Uncertainty, Energy Eigenstates & Spin Precession

Part I — The Uncertainty Principle (Conceptual)

Two observables $A$ and $B$ are called compatible if their commutator vanishes, $[A,B]=0$, and incompatible if $[A,B]\neq 0$. This algebraic distinction carries deep physical consequences.

Compatible observables can be measured simultaneously without disturbance: knowing the result of $A$ does not affect the statistics of $B$. They share a common set of eigenstates, so a state can be simultaneously an eigenstate of both.

Incompatible observables are fundamentally different: measuring one necessarily disturbs the other. This is not a limitation of the apparatus — it is a theorem of the mathematics.

The key example: $\sigma_z$ and $\sigma_x$ do not commute. Measuring $\sigma_x$ on a state that had a definite value of $\sigma_z$ destroys that definite value. After the $\sigma_x$ measurement, $\sigma_z$ is random — even though no physical disturbance was applied between measurements.

The formal quantitative statement is:

For the spin operators $\sigma_z$ and $\sigma_x$: since $[\sigma_z,\sigma_x] = 2i\sigma_y$, the relation gives $\Delta\sigma_z\,\Delta\sigma_x \geq |\langle\sigma_y\rangle|$. In an eigenstate of $\sigma_z$ one has $\langle\sigma_y\rangle = 0$, so the bound is trivially zero — but $\sigma_x$ is still completely random with a 50/50 outcome distribution. The inequality is satisfied as $1 \geq 0$; the uncertainty is real even when the bound is not tight.

Part II — Energy Eigenstates and Stationary States

The eigenvalue equation for the Hamiltonian defines the energy eigenstates:

Because $H$ is Hermitian, the eigenvalues $E_n$ are real and the eigenstates form an orthonormal basis. The time evolution of an energy eigenstate is particularly simple:

The state acquires a time-dependent global phase — but global phases are unobservable. All probabilities are therefore time-independent:

Similarly, all expectation values $\langle A\rangle$ are constant. This is why energy eigenstates are called stationary states: no measurement statistics change with time.

For a general state expanded in the energy basis:

the different energy components accumulate different phases. The amplitude $\langle\phi|\psi(t)\rangle$ is a sum of rotating complex numbers — and their interference makes probabilities oscillate at the beat frequencies $(E_n - E_m)/\hbar$ between levels.

Part III — The Spin Hamiltonian in a Magnetic Field

The simplest dynamical quantum system: a spin-$\tfrac{1}{2}$ particle in a uniform magnetic field $\mathbf{B}$ pointing along the $z$-axis. The Hamiltonian is:

where $\omega = eB/mc$ is the Larmor frequency, and we work in natural units with $\hbar = 1$ throughout. The energy eigenvalues are:

The energy eigenstates are $|u\rangle = (1,0)^T$ and $|d\rangle = (0,1)^T$ — identical to the $\sigma_z$ eigenstates. This is to be expected: $H$ is proportional to $\sigma_z$, so they share eigenstates.

Part IV — Time Evolution of an Arbitrary Spin State

Starting from a general initial state $|\psi(0)\rangle = \alpha|u\rangle + \beta|d\rangle$ with $|\alpha|^2+|\beta|^2=1$, the Schrödinger equation gives:

To see the geometric meaning, write $\alpha = \cos(\theta/2)$ and $\beta = e^{i\varphi}\sin(\theta/2)$ (the Bloch sphere parametrization). After time $t$:

Factoring out a global phase $e^{-i\omega t/2}$ (unobservable), the azimuthal angle evolves as $\varphi \to \varphi + \omega t$. This is precisely precession: the Bloch vector rotates around the $z$-axis.

Computing expectation values explicitly confirms this picture:

The $z$-component of the Bloch vector is unchanged; the transverse component rotates at angular frequency $\omega$. The magnitude of the Bloch vector is preserved.

The Bloch vector precesses around the magnetic field axis at the Larmor frequency $\omega$ — identical to classical Larmor precession. Quantum mechanics gives the same average behavior as classical mechanics.

Part V — Transition Probabilities

Consider starting in $|u\rangle$ and measuring $\sigma_x$ at time $t$. The evolved state is $|\psi(t)\rangle = e^{-i\omega t/2}|u\rangle$. The probability of finding $\sigma_x = +1$ is:

An energy eigenstate shows no oscillating probabilities. Now start instead in the superposition $|\psi(0)\rangle = \tfrac{1}{\sqrt{2}}(|u\rangle+|d\rangle) = |r\rangle$:

This oscillates between $0$ and $1$ with period $2\pi/\omega$ — these are Rabi oscillations, the quantum beating between two energy levels.

Part VI — The Pauli Matrices: Full Algebra

The three Pauli matrices:

satisfy a rich set of algebraic identities:

All four identities are unified by the single master formula:

From the Pauli algebra one can derive the rotation formula. The exponential of a Pauli matrix is:

This is the operator that rotates the spin state around the $z$-axis by angle $\theta$. The time evolution operator $e^{-iHt} = e^{-i\omega t\sigma_z/2}$ is precisely a rotation by angle $\omega t$ — confirming that time evolution in the magnetic field is nothing but rotation.

Part VII — Commutators and the Uncertainty Principle: Formal Statement

The Robertson uncertainty relation gives a precise lower bound on the product of uncertainties:

where $\Delta A^2 = \langle A^2\rangle - \langle A\rangle^2$ is the variance. The derivation applies the Cauchy-Schwarz inequality to the vectors $(A - \langle A\rangle)|\psi\rangle$ and $(B - \langle B\rangle)|\psi\rangle$. Since $A$ and $B$ are Hermitian, their commutator $[A,B]$ is anti-Hermitian, so $\langle[A,B]\rangle$ is purely imaginary — and $|\langle[A,B]\rangle|$ is real and well-defined.

For position and momentum: $[x,p] = i\hbar$, so the bound gives the celebrated Heisenberg relation:

For spin: $[\sigma_x,\sigma_y] = 2i\sigma_z$, so $\Delta\sigma_x\,\Delta\sigma_y \geq |\langle\sigma_z\rangle|$. The uncertainty is state-dependent — it vanishes when $\langle\sigma_z\rangle = 0$ and saturates when the state is an eigenstate of $\sigma_z$.