Larmor Precession

Larmor precession is the rotation of a magnetic moment, spin expectation value, or Bloch vector around a static magnetic field. For a spin with

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

the Zeeman Hamiltonian is

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

In a uniform static field of magnitude $B_0$, the precession frequency magnitude is

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

This page derives the operator and spinor forms of the precession. The static Hamiltonian, energy splitting, and sign conventions are introduced in Spin in Magnetic Fields.

Spin Hamiltonian in a Static Field

Take a constant magnetic field with $B_0>0$,

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

The Hamiltonian is

$$H = -\gamma B_0S_z.$$

For spin-$1/2$,

$$S_z=\frac{\hbar}{2}\sigma_z,$$

so

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

The signed coefficient

$$\omega=\gamma B_0$$

is useful in algebra. The positive precession frequency is $|\omega|$. The actual right-handed angular-velocity vector is $-\gamma\mathbf B$, since $\dot{\mathbf S}=(-\gamma\mathbf B)\times\mathbf S$. This distinction fixes the sense of the motion before any convention about the name “Larmor frequency” is chosen.

Heisenberg Derivation

The Heisenberg equation is

$$\frac{dA}{dt} = \frac{i}{\hbar}[H,A]$$

for an operator with no explicit time dependence. Using

$$[S_i,S_j] = i\hbar\sum_k\epsilon_{ijk}S_k,$$

and

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

one obtains

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

For $\mathbf B=B_0\hat{\mathbf z}$, this gives

$$\frac{dS_x}{dt} = \omega S_y,$$

and

$$\frac{dS_y}{dt} = -\omega S_x,$$

while

$$\frac{dS_z}{dt}=0.$$

Thus the component along the field is conserved and the transverse components rotate.

Operator Solution

The coupled equations imply

$$\frac{d^2S_x}{dt^2} = -\omega^2S_x, \qquad \frac{d^2S_y}{dt^2} = -\omega^2S_y.$$

With operators evaluated at $t=0$, the solution is

$$S_x(t) = S_x(0)\cos\omega t + S_y(0)\sin\omega t,$$

and

$$S_y(t) = S_y(0)\cos\omega t - S_x(0)\sin\omega t.$$

Also

$$S_z(t)=S_z(0).$$

The sign of $\omega=\gamma B_0$ determines the sense of rotation. Many authors quote only the magnitude $|\omega|$ and handle the sense of precession by convention.

Spinor Phase Derivation

The same result appears in the Schrödinger picture. Write an initial spinor in the $S_z$ basis:

$$|\psi(0)\rangle = \alpha|+z\rangle+\beta|-z\rangle.$$

The energies are

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

Therefore

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

For $\alpha\ne0$, the relative amplitude changes as

$$\frac{\beta(t)}{\alpha(t)} = \frac{\beta}{\alpha}e^{-i\omega t}.$$

If the initial state is written on the Bloch sphere as

$$|\psi(0)\rangle = \cos\frac{\theta}{2}|+z\rangle + e^{i\phi_0}\sin\frac{\theta}{2}|-z\rangle,$$

then

$$|\psi(t)\rangle \sim \cos\frac{\theta}{2}|+z\rangle + e^{i(\phi_0-\omega t)} \sin\frac{\theta}{2}|-z\rangle.$$

The polar angle $\theta$ is constant, while the azimuth changes:

$$\phi(t)=\phi_0-\omega t.$$

The Bloch vector therefore precesses around the $z$ axis.

Bloch-Vector Equation

For a pure or mixed spin-$1/2$ state,

$$\rho = \frac12 \left( I+\mathbf r\cdot\boldsymbol\sigma \right).$$

Under

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

the Bloch vector obeys

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

This is the same precession equation as for $\langle\mathbf S\rangle$, since

$$\langle\mathbf S\rangle = \frac{\hbar}{2}\mathbf r.$$

For a nonzero Bloch vector, the trajectory is a circle of fixed radius and polar angle around the field direction when the field is static and the system is closed. Vectors on the axis are stationary, including the maximally mixed state at the origin. Unitary precession changes neither purity nor the field-axis populations.

Energy Eigenstates Do Not Precess

If the spin is prepared in $|+z\rangle$ or $|-z\rangle$ for a field along $z$, the state only accumulates a phase. Its Bloch vector sits at the north or south pole, so there is no visible precession of the expectation value.

Precession requires transverse coherence. For example,

$$|+x\rangle = \frac{1}{\sqrt2} \left( |+z\rangle+|-z\rangle \right)$$

has a Bloch vector in the equatorial plane. The relative phase between the two $S_z$ components changes in time, so the vector rotates around the $z$ axis.

Relation to Magnetic Resonance

For spin $1/2$, the Larmor frequency is also the transition angular frequency between the two spin projections in a static field:

$$\Delta E=\hbar\omega_L.$$

An oscillating transverse field near this frequency can drive transitions. That driven problem is not merely free precession; it introduces an explicitly time-dependent Hamiltonian and, after approximations, Rabi oscillations. The historical technique entry is Magnetic Resonance, the spectroscopy-facing measurement map is Magnetic Resonance Overview, and the two-level driven model is Rabi Oscillations: First Encounter. Magnetometry places the Larmor law inside a calibrated field measurement with finite bandwidth, spatial weighting, noise, and systematic effects.

Sign Conventions

Three signs are easy to mix:

• the sign of the particle charge inside $\gamma$;
• the sign convention for $\boldsymbol\mu=\gamma\mathbf S$;
• the orientation chosen for the positive magnetic field axis.

For an electron, using $e>0$ and

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

one has $\gamma<0$. The spin expectation therefore precesses in the opposite sense compared with a positive-$\gamma$ spin in the same $\mathbf B$. The frequency magnitude is still $|\gamma|B_0$.

When comparing books, check whether they define the Larmor frequency as $\omega_L=\gamma B_0$, $\omega_L=-\gamma B_0$, or only as the positive magnitude.

Common Mistakes

• Forgetting that the sign of $\gamma$ changes the sense of precession.
• Saying that an energy eigenstate “precesses” when only its global phase changes.
• Confusing Larmor precession in a static field with Rabi oscillations driven by a transverse field.
• Treating the Bloch vector as a literal classical spinning object rather than an expectation-value representation.
• Dropping the distinction between angular frequency $\omega$ and ordinary frequency $f=\omega/(2\pi)$.
• Assuming damping or relaxation is present in the closed-system Larmor equation; relaxation requires environmental or open-system physics.

Exercises

1. Starting from $H=-\gamma B_0S_z$, derive $dS_x/dt=\gamma B_0S_y$.

Solution

Use the Heisenberg equation:

$$\frac{dS_x}{dt} = \frac{i}{\hbar}[H,S_x].$$

With $H=-\gamma B_0S_z$,

$$\frac{dS_x}{dt} = -\frac{i\gamma B_0}{\hbar}[S_z,S_x].$$

Since

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

one obtains

$$\frac{dS_x}{dt} = \gamma B_0S_y.$$

2. A spin starts in $|+x\rangle$ in a field $\mathbf B=B_0\hat{\mathbf z}$. Write the state at time $t$ up to global phase.

Solution

Use

$$|+x\rangle = \frac{1}{\sqrt2} \left( |+z\rangle+|-z\rangle \right).$$

Under the Hamiltonian $H=-(\hbar\omega/2)\sigma_z$, the state becomes

$$|\psi(t)\rangle = \frac{1}{\sqrt2} \left( e^{i\omega t/2}|+z\rangle + e^{-i\omega t/2}|-z\rangle \right).$$

Removing the global phase $e^{i\omega t/2}$ gives

$$|\psi(t)\rangle \sim \frac{1}{\sqrt2} \left( |+z\rangle + e^{-i\omega t}|-z\rangle \right).$$

3. For $\gamma<0$ and $B_0>0$, does the Bloch azimuth $\phi(t)$ increase or decrease in the convention of this page?

Solution

This page uses

$$\phi(t)=\phi_0-\omega t, \qquad \omega=\gamma B_0.$$

If $\gamma<0$ and $B_0>0$, then $\omega<0$. Therefore

$$\phi(t)=\phi_0+|\omega|t,$$

so the azimuth increases.

4. Which spin-$1/2$ density operators remain stationary in this field, including mixed states?

Solution

For nonzero $\gamma B_0$, stationarity requires $[\rho,\sigma_z]=0$. Thus

$$\rho=\tfrac12(I+r_z\sigma_z),\qquad -1\leq r_z\leq1.$$

Every diagonal mixture of the two energy states is stationary, not only the pure poles. Such states have no transverse coherence. In particular $I/2$ remains at the center of the Bloch ball.

References

• A. Abragam, The Principles of Nuclear Magnetism, Oxford University Press, 1961.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• 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.
• C. P. Slichter, Principles of Magnetic Resonance, 3rd ed., Springer, 1990.