1.3.2 Schrödinger Picture#

Prompts

What does it mean for a quantum state to evolve while observables stay fixed? How is this view different from the Heisenberg picture?
How does a qubit respond to a magnetic field? Describe the motion of the state on the Bloch sphere and explain what determines the speed of precession.
A microwave pulse can drive a qubit from \(\vert 0\rangle\) to a superposition and back. How does the pulse strength and duration control the qubit’s trajectory on the Bloch sphere?
What are the key properties of the time-evolution operator \(\hat{U}(t) = \mathrm{e}^{-\mathrm{i}\hat{H}t/\hbar}\)? Why must it be unitary?

Lecture Notes#

Overview#

In quantum mechanics, we must decide: does the state evolve in time, or the observables? The Schrödinger picture answers: the state evolves. This is the most intuitive framework for understanding how to control quantum systems—by shaping the state via carefully engineered Hamiltonians.

The heart of the Schrödinger picture is the time-evolution operator \(\hat{U}(t)\), which encodes how any initial state develops. For spin systems, this operator describes both natural precession in a magnetic field and coherent control via microwave drives. Understanding these two regimes—static precession and resonant driving—is the key to quantum engineering.

Notation

Two notation systems appear in this lesson. The Larmor-precession example (and HW 3) use spin notation \(\hat{\sigma}^z, \hat{\sigma}^x\) with kets \(\vert\uparrow\rangle, \vert\downarrow\rangle\) to keep contact with the magnetic-field physics. The Rabi-oscillation example (and HW 5, 6, 9) use QI notation \(\hat{X}, \hat{Y}, \hat{Z}\) with kets \(\vert 0\rangle, \vert 1\rangle\) to match the operational/gate framing. Within any single equation, only one system appears.

The Time-Evolution Operator#

Schrödinger Equation of Motion

The Schrödinger equation governs state evolution:

(17)#\[ \boxed{\mathrm{i}\hbar \frac{\partial\vert\psi(t)\rangle}{\partial t} = \hat{H}\vert\psi(t)\rangle} \]

For a time-independent Hamiltonian \(\hat{H}\), this has a formal solution:

(18)#\[ \hat{U}(t) = \mathrm{e}^{-\mathrm{i}\hat{H}t/\hbar} \]

and the state evolves as

(19)#\[ \vert\psi(t)\rangle = \hat{U}(t)\vert\psi(0)\rangle \]

Properties of the Time-Evolution Operator

Composition: \(\hat{U}(t_1)\hat{U}(t_2) = \hat{U}(t_1 + t_2)\). Multiple evolution steps concatenate.
Unitarity: \(\hat{U}^\dagger(t)\hat{U}(t) = \hat{I}\). The operator preserves inner products and probabilities (probability conservation).
Initial condition: \(\hat{U}(0) = \hat{I}\). At \(t=0\), no evolution has occurred.

Derivation: Unitarity

Since \(\hat{H}\) is Hermitian (\(\hat{H}^\dagger = \hat{H}\)):

\[ \hat{U}^\dagger(t) = (\mathrm{e}^{-\mathrm{i}\hat{H}t/\hbar})^\dagger = \mathrm{e}^{+\mathrm{i}\hat{H}t/\hbar} \]

Therefore:

\[ \hat{U}^\dagger(t)\hat{U}(t) = \mathrm{e}^{+\mathrm{i}\hat{H}t/\hbar} \mathrm{e}^{-\mathrm{i}\hat{H}t/\hbar} = \hat{I} \]

Example: Spin Precession in a Static Magnetic Field#

Consider a spin-1/2 particle in a static magnetic field \(\boldsymbol{B} = B_0\hat{z}\). The Hamiltonian is

(20)#\[ \hat{H} = \frac{\hbar\omega_0}{2}\hat{\sigma}^z \]

where \(\omega_0 = \gamma B_0\) is the Larmor frequency (the natural precession frequency).

Since \(\hat{\sigma}^z\) is diagonal with eigenvalues \(\pm 1\), the time-evolution operator is:

(21)#\[\begin{split} \hat{U}(t) = \mathrm{e}^{-\mathrm{i}\omega_0 t\hat{\sigma}^z/2} = \begin{pmatrix} \mathrm{e}^{-\mathrm{i}\omega_0 t/2} & 0 \\ 0 & \mathrm{e}^{+\mathrm{i}\omega_0 t/2} \end{pmatrix} \end{split}\]

Starting from spin-up \(\vert\uparrow\rangle\).

\[ \vert\psi(t)\rangle = \hat{U}(t)\vert\uparrow\rangle = \mathrm{e}^{-\mathrm{i}\omega_0 t/2}\vert\uparrow\rangle \]

The state remains spin-up but acquires a global phase \(\mathrm{e}^{-\mathrm{i}\omega_0 t/2}\). Global phases are unobservable, so the measurement outcomes never change.

Starting from an equal superposition.

\[ \vert\psi(0)\rangle = \frac{1}{\sqrt{2}}(\vert\uparrow\rangle + \vert\downarrow\rangle) = \vert+\rangle \]

After time evolution:

\[ \vert\psi(t)\rangle = \frac{1}{\sqrt{2}}(\mathrm{e}^{-\mathrm{i}\omega_0 t/2}\vert\uparrow\rangle + \mathrm{e}^{+\mathrm{i}\omega_0 t/2}\vert\downarrow\rangle) \]

The relative phase between the two components is \(\mathrm{e}^{-\mathrm{i}\omega_0 t}\). This causes the Bloch vector to rotate around the z-axis.

Bloch Vector Motion

The expectation values are:

\[ \langle \hat{\sigma}^x \rangle(t) = \cos(\omega_0 t) \]

\[ \langle \hat{\sigma}^y \rangle(t) = \sin(\omega_0 t) \]

\[ \langle \hat{\sigma}^z \rangle(t) = 0 \]

The Bloch vector precesses around the z-axis at the Larmor frequency \(\omega_0\).

Example: Driven Evolution — Rabi Oscillations#

Now suppose we apply a microwave drive resonant with the Larmor frequency. In the rotating frame and under the rotating-wave approximation, the Hamiltonian becomes

(22)#\[ \hat{H}_{\text{RWA}} = \frac{\hbar\Omega}{2}\hat{X} \]

where \(\Omega\) is the Rabi frequency (proportional to the drive field amplitude). Here we switch to QI notation (\(\hat{X}\), \(\hat{Z}\) with \(\vert 0\rangle\), \(\vert 1\rangle\)) since Rabi oscillations are the basis of quantum gate operations.

The time-evolution operator is

(23)#\[ \hat{U}(t) = \mathrm{e}^{-\mathrm{i}\Omega t\hat{X}/2} = \cos(\Omega t/2)\hat{I} - \mathrm{i}\sin(\Omega t/2)\hat{X} \]

This is a rotation around the x-axis by angle \(\Omega t\) on the Bloch sphere.

Starting from \(\vert 0\rangle\). The evolved state is

\[ \vert\psi(t)\rangle = \hat{U}(t)\vert 0\rangle = \cos(\Omega t/2)\vert 0\rangle - \mathrm{i}\sin(\Omega t/2)\vert 1\rangle \]

The probability of measuring the system in state \(\vert 1\rangle\) is:

(24)#\[ P_1(t) = \sin^2(\Omega t/2) \]

This oscillates sinusoidally between 0 and 1 at frequency \(\Omega\) — hence “Rabi oscillations.”

Standard Pulse Durations

\(\pi/2\)-pulse (\(t = \pi/(2\Omega)\)): \(P_1 = \sin^2(\pi/4) = 1/2\) → creates equal superposition
\(\pi\)-pulse (\(t = \pi/\Omega\)): \(P_1 = \sin^2(\pi/2) = 1\) → complete population inversion (flips the state)
\(2\pi\)-pulse (\(t = 2\pi/\Omega\)): \(P_1 = \sin^2(\pi) = 0\) → returns to initial state (one full cycle)

Physical Interpretation

Rabi oscillations are the mechanism of quantum gates. By timing the microwave pulse to duration \(t = \pi/(2\Omega)\) or \(t = \pi/\Omega\), we can create any rotation we want on the Bloch sphere. This is how quantum computers manipulate qubits.

Discussion: limits of quantum control

What limits quantum control in practice?

In theory, we can implement any unitary gate on a qubit by choosing the right Hamiltonian and pulse duration. A \(\pi\)-pulse flips \(\vert 0\rangle \to \vert 1\rangle\); a \(\pi/2\)-pulse creates a perfect superposition.

But real qubits are never perfectly isolated:

Decoherence destroys superpositions on a timescale \(T_2\). If the gate time \(t_{\text{gate}} \sim 1/\Omega\) is not much shorter than \(T_2\), the operation is imperfect.
Leakage: real atoms have more than two levels. A strong drive can excite higher states outside the qubit subspace.
Calibration errors: the Rabi frequency \(\Omega\) depends on the drive amplitude, which fluctuates.

For a superconducting transmon qubit with \(T_2 \sim 100\,\mu\)s and gate time \(\sim 20\) ns, roughly how many gates can you perform before decoherence ruins the computation? What does this imply for the feasibility of quantum error correction?

Poll: Constant observable in Schrödinger picture

In the Schrödinger picture, a state \(\vert\psi(t)\rangle\) evolves while observable \(\hat{A}\) stays constant. If \(\vert\psi(t)\rangle = \frac{1}{\sqrt{2}}(\vert a_1\rangle + \vert a_2\rangle)\) at all times (where \(\hat{A}\vert a_i\rangle = a_i\vert a_i\rangle\)), what must be true?

(A) \([\hat{H}, \hat{A}] = 0\) (the observable commutes with the Hamiltonian).

(B) The expectation value \(\langle\hat{A}\rangle(t)\) is constant in time.

(D) Both (A) and (B).

Summary#

Schrödinger picture: The quantum state \(|\psi(t)\rangle\) evolves via the unitary time-evolution operator \(\hat{U}(t) = \mathrm{e}^{-\mathrm{i}\hat{H}t/\hbar}\); observables are fixed.
Larmor precession: In a static magnetic field, spin precesses around the field direction at the Larmor frequency \(\omega_0 = \gamma B_0\); the Bloch vector traces circles on the Bloch sphere.
Rabi oscillations: A resonant microwave drive oscillates the state between \(|0\rangle\) and \(|1\rangle\) at the Rabi frequency \(\Omega\), with \(P_1(t) = \sin^2(\Omega t/2)\).
Quantum gates via pulse timing: Precise control of pulse duration and drive frequency enables arbitrary rotations on the Bloch sphere and implementation of quantum logic gates.

Homework#

1. General Bloch precession. A qubit has Hamiltonian \(\hat H = \tfrac{\hbar}{2}\,\boldsymbol\omega\cdot\hat{\boldsymbol\sigma}\) where \(\boldsymbol\omega = (\omega_x, \omega_y, \omega_z)\) is a constant 3-vector with magnitude \(\omega = \vert\boldsymbol\omega\vert\) and direction \(\boldsymbol{\omega}/\omega = \boldsymbol\omega/\omega\).

(a) Using \((\hat{\boldsymbol n}\cdot\hat{\boldsymbol\sigma})^2 = \hat I\) from 1.1.3 Problem 7, show that the time-evolution operator has the closed form

\[ \hat U(t) = \cos(\omega t/2)\,\hat I - \mathrm{i}\sin(\omega t/2)\,(\boldsymbol{\omega}/\omega\cdot\hat{\boldsymbol\sigma}). \]

(b) Using the result of 1.3.1 Problem 1 (conservation of observables commuting with \(\hat H\)), argue that \(\langle\boldsymbol{\omega}/\omega\cdot\hat{\boldsymbol\sigma}\rangle\) is conserved in time on every state. Conclude that the Bloch vector’s projection onto the rotation axis \(\boldsymbol{\omega}/\omega\) is constant — the Bloch trajectory lies on a cone around \(\boldsymbol{\omega}/\omega\).

(c) Verify two specializations of this picture against the lecture’s worked examples: (i) \(\boldsymbol\omega = \omega_0\boldsymbol{e}_z\) reproduces Larmor precession of the Bloch vector about \(\boldsymbol{e}_z\) at angular speed \(\omega_0\); (ii) \(\boldsymbol\omega = \Omega\boldsymbol{e}_x\) reproduces resonant Rabi rotation about \(\boldsymbol{e}_x\) at angular speed \(\Omega\). State the rotation axis and angular speed in each case.

2. Larmor evolution of a Y-basis eigenstate. The lecture computes Larmor evolution starting from the \(\hat X\) eigenstate \(\vert+\rangle\). Repeat the computation with the same Hamiltonian \(\hat H = \tfrac{\hbar\omega_0}{2}\hat Z\) but starting from the \(\hat Y\) eigenstate

\[ \vert\mathrm{i}\rangle = \tfrac{1}{\sqrt 2}(\vert 0\rangle + \mathrm{i}\vert 1\rangle). \]

(a) Compute \(\vert\psi(t)\rangle\).

(b) Compute the three Pauli expectation values \(\langle\hat X\rangle(t),\,\langle\hat Y\rangle(t),\,\langle\hat Z\rangle(t)\), and identify the Bloch trajectory.

(c) Compare with the lecture’s \(\vert+\rangle\) initial-condition result: \(\boldsymbol n_+(t) = (\cos\omega_0 t,\,\sin\omega_0 t,\,0)\). What’s the same? What’s the geometric difference between the two trajectories?

(d) Find the first time \(t^*\) at which the Bloch vector reaches \(-\boldsymbol{e}_x\), and identify the physical fraction of a Larmor period this represents.

3. Survival probability. For a system with Hamiltonian \(\hat H\) and initial state \(\vert\psi_0\rangle\), the survival probability is

\[ P_s(t) = \vert\langle\psi_0\vert\psi(t)\rangle\vert^2. \]

(a) Expand \(\vert\psi_0\rangle = \sum_n c_n\vert E_n\rangle\) in energy eigenstates. Show that

\[ \langle\psi_0\vert\psi(t)\rangle = \sum_n \vert c_n\vert^2\,\mathrm{e}^{-\mathrm{i}E_n t/\hbar}. \]

(b) Specialise to a two-level system with eigenvalues \(E_\pm = \pm\Delta\), and initial state \(\vert\psi_0\rangle = \cos(\theta_0/2)\vert E_+\rangle + \sin(\theta_0/2)\vert E_-\rangle\) (real coefficients). Compute \(P_s(t)\) in closed form.

(c) For what initial states does the survival probability oscillate between \(1\) and \(0\)? For what initial states is \(P_s(t) = 1\) identically? Identify the angular frequency of the survival oscillation.

(d) Briefly interpret physically: \(P_s(t)\) measures how much the evolved state has rotated away from its starting direction. What does the minimum survival probability \(1 - \sin^2\theta_0\) tell us about the “distance” between \(\vert\psi_0\rangle\) and the (orthogonal) state reached at the worst point of the oscillation?

4. Two-pulse phase-accumulation sequence. Apply the following three-step sequence to the initial state \(\vert\psi(0)\rangle = \vert 0\rangle\):

A Rabi \(\pi/2\)-pulse: \(\hat U_1 = \cos(\pi/4)\hat I - \mathrm{i}\sin(\pi/4)\hat X\).
Free evolution under \(\hat H_0 = \tfrac{\hbar\omega_0}{2}\hat Z\) for time \(T\): \(\hat U_2(T) = \mathrm{e}^{-\mathrm{i}\omega_0 T\hat Z/2}\).
A second Rabi \(\pi/2\)-pulse, identical to step 1.

(a) Compute the state after each of the three steps.

(b) Show that \(P_1\) — the probability of finding the qubit in \(\vert 1\rangle\) at the end — is

\[ P_1(T) = \cos^2(\omega_0 T/2). \]

(c) The free-evolution stage accumulated only an unobservable phase (it acts trivially on populations in the \(\hat Z\) basis), yet the final \(P_1\) depends on \(T\). Explain how the two \(\pi/2\)-pulses convert the phase \(\omega_0 T\) into a measurable population oscillation.

(d) Use this principle to estimate the precision with which the level splitting \(\omega_0\) can be measured by varying \(T\) and counting the population. Compare with measuring \(\omega_0\) directly from a single Larmor precession (which would require measuring \(\langle\hat X\rangle\) at one fixed time).

5. Pi-pulse implements the NOT gate. From the lecture, the Rabi evolution operator is \(\hat U(t) = \cos(\Omega t/2)\hat I - \mathrm{i}\sin(\Omega t/2)\hat X\).

(a) Evaluate \(\hat U(\pi/\Omega)\) and show \(\hat U(\pi/\Omega) = -\mathrm{i}\hat X\).

(b) Apply \(\hat U(\pi/\Omega)\) to each of \(\vert 0\rangle\), \(\vert 1\rangle\), \(\vert+\rangle\), and \(\vert-\rangle\). For each input, identify the output and confirm it equals \(\hat X\) applied to that state (up to a global phase \(-\mathrm{i}\)).

(c) Compute \(\hat U(2\pi/\Omega)\). Show that the result is \(-\hat I\), not \(+\hat I\). Reference 1.3.1 Problem 6: explain why the half-angle \(\Omega t/2\) in the exponent produces this sign, and why the \(-\hat I\) is unobservable for a single qubit but can become physical in interferometric setups built later.

6. Sequential pulses about different axes. Two pulse sequences are applied to the initial state \(\vert 0\rangle\):

Sequence A. First a \(\pi/2\)-pulse about \(\boldsymbol{e}_x\) (\(\hat R_x = \tfrac{1}{\sqrt 2}(\hat I - \mathrm{i}\hat X)\)), then a \(\pi/2\)-pulse about \(\boldsymbol{e}_z\) (\(\hat R_z = \tfrac{1}{\sqrt 2}(\hat I - \mathrm{i}\hat Z)\)).
Sequence B. The same two pulses in the opposite order: \(\hat R_z\) first, then \(\hat R_x\).

(a) Compute the final state under sequence A.

(b) Compute the final state under sequence B.

(d) Conclude: rotations about different axes do not commute as quantum gates. Why does this matter for the construction of arbitrary qubit operations?

7. Larmor precession vs Rabi oscillation. Compare Larmor precession (static field \(B_0\) along \(\boldsymbol{e}_z\)) and Rabi oscillation (resonant drive along \(\boldsymbol{e}_x\) in the rotating frame).

(a) In what sense are both Bloch-sphere rotations? Identify the rotation axis and angular speed for each.

(b) Larmor precession occurs even with no external control — it is the free evolution of a qubit in a static field. Rabi oscillation requires an applied resonant drive. State, in one sentence, why a static field alone (Larmor) cannot drive population transfer between \(\vert 0\rangle\) and \(\vert 1\rangle\), whereas a resonant drive (Rabi) can.

(c) The two rotations have very different angular speeds in typical experiments: \(\omega_0 \gg \Omega\). Describe the physical separation of these scales — what does \(\omega_0\) measure, and what does \(\Omega\) measure? Why is the small-\(\Omega\) limit (\(\Omega \ll \omega_0\)) the regime in which the rotating-wave approximation is justified?

★ 8. Three-state evolution. Consider a 3-level system with basis \(\{\vert 1\rangle, \vert 2\rangle, \vert 3\rangle\}\) and Hamiltonian \(\hat{H} = \vert 1\rangle\langle 2\vert + \vert 2\rangle\langle 1\vert - \vert 1\rangle\langle 3\vert - \vert 3\rangle\langle 1\vert\) (in units where the coupling is 1, and \(\hbar = 1\)).

(a) Find the eigenvalues and eigenstates of \(\hat{H}\).

(b) The system starts in \(\vert\psi(0)\rangle = \vert 1\rangle\). Compute \(\vert\psi(t)\rangle\).

(d) For a measurement of \(\hat{H}^2\) (note: not \(\hat{H}\)) on \(\vert\psi(t)\rangle\), list the possible outcomes and their corresponding probabilities.