Spin Precession in Three Pictures Notebook

This notebook guide specifies a reproducible spin-precession calculation in three equivalent pictures of quantum mechanics. The purpose is to make picture equivalence computationally visible: states and operators may carry time dependence differently, but physical expectation values agree.

The existing source notebook notebooks/wave-mechanics-canonical-systems/two-level-systems/two-level-system-dynamics.ipynb provides a compatible two-level dynamics benchmark. A Dynamics-specific notebook should extend that style by computing the same spin expectation values in the Schrödinger, Heisenberg, and interaction pictures and comparing them numerically.

Purpose

The notebook should show:

• spin-half precession in a constant magnetic-field-like Hamiltonian;
• state evolution in the Schrödinger picture;
• Pauli-operator evolution in the Heisenberg picture;
• an interaction-picture split with matching expectation values;
• numerical equality checks between all three pictures.

The central validation target is not a plot. It is the equality

$$\langle A\rangle_S(t) = \langle A\rangle_H(t) = \langle A\rangle_I(t)$$

for the same physical observable $A$.

Physics Goal

Use a spin-half Hamiltonian written in Pauli-matrix form:

$$H = \frac{\hbar}{2} \boldsymbol\Omega\cdot\boldsymbol\sigma,$$

where

$$\boldsymbol\sigma=(\sigma_x,\sigma_y,\sigma_z),$$

and $\boldsymbol\Omega$ is a constant vector with units of angular frequency. A useful nontrivial benchmark is

$$\boldsymbol\Omega=(\Omega_x,0,\Omega_z),$$

so the interaction-picture split can separate the $z$ part from the transverse $x$ part.

The exact unitary is

$$U(t) = \cos\left(\frac{\Omega t}{2}\right)I - i\sin\left(\frac{\Omega t}{2}\right) \hat{\boldsymbol\Omega}\cdot\boldsymbol\sigma,$$

where

$$\Omega=\lvert\boldsymbol\Omega\rvert, \qquad \hat{\boldsymbol\Omega} = \frac{\boldsymbol\Omega}{\Omega}.$$

This closed form is the analytic benchmark for the notebook.

Mathematical Model

In the Schrödinger picture,

$$\lvert\psi_S(t)\rangle = U(t)\lvert\psi(0)\rangle,$$

and

$$\langle\sigma_j\rangle_S(t) = \langle\psi_S(t)\rvert\sigma_j\lvert\psi_S(t)\rangle.$$

In the Heisenberg picture,

$$\sigma_{j,H}(t) = U^\dagger(t)\sigma_jU(t),$$

while the state is fixed:

$$\lvert\psi_H\rangle=\lvert\psi(0)\rangle.$$

The expectation value is

$$\langle\sigma_j\rangle_H(t) = \langle\psi_H\rvert\sigma_{j,H}(t)\lvert\psi_H\rangle.$$

The Heisenberg equation gives the vector precession law

$$\frac{d\boldsymbol\sigma_H}{dt} = \boldsymbol\Omega\times\boldsymbol\sigma_H.$$

For the interaction picture, split

$$H=H_0+V,$$

with

$$H_0=\frac{\hbar\Omega_z}{2}\sigma_z, \qquad V=\frac{\hbar\Omega_x}{2}\sigma_x.$$

Then

$$U_0(t)=e^{-iH_0t/\hbar},$$

and the interaction-picture objects are

$$\lvert\psi_I(t)\rangle = U_0^\dagger(t)\lvert\psi_S(t)\rangle,$$

and

$$\sigma_{j,I}(t) = U_0^\dagger(t)\sigma_jU_0(t).$$

The interaction-picture expectation value is

$$\langle\sigma_j\rangle_I(t) = \langle\psi_I(t)\rvert \sigma_{j,I}(t) \lvert\psi_I(t)\rangle.$$

This expression should match the Schrödinger and Heisenberg results at every sampled time.

Numerical Method

Use explicit $2\times2$ matrices:

$$\sigma_x= \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}, \quad \sigma_y= \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}, \quad \sigma_z= \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}.$$

Set $\hbar=1$ in the code unless physical units are part of the exercise. Choose an initial state, such as the $+x$ spin state

$$\lvert +x\rangle = \frac{1}{\sqrt2} \begin{pmatrix} 1\\ 1 \end{pmatrix}.$$

For each time $t_n$:

1. Build $U(t_n)$ from the exact closed form.
2. Compute $\lvert\psi_S(t_n)\rangle=U(t_n)\lvert\psi(0)\rangle$.
3. Compute $\sigma_{j,H}(t_n)=U^\dagger(t_n)\sigma_jU(t_n)$.
4. Build $U_0(t_n)$ and compute the interaction-picture state and operators.
5. Evaluate $\langle\sigma_x\rangle$, $\langle\sigma_y\rangle$, and $\langle\sigma_z\rangle$ in all three pictures.
6. Assert that the three answers agree within tolerance.

No numerical ODE solver is required for the baseline notebook. An optional extension can integrate the interaction-picture equation and compare the result with the exact $U_0^\dagger(t)U(t)$.

How to Run the Notebook

Use a NumPy-only baseline. A reliable first parameter set is

$$\Omega_x=0.7, \qquad \Omega_z=1.3, \qquad \hbar=1.$$

Sample several periods of the total precession frequency

$$\Omega=\sqrt{\Omega_x^2+\Omega_z^2}.$$

The notebook should print or assert:

• unitarity residuals for $U(t)$ and $U_0(t)$;
• equality residuals between pictures;
• Bloch-vector norm;
• conservation of the Bloch-vector component along $\boldsymbol\Omega$;
• Pauli algebra checks at initialization.

Plots are useful after the assertions pass. A good plot shows the three components of the Bloch vector versus time, but the validation cells should remain independent of plotting.

Expected Results

The Bloch vector

$$\mathbf r(t) = \langle\boldsymbol\sigma\rangle(t)$$

rotates about $\boldsymbol\Omega$. Its norm is preserved for a pure state:

$$\lvert\mathbf r(t)\rvert=1.$$

The component along the precession axis is conserved:

$$\mathbf r(t)\cdot\hat{\boldsymbol\Omega} = \mathbf r(0)\cdot\hat{\boldsymbol\Omega}.$$

For the special case $\boldsymbol\Omega=(0,0,\Omega)$ and initial state $\lvert +x\rangle$,

$$\langle\sigma_x\rangle(t)=\cos\Omega t, \qquad \langle\sigma_y\rangle(t)=\sin\Omega t, \qquad \langle\sigma_z\rangle(t)=0.$$

This simple case is a useful sign-convention check.

Validation Checks

The minimum validation suite should include:

Check	Target
Pauli algebra	$\sigma_i\sigma_j=\delta_{ij}I+i\epsilon_{ijk}\sigma_k$
Unitarity	$U^\dagger U=I$ and $U_0^\dagger U_0=I$
Picture equality	all three $\langle\sigma_j\rangle(t)$ values agree
Bloch norm	$\lvert\mathbf r(t)\rvert=1$ for pure states
Axis projection	$\mathbf r(t)\cdot\hat{\boldsymbol\Omega}$ is constant
Special-case signs	$z$-axis precession matches the analytic sine and cosine formulas

The equality check should compare arrays over all sampled times, not just one time point.

Convergence Checks

For the exact-unitary baseline, convergence mainly means verifying floating-point stability and sampling resolution. If the notebook adds numerical integration of the interaction-picture equation, convergence should be checked by reducing the time step and comparing with the exact unitary.

For a time-stepper, useful residuals are:

$$\max_t \left\lVert U_{\rm num}^\dagger(t)U_{\rm num}(t)-I \right\rVert,$$

and

$$\max_t \left\lVert U_{\rm num}(t)-U_{\rm exact}(t) \right\rVert.$$

If the integrator is not exactly unitary, the Bloch-vector norm may drift. That drift should be reported, not hidden by normalizing after every step unless the method explicitly includes such a projection.

Common Numerical Failures

• Changing the sign convention for $H$ without updating the expected precession direction.
• Comparing a Schrödinger-picture state with a Heisenberg-picture operator incorrectly.
• Treating a global phase difference between state vectors as a physical disagreement.
• Forgetting complex conjugation in expectation values.
• Using a nonunitary time stepper and missing Bloch-norm drift.
• Testing only $\sigma_z$ when $\sigma_z$ happens to be conserved.
• Mixing conventions for spin operators $S_j=\hbar\sigma_j/2$ and Pauli matrices $\sigma_j$.

Extensions

Natural extensions include:

• add a rotating frame and connect it to the interaction picture;
• use a time-dependent transverse drive and compare with a time-ordered numerical solution;
• add a Rabi-oscillation benchmark;
• compare exact unitary evolution with first-order and second-order time stepping;
• add mixed states and verify density-matrix evolution;
• add dephasing only after the closed-system picture equivalence is validated.

Cross-Links

• Schrödinger Picture
• Heisenberg Picture
• Interaction Picture
• Two-State Hamiltonians
• Pauli Matrices
• Bloch Sphere
• Spin and Pauli Matrix Conventions
• Validation Tests

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• J. M. Thijssen, Computational Physics, 2nd ed., Cambridge University Press, 2007.

Exercises

1. For $H=\hbar\Omega\sigma_z/2$ and initial state $\lvert +x\rangle$, compute $\langle\sigma_x\rangle(t)$ and $\langle\sigma_y\rangle(t)$.

Solution

The unitary is

$$U(t)=e^{-i\Omega t\sigma_z/2}.$$

The state evolves to

$$\lvert\psi(t)\rangle = \frac{1}{\sqrt2} \begin{pmatrix} e^{-i\Omega t/2}\\ e^{i\Omega t/2} \end{pmatrix}.$$

Direct evaluation gives

$$\langle\sigma_x\rangle(t)=\cos\Omega t, \qquad \langle\sigma_y\rangle(t)=\sin\Omega t.$$

2. Why should the three pictures give the same spin expectation values?

Solution

The pictures are related by unitary transformations that move time dependence between states and operators. For example,

$$\langle\psi_S(t)\rvert A_S\lvert\psi_S(t)\rangle = \langle\psi(0)\rvert U^\dagger(t)A_SU(t)\lvert\psi(0)\rangle.$$

The right-hand side is the Heisenberg-picture expectation value. The interaction picture is another unitary redistribution of the same time dependence, so the scalar expectation value is unchanged.

3. What validation check detects a sign error in the Hamiltonian convention?

Solution

The special case $\boldsymbol\Omega=(0,0,\Omega)$ with initial state $\lvert +x\rangle$ detects the sign. With the convention $H=\hbar\Omega\sigma_z/2$, the expected result is

$$\langle\sigma_x\rangle=\cos\Omega t, \qquad \langle\sigma_y\rangle=\sin\Omega t.$$

Changing the sign of $H$ reverses the sign of the $\sigma_y$ expectation.

4. Why is comparing state-vector components across pictures a poor validation test?

Solution

Different pictures intentionally assign time dependence to different objects. State vectors in different pictures need not have the same components, and they may differ by unitary transformations or global phases. Physical validation should compare expectation values, transition probabilities, unitarity, and invariant quantities such as Bloch-vector norm.