Spin in Magnetic Fields

Spin becomes experimentally visible because spin often carries a magnetic moment. In a magnetic field, the basic coupling is

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

For a spin degree of freedom with

$$\boldsymbol\mu=\gamma\mathbf S,$$

the Hamiltonian is

$$H = -\gamma\mathbf S\cdot\mathbf B.$$

For spin-$1/2$,

$$\mathbf S=\frac{\hbar}{2}\boldsymbol\sigma,$$

so

$$H = -\frac{\gamma\hbar}{2}\, \mathbf B\cdot\boldsymbol\sigma.$$

This page explains the static-field Hamiltonian, sign conventions, energy splitting, and symmetry meaning. The actual precession dynamics is treated separately in Larmor Precession.

What the Field Couples To

A magnetic field couples to a magnetic moment, not to the abstract word “spin” by itself. The proportionality between magnetic moment and spin is model-dependent:

$$\boldsymbol\mu = \gamma\mathbf S.$$

The gyromagnetic ratio $\gamma$ contains the charge, mass, $g$ factor, and sign convention. For a particle of charge $q$, mass $m$, and spin $g$ factor $g$, one often writes

$$\gamma = g\frac{q}{2m}.$$

For an electron, it is common to define $e>0$ and $q=-e$. The Bohr magneton is

$$\mu_B=\frac{e\hbar}{2m_e}.$$

Then the electron spin magnetic moment is conventionally written

$$\boldsymbol\mu_S = -g_s\mu_B \frac{\mathbf S}{\hbar},$$

with $g_s\simeq2$ at leading order. The minus sign is physical: the electron magnetic moment points opposite to its spin angular momentum.

Static Field Along z

Take a uniform field

$$\mathbf B=B_0\hat{\mathbf z}.$$

The spin Hamiltonian is

$$H = -\gamma B_0S_z = -\frac{\gamma\hbar B_0}{2}\sigma_z.$$

The $S_z$ eigenstates are also energy eigenstates:

$$S_z|+z\rangle = \frac{\hbar}{2}|+z\rangle, \qquad S_z|-z\rangle = -\frac{\hbar}{2}|-z\rangle.$$

Their energies are

$$E_{+z} = -\frac{\gamma\hbar B_0}{2}, \qquad E_{-z} = \frac{\gamma\hbar B_0}{2}.$$

The positive energy splitting is

$$\Delta E = \hbar|\gamma|B_0.$$

Which state is lower depends on the sign of $\gamma B_0$. For an electron with $B_0>0$ and the convention $\boldsymbol\mu_S=-g_s\mu_B\mathbf S/\hbar$, the lower-energy state has spin anti-aligned with $\mathbf B$ because the magnetic moment is aligned with $\mathbf B$.

Pauli-Vector Form

Any spin-$1/2$ Hamiltonian of the form

$$H=\mathbf b\cdot\boldsymbol\sigma$$

has eigenvalues $\pm|\mathbf b|$. A spin in a magnetic field corresponds to

$$\mathbf b = -\frac{\gamma\hbar}{2}\mathbf B.$$

Therefore the energy eigenstates are spin states along the direction of $\mathbf b$, not always along the direction of $\mathbf B$. If $\gamma>0$, the lower-energy state has spin along $\mathbf B$. If $\gamma<0$, the lower-energy state has spin opposite $\mathbf B$.

This is the clean two-level-system viewpoint. It is developed more generally in Spin-1/2 as a Canonical System and Pauli-Matrix Hamiltonians.

Zeeman Splitting

The splitting of spin or angular-momentum energy levels by a magnetic field is a Zeeman effect. For the simple spin-$1/2$ Hamiltonian above, the transition angular frequency associated with the two levels is

$$\omega_0 = \frac{\Delta E}{\hbar} = |\gamma|B_0.$$

In atomic spectroscopy, the same magnetic-dipole idea appears in more elaborate forms:

$$H_Z = -\boldsymbol\mu\cdot\mathbf B,$$

but $\boldsymbol\mu$ may include orbital, spin, total-angular-momentum, nuclear, and effective-medium contributions. Weak-field atomic shifts are often written schematically as

$$\Delta E = g_J\mu_BBm_J,$$

under assumptions about the coupling regime and sign conventions. There is no universal Zeeman formula without specifying the relevant magnetic moment.

For magnetic-moment and $g$-factor conventions, see Magnetic Moments and g-Factors. For historical context and atomic spectroscopy, see Magnetic Moments and Zeeman Effect Revisited. For degenerate level splitting and the weak-field to Paschen–Back hierarchy, see Zeeman Effect as a Perturbation Example. The compact Hamiltonian card is Spin in Magnetic Field Hamiltonian.

Symmetry of a Static Field

With no magnetic field, an isolated spin Hamiltonian proportional to the identity has full rotational symmetry in spin space. A fixed field $\mathbf B=B_0\hat{\mathbf z}$ selects an axis and leaves only rotations about that axis as symmetries.

For

$$H=-\gamma B_0S_z,$$

one has

$$[H,S_z]=0,$$

but

$$[H,S_x]\ne0, \qquad [H,S_y]\ne0$$

when $\gamma B_0\ne0$. The component along the field is conserved; transverse components are not stationary. This is the operator reason a spin expectation value not aligned with the field precesses.

The magnitude remains fixed:

$$[H,S^2]=0.$$

Thus the field splits orientations within a spin multiplet but does not change the spin quantum number $s$.

For spin-$1/2$, the time-reversal operator flips $\mathbf S$. A fixed magnetic field is therefore time-reversal breaking; the comparison becomes covariant only when $\mathbf B$ is reversed as an external field. See Time Reversal for Spin-1/2 Particles.

Precession Preview

The Heisenberg equation gives

$$\frac{d\mathbf S}{dt} = \gamma\,\mathbf S\times\mathbf B,$$

up to the sign convention in $\gamma$. The expectation value of spin therefore rotates around the magnetic field with angular frequency magnitude

$$\omega_L=|\gamma|B_0.$$

This is Larmor precession. The present page records the Hamiltonian and the frequency scale; Larmor Precession treats the time evolution, Bloch-sphere trajectories, and sign conventions in more detail.

Driven Fields and Resonance Preview

A static field creates a splitting. A transverse oscillating field can drive transitions between the two levels. For example,

$$\mathbf B(t) = B_0\hat{\mathbf z} + B_1\cos(\omega t)\hat{\mathbf x}$$

gives a Hamiltonian with a diagonal static term and a transverse term proportional to $\sigma_x$. Transitions are strongest near

$$\omega\simeq\omega_0.$$

This is the basic idea behind magnetic resonance. The experimental technique is introduced in Magnetic Resonance, and the driven two-level-system dynamics belongs to Rabi Oscillations: First Encounter.

Uniform Versus Inhomogeneous Fields

A uniform magnetic field changes spin energies and causes precession. It does not by itself spatially split a neutral beam. Stern–Gerlach separation requires a field gradient, because the force on a magnetic moment depends on the spatial derivative of the interaction energy:

$$F_z \simeq \mu_z \frac{\partial B_z}{\partial z}.$$

Thus two related but distinct uses of magnetic fields should be kept separate:

• a uniform field defines a spin Hamiltonian and Zeeman splitting;
• an inhomogeneous field can correlate spin or magnetic-moment projection with spatial path.

The second is the apparatus logic behind Stern–Gerlach Revisited.

Common Mistakes

• Forgetting that the magnetic field couples to $\boldsymbol\mu$, whose relation to $\mathbf S$ includes sign and $g$ factor.
• Treating the electron magnetic moment as parallel to electron spin.
• Writing $H=-\gamma\mathbf S\cdot\mathbf B$ without specifying the sign convention for $\gamma$.
• Assuming the lower-energy state is always $|+z\rangle$ when $\mathbf B=B_0\hat{\mathbf z}$.
• Confusing Zeeman splitting in a static field with Rabi oscillations driven by a transverse time-dependent field.
• Using a spin-only Hamiltonian when orbital magnetic coupling or spatial motion in a vector potential is important.
• Assuming a uniform field produces Stern–Gerlach beam splitting.

Cross-Links

• Spin Problems
• What Spin Is and Is Not
• Spin as Intrinsic Angular Momentum
• Higher Spin Systems
• Magnetic Moments and g-Factors
• Pauli Matrices
• Bloch Sphere
• Stern–Gerlach Revisited
• Spin Rotations
• States, Observables, and Hamiltonians
• Magnetic Moments from Orbital Motion
• Larmor Precession
• Berry Phase for Spin-1/2
• Time Reversal for Spin-1/2 Particles
• Kramers Degeneracy
• Explicit Symmetry Breaking
• Spin in Magnetic Field Hamiltonian
• Magnetic Moments
• Zeeman Effect Revisited
• Zeeman Effect as a Perturbation Example
• Zeeman Effect
• Magnetic Resonance
• Spin-1/2 as a Canonical System
• Rabi Oscillations: First Encounter
• Spin and Pauli Matrix Conventions

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• C. P. Slichter, Principles of Magnetic Resonance, 3rd ed., Springer, 1990.
• A. Abragam, The Principles of Nuclear Magnetism, Oxford University Press, 1961.

Exercises

1. For $H=-(\gamma\hbar B_0/2)\sigma_z$, find the two energies and the positive splitting.

Solution

The $\sigma_z$ eigenvalues are $+1$ and $-1$. Therefore

$$E_{+z} = -\frac{\gamma\hbar B_0}{2}, \qquad E_{-z} = \frac{\gamma\hbar B_0}{2}.$$

The signed difference is

$$E_{-z}-E_{+z} = \gamma\hbar B_0.$$

The positive splitting is

$$\Delta E = \hbar|\gamma|B_0.$$

2. For an electron with $e>0$, $\boldsymbol\mu_S=-g_s\mu_B\mathbf S/\hbar$, and $\mathbf B=B_0\hat{\mathbf z}$ with $B_0>0$, which spin state is lower in energy?

Solution

The Hamiltonian is

$$H = -\boldsymbol\mu_S\cdot\mathbf B = g_s\mu_BB_0 \frac{S_z}{\hbar}.$$

For $|+z\rangle$,

$$E_{+z} = \frac{g_s\mu_BB_0}{2}.$$

For $|-z\rangle$,

$$E_{-z} = -\frac{g_s\mu_BB_0}{2}.$$

Thus $|-z\rangle$ is lower. The spin is anti-aligned with the field, while the magnetic moment is aligned with the field.

3. Show that $S_z$ is conserved but $S_x$ is not for $H=-\gamma B_0S_z$.

Solution

Since $H$ is proportional to $S_z$,

$$[H,S_z]=-\gamma B_0[S_z,S_z]=0.$$

For $S_x$,

$$[H,S_x] = -\gamma B_0[S_z,S_x].$$

Using

$$[S_z,S_x]=i\hbar S_y,$$

one gets

$$[H,S_x] = -i\hbar\gamma B_0S_y,$$

which is generally nonzero.

4. A spin has $|\gamma|/(2\pi)=42.58\,\mathrm{MHz/T}$ in a field of $3.0\,\mathrm T$. What is the transition frequency?

Solution

The transition frequency in cycles per second is

$$f_0 = \frac{\omega_0}{2\pi} = \left( 42.58\,\mathrm{MHz/T} \right) (3.0\,\mathrm T).$$

Thus

$$f_0 \approx 127.7\,\mathrm{MHz}.$$