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. Hamiltonian eigenstate evolution. For a Hamiltonian \(\hat{H}\) with eigenstates \(\vert n\rangle\) and eigenvalues \(E_n\), show that \(\hat{U}(t)\vert n\rangle = \mathrm{e}^{-\mathrm{i}E_n t/\hbar}\vert n\rangle\). Why does the time-evolution operator have the form \(\hat{U}(t) = \sum_n \mathrm{e}^{-\mathrm{i}E_n t/\hbar}\vert n\rangle\langle n\vert\)?

2. Schrödinger equation solution. Starting from the Schrödinger equation \(\mathrm{i}\hbar \partial_t\vert\psi\rangle = \hat{H}\vert\psi\rangle\), show that \(\langle\psi(t)\vert\psi(t)\rangle\) is constant in time. Explain why this implies that \(\hat{U}(t)\) must be unitary.

3. Larmor precession. A spin-1/2 starts in state \(\vert\psi(0)\rangle = \frac{1}{\sqrt{2}}(\vert\uparrow\rangle + \vert\downarrow\rangle)\) in a magnetic field with Larmor frequency \(\omega_0\).

(a) Compute \(\vert\psi(t)\rangle\) using \(\hat{U}(t) = \mathrm{e}^{-\mathrm{i}\omega_0 t\hat{\sigma}^z/2}\).

(b) Calculate \(\langle\hat{\sigma}^x\rangle(t)\), \(\langle\hat{\sigma}^y\rangle(t)\), \(\langle\hat{\sigma}^z\rangle(t)\).

4. Phase freedom in time evolution. Show that if \(\vert\psi(t)\rangle = \mathrm{e}^{\mathrm{i}\phi(t)}\vert\chi(t)\rangle\) (differing by a global phase \(\mathrm{e}^{\mathrm{i}\phi}\)), then all observable expectation values are identical: \(\langle\psi\vert\hat{O}\vert\psi\rangle = \langle\chi\vert\hat{O}\vert\chi\rangle\). Why are global phases unobservable in quantum mechanics?

5. Rabi evolution with detuning. Using \(\hat{U}(t) = \cos(\Omega t/2)\hat{I} - \mathrm{i}\sin(\Omega t/2)\hat{X}\), compute the state \(\vert\psi(\pi/\Omega)\rangle\) starting from \(\vert\psi(0)\rangle = \vert 0\rangle\). Show that \(P_1 = \sin^2(\pi/2) = 1\) (complete population inversion). What is the state up to a global phase?

6. Rabi pi-pulse. Apply Rabi evolution for time \(t = \pi/(2\Omega)\) starting from \(\vert 0\rangle\).

(a) Write \(\vert\psi(\pi/(2\Omega))\rangle\) explicitly.

(b) Compute \(\langle\hat{X}\rangle\), \(\langle\hat{Y}\rangle\), \(\langle\hat{Z}\rangle\) for this state.

(d) Why is this called a “\(\pi/2\) pulse”?

7. Larmor precession vs Rabi. Compare Larmor precession (static field \(B_0\) along z) and Rabi oscillations (resonant drive along x in the RWA). In what sense are they both “rotations” on the Bloch sphere? What is fundamentally different about their timescales and mechanisms?

8. Generalized Rabi frequency. The generalized Rabi oscillation frequency is \(\tilde{\Omega} = \sqrt{\delta^2 + \Omega^2}\) where \(\delta\) is the detuning and \(\Omega\) is the on-resonance Rabi frequency.

(a) For resonant driving (\(\delta = 0\)), show that \(P_1(t) = \sin^2(\Omega t/2)\).

(b) For off-resonant driving (\(\delta \gg \Omega\)), explain why the oscillation amplitude is suppressed.

9. 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).

(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.