time is $\tau = 2m\sigma_0^2/\hbar \approx 1.5 \times 10^{-14}$ s — about 15 femtoseconds. After one nanosecond, the packet has spread to roughly 60 nm. For a baseball ($m = 0.145$ kg) with $\sigma_0 = 1$ cm, $\tau \approx 2.7 \times 10^{27}$ s — roughly $10^{20}$ years, far longer than the age of the universe. This is why you have never seen a baseball spread.

⚠️ Common Misconception: Wave packet spreading does not mean the particle "falls apart" or loses its identity. It means our knowledge of the particle's position becomes less certain over time. If you measure the position, you will find the particle at a single location — and the wave packet will collapse to a localized state around that location.

Phase Velocity vs. Group Velocity

A common source of confusion is the distinction between phase velocity and group velocity for the free particle. Let us be precise.

A single plane wave $e^{i(kx - \omega t)}$ has phase velocity:

$$v_p = \frac{\omega}{k} = \frac{\hbar k}{2m}$$

This is the speed at which the crests of the wave move. For a free nonrelativistic particle, $v_p$ depends on $k$ — the medium is dispersive.

The wave packet, which is a superposition of plane waves, moves at the group velocity:

$$v_g = \frac{d\omega}{dk}\bigg|_{k_0} = \frac{\hbar k_0}{m} = \frac{p_0}{m}$$

Notice that $v_g = 2v_p$. The group velocity equals the classical velocity $p/m$, while the phase velocity is half of that. This is peculiar to the nonrelativistic case and is a consequence of the quadratic dispersion relation $\omega \propto k^2$.

For comparison, photons in vacuum have $\omega = ck$ (linear dispersion), so $v_p = v_g = c$. This is why photon wave packets do not spread in vacuum — all momentum components travel at the same speed.

🔴 Warning: Do not confuse the spreading of the wave packet with the group velocity dispersion (GVD) that occurs in optical fibers and other media. The free-particle spreading we discuss here is a fundamental quantum effect that occurs even in vacuum — it is a consequence of the Schrödinger equation, not the medium. The GVD in optical fibers is an additional classical effect caused by the frequency dependence of the refractive index.

Worked Example 7.3: Spreading Timescales

Let us calculate the spreading timescale $\tau = 2m\sigma_0^2/\hbar$ for several systems to develop physical intuition:

The pattern is clear: the more massive the object and the more localized it is, the slower it spreads. For macroscopic objects, quantum spreading is utterly negligible — wave packet spreading is not why you lose your keys.

✅ Check Your Understanding (CYU-7.2): 1. A particle is in the state $\Psi(x, 0) = (1/\sqrt{2})[\psi_1(x) + \psi_2(x)]$ in an infinite well. Write $\Psi(x, t)$. At what time does $|\Psi(x, t)|^2$ first return to $|\Psi(x, 0)|^2$? 2. Why does a photon wave packet not spread in vacuum? (Hint: what is the dispersion relation for photons?) 3. Spaced review (Ch 3): The infinite well has energies $E_n = n^2 E_1$. What is $E_1$ for a well of width $a$?

7.4 Quantum Revivals

A Remarkable Periodicity

One of the most beautiful phenomena in quantum mechanics is the quantum revival: a wave packet that has spread and distorted beyond recognition spontaneously reassembles into its original shape at a later time.

The possibility of revivals depends entirely on the structure of the energy spectrum. In the infinite square well, $E_n = n^2 E_1$ where $E_1 = \pi^2\hbar^2/(2ma^2)$. A general state evolves as:

$$\Psi(x, t) = \sum_n c_n\, e^{-in^2 E_1 t/\hbar}\, \psi_n(x)$$

The wave function returns to its initial form when all the phase factors return to their initial values — that is, when $e^{-in^2 E_1 T_{\text{rev}}/\hbar} = 1$ for all $n$. This requires $n^2 E_1 T_{\text{rev}}/\hbar = 2\pi m_n$ for integers $m_n$, which is satisfied when:

$$\boxed{T_{\text{rev}} = \frac{2\pi\hbar}{E_1} = \frac{4ma^2}{\pi\hbar}}$$

This is the revival time. After one full revival time, the wave packet is exactly back where it started — every phase factor has completed an integer number of full rotations.

🔗 Spaced Review (Ch 3): The energy levels $E_n = n^2 E_1$ of the infinite square well are the key input here. Notice that the revival phenomenon depends on the energies being integer multiples of a common unit. The QHO, with $E_n = (n + 1/2)\hbar\omega$, gives a revival time $T_{\text{QHO}} = 2\pi/\omega$ — much simpler, because the equally spaced levels rephase simultaneously after exactly one classical period.

Fractional Revivals

Even more remarkable are the fractional revivals that occur at rational fractions of $T_{\text{rev}}$. At time $t = T_{\text{rev}}/q$ (for integer $q$), the wave packet does not return to its original form, but instead splits into a superposition of $q$ copies of the initial wave packet, evenly distributed around the well.

At $t = T_{\text{rev}}/2$, the wave packet reappears at the mirror image position in the well. At $t = T_{\text{rev}}/3$, three copies appear. At $t = T_{\text{rev}}/4$, either four copies or two copies (depending on the initial state) — the arithmetic of number theory enters quantum mechanics.

This is not abstract mathematics. Fractional and full revivals have been observed experimentally in: - Rydberg atoms excited to high principal quantum numbers (where the energy levels approximate $E_n \propto -1/n^2$ and revivals occur in the radial wave function) - Molecular vibrations in cold molecules trapped in optical lattices - Cold atom systems where atoms are confined in optical potentials engineered to mimic the infinite well

🧪 Experiment: In 1996, Yeazell and Stroud observed revival dynamics in Rydberg wave packets in potassium atoms. By exciting a coherent superposition of states near $n \approx 65$, they observed the wave packet spread, fractionally revive into multiple copies, and fully revive to its original form — a stunning confirmation of the quantum theory of time evolution.

📊 By the Numbers: For an electron in a quantum dot of width $a = 10$ nm, $T_{\text{rev}} = 4ma^2/(\pi\hbar) \approx 3.5 \times 10^{-13}$ s $\approx 0.35$ ps. Fast, but within reach of modern ultrafast spectroscopy.

The Hierarchy of Time Scales

For a wave packet centered around quantum number $\bar{n}$ in the infinite well, there are three important time scales:

1. Classical period: $T_{\text{cl}} = \pi\hbar/(\bar{n}E_1) = T_{\text{rev}}/(2\bar{n})$. This is the time for the wave packet to traverse the well once (bouncing between walls like a classical particle).

2. Collapse time: $T_{\text{collapse}} \sim T_{\text{cl}} \cdot \bar{n}/(\Delta n)$ where $\Delta n$ is the spread in quantum numbers. This is the time for the wave packet to spread significantly due to the dephasing of its energy components.

3. Revival time: $T_{\text{rev}} = 4ma^2/(\pi\hbar)$. This is independent of $\bar{n}$ — all wave packets in the infinite well revive at the same time.

The ordering is $T_{\text{cl}} \ll T_{\text{collapse}} \ll T_{\text{rev}}$ for large $\bar{n}$. This explains the three phases of wave packet dynamics: - For $t \lesssim T_{\text{cl}}$: classical-like bouncing - For $T_{\text{collapse}} \lesssim t \lesssim T_{\text{rev}}$: spread out, apparently random - At $t = T_{\text{rev}}$: miraculous reassembly

The ratio $T_{\text{rev}}/T_{\text{cl}} = 2\bar{n}$ tells us how many classical bounces occur before the revival. For $\bar{n} = 50$, that is 100 bounces — the packet bounces back and forth a hundred times, spreads into apparent noise, and then perfectly reassembles. It is as if you shuffled a deck of cards a hundred times and it returned to its original order. In classical mechanics, this would require a miracle. In quantum mechanics, it is a theorem.

7.5 The Schrödinger Picture

States Evolve, Operators Stay Fixed

Everything we have done so far in this textbook lives in the Schrödinger picture — the framework where:

• States carry the time dependence: $\Psi(x, t) = \hat{U}(t)\, \Psi(x, 0)$
• Operators (like $\hat{x}$, $\hat{p}$, $\hat{H}$) are time-independent (unless they have explicit time dependence in their definition)

The physical content — the experimentally measurable expectation values — are computed as:

$$\langle \hat{A} \rangle(t) = \int \Psi^*(x, t)\, \hat{A}\, \Psi(x, t)\, dx$$

Think of this as a movie: the operators are the fixed camera, and the quantum state is the actor moving through the scene. The picture you see (the expectation value) depends on both the camera position and the actor's position, but in the Schrödinger picture, only the actor moves.

This is intuitive. It matches how we think classically: the position of a particle changes; the definition of "position" does not. And it matches how we have set up quantum mechanics from Chapter 2 onward: the wave function evolves according to the Schrödinger equation, and operators are fixed mathematical objects.

The Schrödinger picture has several practical advantages. First, it is the natural framework for solving the time-independent Schrödinger equation — which is how we have found all our energy eigenstates so far. Second, the wave function $\Psi(x,t)$ has a direct physical interpretation via the Born rule: $|\Psi(x,t)|^2$ is the probability density for finding the particle at position $x$ at time $t$. Third, it is the most common framework in textbooks and in the historical development of quantum mechanics.

Its main disadvantage is that every calculation requires knowing $\Psi(x,t)$, which means solving the full time-dependent Schrödinger equation. For time-dependent Hamiltonians, this can be extremely difficult. Moreover, the time dependence of the state can obscure the symmetry structure of the problem — conservation laws, for instance, are less transparent in the Schrödinger picture.

But here is the remarkable fact: this is not the only consistent way to do it.

✅ Check Your Understanding (CYU-7.3): 1. In the Schrödinger picture, what is time-independent: the state, the operator, or the expectation value? 2. Convince yourself that if $\hat{A}$ commutes with $\hat{H}$, then $\langle \hat{A} \rangle$ is time-independent even for a general (non-stationary) state. What physical principle does this reflect? 3. Spaced review (Ch 6): What does $[\hat{A}, \hat{H}] = 0$ tell you about the observables $A$ and $H$?

7.6 The Heisenberg Picture

Operators Evolve, States Stay Fixed

Werner Heisenberg, in his original 1925 formulation of quantum mechanics (which preceded Schrödinger's wave equation by about a year), worked with time-dependent operators and fixed states. In the Heisenberg picture:

• States are frozen at their initial values: $\Psi_H = \Psi(x, 0)$ (the subscript $H$ denotes Heisenberg picture)
• Operators carry all the time dependence:

$$\boxed{\hat{A}_H(t) = \hat{U}^\dagger(t)\, \hat{A}_S\, \hat{U}(t) = e^{i\hat{H}t/\hbar}\, \hat{A}_S\, e^{-i\hat{H}t/\hbar}}$$

where $\hat{A}_S$ is the Schrödinger-picture operator (time-independent).

The expectation value is unchanged — as it must be, since the physics cannot depend on our bookkeeping choice:

$$\langle \hat{A} \rangle(t) = \langle \Psi(t)| \hat{A}_S | \Psi(t) \rangle_S = \langle \Psi(0)| \hat{U}^\dagger \hat{A}_S \hat{U} | \Psi(0) \rangle = \langle \Psi_H | \hat{A}_H(t) | \Psi_H \rangle_H$$

Both pictures give the same experimental predictions. The choice between them is purely a matter of mathematical convenience.

🔵 Historical Note: It was Dirac who showed in 1927 that the Schrödinger and Heisenberg formulations are mathematically equivalent. This was one of the great clarifying moments in the history of quantum mechanics — two apparently incompatible theories (Schrödinger's wave mechanics and Heisenberg's matrix mechanics) turned out to be the same theory in different "pictures."

The Heisenberg Equation of Motion

Differentiating $\hat{A}_H(t)$ with respect to time:

$$\frac{d\hat{A}_H}{dt} = \frac{i}{\hbar}\hat{U}^\dagger \hat{H}\hat{A}_S\hat{U} - \frac{i}{\hbar}\hat{U}^\dagger\hat{A}_S\hat{H}\hat{U} = \frac{i}{\hbar}\hat{U}^\dagger[\hat{H}, \hat{A}_S]\hat{U}$$

Since $[\hat{H}, \hat{A}_S]$ is itself an operator that can be transformed to the Heisenberg picture, and since $\hat{U}^\dagger \hat{H} \hat{U} = \hat{H}$ when $\hat{H}$ is time-independent, we obtain the Heisenberg equation of motion:

$$\boxed{\frac{d\hat{A}_H}{dt} = \frac{i}{\hbar}[\hat{H}, \hat{A}_H(t)] + \left(\frac{\partial \hat{A}}{\partial t}\right)_H}$$

The last term accounts for operators with explicit time dependence (like a potential that is being switched on). For most operators we encounter, $\partial \hat{A}/\partial t = 0$, and we get the elegant result:

$$\frac{d\hat{A}_H}{dt} = \frac{i}{\hbar}[\hat{H}, \hat{A}_H(t)]$$

💡 Key Insight: This equation is the quantum analogue of Hamilton's equations in classical mechanics. In classical mechanics, $dA/dt = \{A, H\}$ where $\{A, H\}$ is the Poisson bracket. The correspondence is:

$$\{A, H\}_{\text{classical}} \longleftrightarrow \frac{1}{i\hbar}[\hat{A}, \hat{H}]_{\text{quantum}}$$

This is Dirac's quantization rule: replace Poisson brackets with commutators (divided by $i\hbar$). The Heisenberg equation makes this connection transparent.

Example: Free Particle in the Heisenberg Picture

For a free particle, $\hat{H} = \hat{p}^2/(2m)$. Using the commutator results from Chapter 6:

$$\frac{d\hat{x}_H}{dt} = \frac{i}{\hbar}\left[\frac{\hat{p}_H^2}{2m}, \hat{x}_H\right] = \frac{\hat{p}_H}{m}$$

$$\frac{d\hat{p}_H}{dt} = \frac{i}{\hbar}\left[\frac{\hat{p}_H^2}{2m}, \hat{p}_H\right] = 0$$

The momentum is constant. The position evolves linearly:

$$\hat{x}_H(t) = \hat{x}_H(0) + \frac{\hat{p}_H}{m}t$$

These are exactly Newton's equations — but now they are operator equations. Taking expectation values: $\langle \hat{x} \rangle(t) = \langle \hat{x} \rangle(0) + \langle \hat{p} \rangle t/m$ and $\langle \hat{p} \rangle(t) = \langle \hat{p} \rangle(0)$. Classical mechanics emerges from the operator algebra.

Harmonic Oscillator in the Heisenberg Picture

For the QHO, $\hat{H} = \hat{p}^2/(2m) + m\omega^2\hat{x}^2/2$:

$$\frac{d\hat{x}_H}{dt} = \frac{\hat{p}_H}{m}, \qquad \frac{d\hat{p}_H}{dt} = -m\omega^2 \hat{x}_H$$

These are coupled first-order equations. Differentiating the first and substituting the second:

$$\frac{d^2\hat{x}_H}{dt^2} = -\omega^2 \hat{x}_H$$

This is the classical equation of motion for a harmonic oscillator — but for the operator. The solution is:

$$\hat{x}_H(t) = \hat{x}_H(0)\cos(\omega t) + \frac{\hat{p}_H(0)}{m\omega}\sin(\omega t)$$

🔗 Spaced Review (Ch 4): This connects beautifully to the coherent states of the QHO. A coherent state $|\alpha\rangle$ has $\langle \hat{x} \rangle(t) = \sqrt{2\hbar/(m\omega)}\,\text{Re}(\alpha e^{-i\omega t})$ — it oscillates sinusoidally just like a classical oscillator, which is why coherent states are the "most classical" quantum states.

Worked Example 7.4: Conservation Laws in the Heisenberg Picture

The Heisenberg equation provides the most elegant way to derive conservation laws. An observable $\hat{A}$ is conserved (constant of the motion) if and only if $[\hat{H}, \hat{A}] = 0$.

Example: Momentum conservation for a free particle. For $\hat{H} = \hat{p}^2/(2m)$:

$$\frac{d\hat{p}_H}{dt} = \frac{i}{\hbar}[\hat{H}, \hat{p}_H] = \frac{i}{\hbar}\frac{1}{2m}[\hat{p}^2, \hat{p}] = 0$$

since any operator commutes with a function of itself. Momentum is conserved — exactly as in classical mechanics for a free particle.

Example: Energy conservation. For any time-independent Hamiltonian:

$$\frac{d\hat{H}_H}{dt} = \frac{i}{\hbar}[\hat{H}, \hat{H}] = 0$$

Energy is always conserved when the Hamiltonian has no explicit time dependence. This is the quantum version of the classical result that the Hamiltonian is a constant of the motion when it has no explicit time dependence.

Example: Angular momentum in a central potential. If $V = V(r)$ depends only on the radial distance, then $[\hat{H}, \hat{L}^2] = 0$ and $[\hat{H}, \hat{L}_z] = 0$ (we will prove this carefully in Chapter 10). The Heisenberg equation immediately tells us $d\hat{L}^2_H/dt = 0$ and $d(\hat{L}_z)_H/dt = 0$: angular momentum is conserved. In the Schrödinger picture, deriving the same result requires more work — you have to show that the expectation values $\langle\hat{L}^2\rangle$ and $\langle\hat{L}_z\rangle$ are time-independent for any state, which is equivalent but less transparent.

This is a powerful illustration of why physicists love the Heisenberg picture: conservation laws are commutators, and commutators are easy to compute. The connection between symmetry and conservation — the most profound theme in modern physics — is most naturally expressed in Heisenberg's language.

7.7 The Interaction Picture

The Best of Both Worlds

In many practical problems, the Hamiltonian naturally splits into two parts:

$$\hat{H} = \hat{H}_0 + \hat{V}(t)$$

where $\hat{H}_0$ is a "simple" Hamiltonian we can solve exactly (the free particle, the harmonic oscillator, the hydrogen atom) and $\hat{V}(t)$ is a "perturbation" — perhaps a time-dependent external field.

The interaction picture (also called the Dirac picture) splits the time evolution between states and operators:

• Operators evolve under $\hat{H}_0$ (the "easy" part):

$$\hat{A}_I(t) = e^{i\hat{H}_0 t/\hbar}\, \hat{A}_S\, e^{-i\hat{H}_0 t/\hbar}$$

• States evolve under $\hat{V}$ (the "interesting" part):

$$i\hbar \frac{\partial}{\partial t}|\Psi_I(t)\rangle = \hat{V}_I(t)\, |\Psi_I(t)\rangle$$

where $\hat{V}_I(t) = e^{i\hat{H}_0 t/\hbar}\, \hat{V}(t)\, e^{-i\hat{H}_0 t/\hbar}$ is the perturbation in the interaction picture.

The interaction picture state is related to the Schrödinger picture state by:

$$|\Psi_I(t)\rangle = e^{i\hat{H}_0 t/\hbar}\, |\Psi_S(t)\rangle$$

In words: we "factor out" the known, boring time evolution due to $\hat{H}_0$ and focus on the new, interesting dynamics caused by $\hat{V}$.

Why the Interaction Picture Matters

The interaction picture is the natural setting for perturbation theory. When $\hat{V}$ is small compared to $\hat{H}_0$, the interaction-picture state changes slowly, and we can expand the time evolution perturbatively — leading to Fermi's golden rule, transition rates, and the entire apparatus of time-dependent perturbation theory that we will develop in Chapter 21.

To see how perturbation theory emerges, note that the interaction-picture state satisfies:

$$|\Psi_I(t)\rangle = |\Psi_I(0)\rangle - \frac{i}{\hbar}\int_0^t \hat{V}_I(t')\,|\Psi_I(t')\rangle\, dt'$$

If $\hat{V}$ is small, $|\Psi_I(t)\rangle \approx |\Psi_I(0)\rangle$ to zeroth order, and we can iterate:

$$|\Psi_I(t)\rangle \approx |\Psi_I(0)\rangle - \frac{i}{\hbar}\int_0^t \hat{V}_I(t')\,|\Psi_I(0)\rangle\, dt' + \left(-\frac{i}{\hbar}\right)^2\int_0^t dt'\int_0^{t'} dt''\, \hat{V}_I(t')\hat{V}_I(t'')\,|\Psi_I(0)\rangle + \cdots$$

This is the Dyson series — the backbone of time-dependent perturbation theory. The first-order term gives Fermi's golden rule; the second-order term gives scattering amplitudes. We will develop all of this systematically in Chapter 21, but it is important to see where it comes from: the interaction picture isolates the "interesting" dynamics, and the perturbation expansion organizes them order by order.

It is also the standard language of quantum field theory and scattering theory (Chapter 22). When you see the S-matrix, Feynman diagrams, or the Dyson series in advanced physics, you are working in the interaction picture.

Comparison: Three Pictures at a Glance

The following table summarizes the three pictures. All three give identical physics — identical expectation values for all observables at all times.

💡 Key Insight: The three pictures are related by unitary transformations. Going from Schrödinger to Heisenberg: multiply operators by $\hat{U}^\dagger$ from the left and $\hat{U}$ from the right. Going from Schrödinger to interaction: factor $\hat{U}$ into $\hat{U}_0 \hat{U}_I$ where $\hat{U}_0 = e^{-i\hat{H}_0 t/\hbar}$. The physics is invariant under these transformations because expectation values are preserved by unitary conjugation.

⚖️ Interpretation: The existence of multiple equivalent pictures reflects something deep about quantum mechanics: the division between "system" and "measurement apparatus," between "what evolves" and "what stays fixed," is not physically determined. It is a choice — a gauge freedom, if you will. This flexibility becomes even more profound in quantum field theory and in discussions of the measurement problem (Chapter 33).

✅ Check Your Understanding (CYU-7.4): 1. At $t = 0$, all three pictures coincide. Why? 2. In the interaction picture, if $\hat{V} = 0$ (no perturbation), what happens to the states? What happens to the operators? Which standard picture does this reduce to? 3. Explain in one sentence why the interaction picture is ideal for perturbation theory.

7.8 Ehrenfest's Theorem: The Classical Limit

From Quantum to Classical

One of the most important questions in physics: under what conditions does quantum mechanics reduce to classical mechanics? After all, classical mechanics works beautifully for planets, baseballs, and bridges. It must be hiding inside quantum mechanics somewhere.

Ehrenfest's theorem provides the bridge. Starting from the Heisenberg equation of motion, take the expectation value:

$$\frac{d}{dt}\langle \hat{A} \rangle = \left\langle \frac{i}{\hbar}[\hat{H}, \hat{A}] \right\rangle + \left\langle \frac{\partial \hat{A}}{\partial t} \right\rangle$$

Deriving Ehrenfest's Equations

For a particle in a potential $V(x)$ with Hamiltonian $\hat{H} = \hat{p}^2/(2m) + V(\hat{x})$, apply this to $\hat{x}$ and $\hat{p}$:

Position:

$$\frac{d\langle \hat{x} \rangle}{dt} = \frac{i}{\hbar}\left\langle \left[\frac{\hat{p}^2}{2m} + V(\hat{x}),\, \hat{x}\right] \right\rangle$$

Using $[\hat{p}^2, \hat{x}] = -2i\hbar\hat{p}$ (from the fundamental commutator $[\hat{x}, \hat{p}] = i\hbar$ — recall Chapter 6) and $[V(\hat{x}), \hat{x}] = 0$:

$$\boxed{\frac{d\langle \hat{x} \rangle}{dt} = \frac{\langle \hat{p} \rangle}{m}}$$

This is Newton's first law in disguise: the velocity of the center of the wave packet equals the average momentum divided by the mass.

Momentum:

$$\frac{d\langle \hat{p} \rangle}{dt} = \frac{i}{\hbar}\left\langle \left[\frac{\hat{p}^2}{2m} + V(\hat{x}),\, \hat{p}\right] \right\rangle$$

Using $[\hat{p}^2, \hat{p}] = 0$ and $[V(\hat{x}), \hat{p}] = i\hbar V'(\hat{x})$ (this follows from $[\hat{x}^n, \hat{p}] = i\hbar n\hat{x}^{n-1}$ and extending to Taylor-expandable functions):

$$\boxed{\frac{d\langle \hat{p} \rangle}{dt} = -\left\langle \frac{dV}{dx}(\hat{x}) \right\rangle = \langle \hat{F}(\hat{x}) \rangle}$$

This is Newton's second law — almost. The rate of change of the average momentum equals the average of the force.

The "Almost" That Matters

The crucial subtlety is the difference between $\langle F(\hat{x}) \rangle$ and $F(\langle \hat{x} \rangle)$. Ehrenfest's theorem gives:

$$\frac{d\langle \hat{p} \rangle}{dt} = \langle F(\hat{x}) \rangle$$

Classical mechanics gives:

$$\frac{dp}{dt} = F(x)$$

These are the same only if $\langle F(\hat{x}) \rangle = F(\langle \hat{x} \rangle)$. When is this true?

For a linear force ($F = -kx$, i.e., the harmonic oscillator), $\langle F(\hat{x}) \rangle = -k\langle \hat{x} \rangle = F(\langle \hat{x} \rangle)$ exactly. The expectation values obey classical equations exactly, for any state — not just narrow wave packets.

For a general potential, $\langle F(\hat{x}) \rangle \approx F(\langle \hat{x} \rangle)$ when the wave packet is narrow compared to the length scale over which $F$ varies. This is the classical limit: quantum mechanics reduces to classical mechanics when the wave packet is so localized that the force does not vary appreciably across its width.

Formally, expand $F(\hat{x})$ around $\langle \hat{x} \rangle$:

$$F(\hat{x}) = F(\langle \hat{x} \rangle) + F'(\langle \hat{x} \rangle)(\hat{x} - \langle \hat{x} \rangle) + \frac{1}{2}F''(\langle \hat{x} \rangle)(\hat{x} - \langle \hat{x} \rangle)^2 + \cdots$$

Taking expectation values:

$$\langle F(\hat{x}) \rangle = F(\langle \hat{x} \rangle) + \frac{1}{2}F''(\langle \hat{x} \rangle)\, (\Delta x)^2 + \cdots$$

The correction term involves $(\Delta x)^2 = \langle (\hat{x} - \langle \hat{x} \rangle)^2 \rangle$, the position uncertainty squared. Classical mechanics is recovered when $(\Delta x)^2 |F''|/|F| \ll 1$ — that is, when the curvature of the force does not matter over the width of the wave packet.

🔴 Warning: Ehrenfest's theorem does not say that quantum mechanics always reduces to classical mechanics in some limit. It says that the expectation values of position and momentum satisfy classical-looking equations. But the full quantum state contains much more information than just the expectation values — it encodes all the higher moments, interference, entanglement, and so on. Classical mechanics emerges when all this extra structure becomes irrelevant, which is a statement about the state (narrow, decoherent) not just the equations.

🔗 Connection (Ch 6): Ehrenfest's theorem is ultimately a consequence of the commutation relation $[\hat{x}, \hat{p}] = i\hbar$. If this commutator were zero (as it is classically, where position and momentum are just numbers), the Heisenberg equation would trivially reduce to Hamilton's equations. The nonzero commutator is what makes quantum mechanics different — and Ehrenfest's theorem tells us when that difference stops mattering for the average behavior.

Worked Example 7.5: The Classical Limit of the Harmonic Oscillator

For the QHO with $V(x) = \frac{1}{2}m\omega^2 x^2$, Ehrenfest's equations give:

$$\frac{d\langle\hat{x}\rangle}{dt} = \frac{\langle\hat{p}\rangle}{m}$$

$$\frac{d\langle\hat{p}\rangle}{dt} = -\langle m\omega^2 \hat{x}\rangle = -m\omega^2\langle\hat{x}\rangle$$

These are exact — no approximation required — because $F(x) = -m\omega^2 x$ is linear, so $\langle F(\hat{x})\rangle = F(\langle\hat{x}\rangle)$ identically. Combining:

$$\frac{d^2\langle\hat{x}\rangle}{dt^2} = -\omega^2\langle\hat{x}\rangle$$

This is exactly Newton's equation for a classical harmonic oscillator. The solution is:

$$\langle\hat{x}\rangle(t) = \langle\hat{x}\rangle(0)\cos(\omega t) + \frac{\langle\hat{p}\rangle(0)}{m\omega}\sin(\omega t)$$

For a coherent state (Chapter 4) with $\langle\hat{x}\rangle(0) = x_0$ and $\langle\hat{p}\rangle(0) = p_0$, this traces out the classical trajectory exactly. But even for a bizarre, highly non-classical state (say, a superposition of the $n = 0$ and $n = 100$ eigenstates), the expectation values still follow the classical equation. The QHO is special in this regard — it is the only potential for which Ehrenfest's theorem is exact for all states.

This is closely related to a result you may recall from classical mechanics: the harmonic oscillator is the only nonlinear system whose period is independent of amplitude. The same mathematical structure — linearity of the restoring force — that makes the classical period universal makes the quantum Ehrenfest equations exact.

When Does the Classical Limit Fail?

Consider a double-well potential $V(x) = \lambda(x^2 - a^2)^2$. A particle initially localized in one well will quantum-mechanically tunnel to the other well with a tunneling time $T_{\text{tunnel}}$ (recall Chapter 3). Ehrenfest's theorem gives:

$$\frac{d\langle\hat{p}\rangle}{dt} = -\langle V'(\hat{x})\rangle$$

For a state localized in the left well, $\langle\hat{x}\rangle \approx -a$ and the classical force $F(\langle\hat{x}\rangle) = 0$ (the particle sits at the bottom of the well). But quantum mechanically, the particle tunnels. The classical trajectory says "stay put"; the quantum expectation value slowly oscillates between the two wells.

What went wrong? The wave function has significant width spanning the classically forbidden region between the wells. The force $F(x) = -V'(x)$ varies enormously across this width — it points leftward on the left side of the barrier and rightward on the right side. The approximation $\langle F(\hat{x})\rangle \approx F(\langle\hat{x}\rangle)$ fails catastrophically. This is tunneling: a phenomenon that Ehrenfest's theorem cannot capture because the wave packet is broad on the scale of the potential's variation.

The general lesson: Ehrenfest's theorem successfully reduces quantum mechanics to classical mechanics only when the wave packet remains narrow compared to the features of the potential for the entire duration of interest. For times long enough that quantum effects (tunneling, interference, spreading) become important, the classical approximation breaks down.

✅ Check Your Understanding (CYU-7.5): 1. Show that for a free particle ($V = 0$), Ehrenfest's theorem gives $d^2\langle \hat{x} \rangle/dt^2 = 0$. What does this mean physically? 2. For the QHO ($V = m\omega^2 x^2/2$), show that $d\langle \hat{p} \rangle/dt = -m\omega^2 \langle \hat{x} \rangle$. Does the classical limit condition matter here? 3. For $V(x) = \lambda x^4$, what is the leading correction to the classical equation of motion? At what width $\Delta x$ does the correction become 10% of the leading term?

7.9 Rabi Oscillations: A Two-Level System in a Driving Field

The Simplest Time-Dependent Problem

We now turn to what is arguably the most important time-dependent problem in all of quantum mechanics: a two-level system driven by an oscillating external field. This single problem describes:

• An atom in a laser field (quantum optics)
• A nuclear spin in a magnetic field (NMR/MRI)
• A qubit being manipulated by microwave pulses (quantum computing)
• A molecule undergoing a stimulated transition (spectroscopy)

The universality of this problem cannot be overstated. Master it, and you have the key to an enormous range of physics.

Setup: The Two-Level Hamiltonian

Consider a quantum system with two energy levels $|1\rangle$ and $|2\rangle$ with energies $E_1$ and $E_2$ (where $E_2 > E_1$). Define the transition frequency:

$$\omega_0 = \frac{E_2 - E_1}{\hbar}$$

The unperturbed Hamiltonian is:

$$\hat{H}_0 = E_1 |1\rangle\langle 1| + E_2 |2\rangle\langle 2|$$

Now apply a time-dependent perturbation — an oscillating field with frequency $\omega$ near resonance ($\omega \approx \omega_0$):

$$\hat{V}(t) = V_0 \cos(\omega t)\left(|1\rangle\langle 2| + |2\rangle\langle 1|\right)$$

The total Hamiltonian is $\hat{H} = \hat{H}_0 + \hat{V}(t)$. The general state is:

$$|\Psi(t)\rangle = c_1(t)\, e^{-iE_1 t/\hbar}\, |1\rangle + c_2(t)\, e^{-iE_2 t/\hbar}\, |2\rangle$$

where the exponentials account for the unperturbed time evolution (this is essentially the interaction picture).

The Rotating Wave Approximation

Substituting into the Schrödinger equation yields coupled equations for $c_1(t)$ and $c_2(t)$:

$$i\hbar \dot{c}_1 = \frac{V_0}{2}\left(e^{i(\omega_0 - \omega)t} + e^{i(\omega_0 + \omega)t}\right) c_2$$

$$i\hbar \dot{c}_2 = \frac{V_0}{2}\left(e^{-i(\omega_0 - \omega)t} + e^{-i(\omega_0 + \omega)t}\right) c_1$$

The terms with $e^{\pm i(\omega_0 + \omega)t}$ oscillate rapidly (at frequency $\omega_0 + \omega \approx 2\omega_0$) and average to nearly zero over the timescale on which $c_1$ and $c_2$ change. Dropping these fast-oscillating terms is the rotating wave approximation (RWA) — one of the most useful approximations in all of physics.

After the RWA, defining the detuning $\delta = \omega - \omega_0$ and the Rabi frequency $\Omega_R = V_0/\hbar$:

$$i\dot{c}_1 = \frac{\Omega_R}{2} e^{-i\delta t}\, c_2, \qquad i\dot{c}_2 = \frac{\Omega_R}{2} e^{i\delta t}\, c_1$$

The Solution: Rabi Oscillations

For the resonant case ($\delta = 0$, meaning the driving frequency exactly matches the transition frequency), the solution with initial condition $c_1(0) = 1$, $c_2(0) = 0$ (system starts in the ground state) is:

$$c_1(t) = \cos\left(\frac{\Omega_R t}{2}\right), \qquad c_2(t) = -i\sin\left(\frac{\Omega_R t}{2}\right)$$

The probabilities are:

$$P_1(t) = \cos^2\!\left(\frac{\Omega_R t}{2}\right), \qquad P_2(t) = \sin^2\!\left(\frac{\Omega_R t}{2}\right)$$

The system oscillates between the two states with frequency $\Omega_R/2$. These are Rabi oscillations — the quantum mechanical analogue of a pendulum swinging between two equivalent positions.

Let us pause and appreciate what this means physically. The system does not gradually transition from $|1\rangle$ to $|2\rangle$ and stay there. Instead, it oscillates coherently — absorbing energy from the driving field, reaching $|2\rangle$, then returning the energy to the field and falling back to $|1\rangle$. This coherent back-and-forth continues indefinitely (in the absence of dissipation). The key word is "coherent": the phase relationship between $c_1$ and $c_2$ is maintained throughout the oscillation. This is fundamentally different from incoherent absorption and emission (which is described by rate equations, not the Schrödinger equation).

Worked Example 7.6: Deriving the Off-Resonant Solution

For the off-resonant case ($\delta \neq 0$), we need to solve the coupled equations:

$$i\dot{c}_1 = \frac{\Omega_R}{2}e^{-i\delta t}c_2, \qquad i\dot{c}_2 = \frac{\Omega_R}{2}e^{i\delta t}c_1$$

Differentiate the second equation:

$$i\ddot{c}_2 = \frac{\Omega_R}{2}\left(i\delta\, e^{i\delta t} c_1 + e^{i\delta t}\dot{c}_1\right)$$

Substitute $\dot{c}_1 = -i(\Omega_R/2)e^{-i\delta t}c_2$ and $c_1 = (2i/\Omega_R)e^{-i\delta t}\dot{c}_2$:

$$i\ddot{c}_2 = \frac{\Omega_R}{2}\left(i\delta \cdot \frac{2i}{\Omega_R}\dot{c}_2 + e^{i\delta t}\cdot\left(-\frac{i\Omega_R}{2}e^{-i\delta t}c_2\right)\right)$$

$$i\ddot{c}_2 = -\delta\dot{c}_2 + \frac{\Omega_R^2}{4}(-i)c_2$$

Rearranging: $\ddot{c}_2 + i\delta\dot{c}_2 + (\Omega_R^2/4)c_2 = 0$.

This is a second-order ODE with constant (complex) coefficients. The characteristic equation $\lambda^2 + i\delta\lambda + \Omega_R^2/4 = 0$ has roots $\lambda = (-i\delta \pm i\Omega)/2$ where $\Omega = \sqrt{\Omega_R^2 + \delta^2}$. With initial conditions $c_2(0) = 0$ and $\dot{c}_2(0) = -i\Omega_R c_1(0)/2 = -i\Omega_R/2$, the solution is:

$$c_2(t) = -i\frac{\Omega_R}{\Omega}\sin\left(\frac{\Omega t}{2}\right)e^{i\delta t/2}$$

The transition probability is then:

$$P_{1\to 2}(t) = |c_2(t)|^2 = \frac{\Omega_R^2}{\Omega^2}\sin^2\!\left(\frac{\Omega t}{2}\right)$$

The generalized Rabi frequency is:

$$\boxed{\Omega = \sqrt{\Omega_R^2 + \delta^2}}$$

and the transition probability is:

$$\boxed{P_{1\to 2}(t) = \frac{\Omega_R^2}{\Omega^2}\sin^2\!\left(\frac{\Omega t}{2}\right)}$$

The maximum transition probability is $\Omega_R^2/\Omega^2 < 1$ when off-resonance: the system never fully transitions to state $|2\rangle$. The amplitude of the oscillation is reduced, and the frequency is increased. This is the quantum mechanical origin of resonance: the driving is most effective when the driving frequency matches the natural frequency.

Special Pulses

Specific pulse durations have enormous practical importance:

• $\pi$-pulse: $\Omega_R t = \pi$. Completely inverts the population: $|1\rangle \to |2\rangle$. This is how you flip a qubit.
• $\pi/2$-pulse: $\Omega_R t = \pi/2$. Creates an equal superposition: $|1\rangle \to (|1\rangle - i|2\rangle)/\sqrt{2}$. This is the quantum analogue of a beam splitter — and is exactly the Hadamard gate in quantum computing (up to phases).
• $2\pi$-pulse: $\Omega_R t = 2\pi$. Returns the system to its original state, but with a sign change: $|1\rangle \to -|1\rangle$. This geometric phase has observable consequences.

📊 By the Numbers: In nuclear magnetic resonance (NMR), a proton in a 1 Tesla magnetic field has a Larmor frequency of $\omega_0/(2\pi) = 42.58$ MHz. Typical Rabi frequencies are $\Omega_R/(2\pi) \sim 10$ kHz, so a $\pi$-pulse takes about 50 microseconds. In trapped-ion quantum computers, Rabi frequencies can exceed $\Omega_R/(2\pi) \sim 100$ kHz, giving $\pi$-pulse times of about 5 microseconds. In superconducting qubits, $\pi$-pulse times can be as short as 10-20 nanoseconds.

🧪 Experiment: Rabi oscillations were first observed by I. I. Rabi in 1938 using molecular beams, for which he received the 1944 Nobel Prize. The experiment measured the magnetic moments of nuclei by observing the resonant absorption of radio-frequency radiation — the technique that would later become nuclear magnetic resonance (NMR) and then magnetic resonance imaging (MRI). Every MRI scan you have ever had relies on the physics of this section.

The Physics of Resonance

The Rabi formula reveals the fundamental physics of resonance. At exact resonance ($\delta = 0$), complete population transfer occurs: $P_{1\to 2}^{\max} = 1$. Away from resonance, the maximum decreases as a Lorentzian:

$$P_{1\to 2}^{\max} = \frac{\Omega_R^2}{\Omega_R^2 + \delta^2}$$

The full width at half maximum (FWHM) of this resonance curve is $2\Omega_R$ — the stronger the driving, the broader the resonance. This is a general feature: strong driving makes the system less selective about the frequency (this is related to the energy-time uncertainty relation, $\Delta E \cdot \Delta t \gtrsim \hbar/2$ — a faster pulse has broader frequency content).

The off-resonant oscillation frequency $\Omega = \sqrt{\Omega_R^2 + \delta^2}$ is always faster than the resonant Rabi frequency $\Omega_R$. This is counterintuitive at first: you might expect off-resonant driving to be slower. But the oscillation is between the dressed states (eigenstates of the driven Hamiltonian), not the bare states $|1\rangle$ and $|2\rangle$. Off-resonance, the dressed states are closer to the bare states, so less population is transferred per cycle — but the cycles are faster.

Connection to the Mach-Zehnder Interferometer

There is a beautiful connection between Rabi oscillations and our anchor example. The beam splitter in a Mach-Zehnder interferometer performs a $\pi/2$ rotation on the two-state system $\{|U\rangle, |L\rangle\}$:

$$\text{Beam splitter:} \quad |L\rangle \to \frac{1}{\sqrt{2}}(|U\rangle + i|L\rangle)$$

This is precisely a $\pi/2$-pulse. The mirror performs a $\pi$-pulse (reflection). The entire interferometer is a sequence of Rabi-like rotations on a two-level system. This is why quantum optics and quantum computing use the same mathematical language — the Bloch sphere (Chapter 13) unifies them all.

⚖️ Interpretation: Rabi oscillations illustrate a core theme of this chapter: time evolution in quantum mechanics is unitary rotation. Whether we are talking about a photon in an interferometer, a spin in a magnetic field, or a qubit in a quantum computer, the time-evolution operator rotates the state vector in Hilbert space. Different physical systems are described by different Hamiltonians, but the mathematical structure — unitary evolution, superposition, interference — is universal.

✅ Check Your Understanding (CYU-7.6): 1. At exact resonance ($\delta = 0$), at what time is the probability of finding the system in state $|2\rangle$ first equal to 1/2? 2. If you double the driving field amplitude (doubling $V_0$ and hence $\Omega_R$), what happens to the $\pi$-pulse time? 3. Why does the maximum transition probability decrease when the driving frequency is detuned from resonance? Give both a mathematical and a physical explanation.

7.10 Summary and Looking Ahead

What We Have Learned

This chapter has covered the heart of quantum dynamics. Let us collect the key results:

The time-evolution operator $\hat{U}(t) = e^{-i\hat{H}t/\hbar}$ (for time-independent $\hat{H}$) is the fundamental object governing how quantum states change in time. It is unitary, preserving probabilities, and the Hamiltonian is its generator.

Stationary states $\psi_n(x)$ acquire only a phase factor $e^{-iE_n t/\hbar}$ under time evolution. All expectation values are constant — these states are genuinely "stationary" in every measurable sense.

Wave packet dynamics arise from the superposition of energy eigenstates with different frequencies. Free-particle wave packets spread due to dispersion. In the infinite well, wave packets undergo quantum revivals at $T_{\text{rev}} = 4ma^2/(\pi\hbar)$, with fractional revivals at rational fractions of $T_{\text{rev}}$.

Three pictures — Schrödinger, Heisenberg, and interaction — describe the same physics with different bookkeeping. The Schrödinger picture evolves states; the Heisenberg picture evolves operators; the interaction picture splits the evolution, making it the natural setting for perturbation theory.

Ehrenfest's theorem connects quantum and classical mechanics: $d\langle \hat{x}\rangle/dt = \langle \hat{p}\rangle/m$ and $d\langle \hat{p}\rangle/dt = \langle \hat{F}\rangle$. Classical mechanics is recovered when wave packets are narrow compared to the scale of force variation.

Rabi oscillations describe the coherent oscillation of a two-level system driven by a resonant field. The Rabi frequency $\Omega_R = V_0/\hbar$ sets the oscillation rate; the generalized Rabi frequency $\Omega = \sqrt{\Omega_R^2 + \delta^2}$ accounts for detuning.

Key Equations at a Glance

Project Checkpoint: Quantum Toolkit v0.7

In this chapter's code component (see code/project-checkpoint.py), you add the following to your growing quantum simulation toolkit:

• time_evolve(psi, H, t) — Apply the time-evolution operator to evolve state psi under Hamiltonian H for time t
• wave_packet(x, x0, sigma, k0) — Construct a Gaussian wave packet with specified center, width, and momentum
• ehrenfest_check(psi_t, x_grid, H, m) — Numerically verify Ehrenfest's theorem by comparing $d\langle\hat{x}\rangle/dt$ with $\langle\hat{p}\rangle/m$

Looking Ahead

Chapter 8 is the bridge chapter — perhaps the most important chapter in the entire book. We will introduce Dirac notation (bras, kets, inner products), which will transform the way we write and think about quantum mechanics. Everything we have done in wave-mechanics notation ($\psi$, $\Psi$, integrals) will be rewritten in the compact, powerful, representation-independent language of $|\psi\rangle$, $\langle\phi|$, and $\langle\phi|\hat{A}|\psi\rangle$. The Heisenberg picture, which already looked clean in wave mechanics, becomes positively elegant in Dirac notation. And the interaction picture — which looked like a mathematical trick here — will become the natural framework for the perturbation theory we develop in Part IV.

The dynamics of this chapter are not merely mathematical tools. They are the physics of lasers, MRI machines, quantum computers, and atomic clocks. Every quantum technology you have ever heard of relies on controlled time evolution — on understanding, predicting, and manipulating how quantum states change.

You now have that understanding. Let us keep building.

New Terms Introduced in This Chapter