Spin in a Magnetic Field

When a spin- $\tfrac{1}{2}$ particle is placed in a uniform magnetic field $\mathbf{B} = B_0\,\hat{z}$ , two things happen simultaneously: the energy levels split into two distinct values (Zeeman splitting), and any spin state that is not an energy eigenstate precesses around the field direction at a characteristic frequency. Both effects flow directly from the Hamiltonian that couples spin to the field.

The spin Hamiltonian

The interaction energy of a magnetic dipole moment $\boldsymbol{\mu}$ with a field $\mathbf{B}$ is

$H = -\boldsymbol{\mu} \cdot \mathbf{B}.$

For an electron the spin magnetic moment is

$\boldsymbol{\mu} = -\frac{g_s e}{2m_e}\,\mathbf{S},$

where $g_s \approx 2$ is the electron $g$ -factor, $e$ is the elementary charge, and $m_e$ is the electron mass. Defining the Larmor angular frequency

$\omega_0 = \frac{g_s e B_0}{2m_e} = \frac{e B_0}{m_e}$

(the last equality uses $g_s = 2$ ), the Hamiltonian for a field along $\hat{z}$ becomes

$H = \omega_0\, S_z,$

where $S_z = \tfrac{\hbar}{2}\,\sigma_z$ and $\sigma_z$ is the Pauli $z$ -matrix.

Zeeman splitting

Because $S_z$ has eigenvalues $\pm\tfrac{\hbar}{2}$ , the Hamiltonian has two energy eigenvalues:

$E_{\pm} = \pm\frac{\hbar\omega_0}{2}.$

The spin-up state $|{+z}\rangle$ (eigenvalue $m_s = +\tfrac{1}{2}$ ) has energy $E_+ = +\tfrac{\hbar\omega_0}{2}$ , and the spin-down state $|{-z}\rangle$ has energy $E_- = -\tfrac{\hbar\omega_0}{2}$ . The gap between them is

$\Delta E = E_+ - E_- = \hbar\omega_0.$

This splitting of a single degenerate level into two sub-levels by an applied magnetic field is called the Zeeman effect. For a proton in a field of $B_0 = 1\,\text{T}$ , the relevant quantity is the proton Larmor frequency $\omega_0 / (2\pi) \approx 42.6\,\text{MHz}$ , which lies in the radio-frequency band — the basis of magnetic resonance imaging (MRI).

Time evolution and Larmor precession

The energy eigenstates $|{\pm z}\rangle$ are stationary: they pick up only an overall phase under time evolution and give constant expectation values for all observables. A spin state that is not aligned with the field, however, is a superposition of the two eigenstates:

$|\psi(0)\rangle = \alpha\,|{+z}\rangle + \beta\,|{-z}\rangle, \qquad |\alpha|^2 + |\beta|^2 = 1.$

Applying the Schrödinger time-evolution operator $e^{-iHt/\hbar}$ independently to each eigenstate gives

$|\psi(t)\rangle = \alpha\,e^{-i\omega_0 t/2}\,|{+z}\rangle + \beta\,e^{+i\omega_0 t/2}\,|{-z}\rangle.$

The relative phase between the two components advances at rate $\omega_0$ . Computing the expectation values of $S_x$ and $S_y$ from this state reveals that the spin vector rotates around the $z$ -axis at the same rate. Specifically, if the spin starts in the $xz$ -plane at polar angle $\theta$ from $+z$ :

$\langle S_x(t) \rangle = \frac{\hbar}{2}\sin\theta\cos(\omega_0 t),$ $\langle S_y(t) \rangle = \frac{\hbar}{2}\sin\theta\sin(\omega_0 t),$ $\langle S_z(t) \rangle = \frac{\hbar}{2}\cos\theta.$

The $z$ -component is constant while the transverse component traces a circle. This is Larmor precession: the spin expectation value revolves around the field axis at angular frequency $\omega_0$ , completing one full revolution in period $T = 2\pi/\omega_0$ .

Resonance

If an oscillating field at frequency $\omega_0$ is applied perpendicular to $\mathbf{B}$ , it drives transitions between $|{+z}\rangle$ and $|{-z}\rangle$ . This resonance condition $\omega = \omega_0$ is the cornerstone of nuclear magnetic resonance (NMR) and electron spin resonance (ESR). A pulse lasting exactly the right time can rotate the spin from $|{+z}\rangle$ to $|{-z}\rangle$ — the quantum analogue of flipping a bit — because the oscillating field continuously accumulates phase with the precessing spin.

Connection to the qubit

The two energy levels $E_\pm$ map directly onto the computational basis states $|0\rangle$ and $|1\rangle$ of a qubit. Single-qubit rotations in a quantum computer are often implemented physically by driving spin resonance: a carefully timed microwave or radio-frequency pulse precesses the spin by a controlled angle, executing a rotation gate. The energy gap $\Delta E = \hbar\omega_0$ sets the qubit transition frequency and must be known precisely for gate calibration.