Larmor Magnetic-Moment Precession

Theory

The spin vector precesses around the magnetic field axis (Z-axis). For an initial spin vector S = (sin(tilt) cos(φ₀), sin(tilt) sin(φ₀), cos(tilt)), the time evolution of the azimuthal angle is φ(t) = φ₀ - Bt. The Z-component of spin (S_z) and the spin vector's norm (|S| = 1) are conserved quantities. The magnetic energy E = -B·S = -B S_z is also conserved. The signed precession frequency is -B (not the absolute value |B|), meaning reversing the magnetic field direction reverses the rotation direction. A tilt of 0 gives a stationary spin despite a nonzero formal precession frequency, and a field of B=0 results in no motion. This module demonstrates pure precession without damping, and is not Larmor radiation, population relaxation, or a driven quantum Rabi experiment.

Worked Example

Consider a system where the magnetic field B = 1, the initial tilt angle is 45 degrees, the initial azimuth is 0 degrees, and the elapsed time is 0.5π (Time over pi = 0.5).

1. The initial azimuth φ₀ is 0, and the field B is 1, so the azimuth at time t = 0.5π is φ(t) = 0 - 1 × (0.5π) = -0.5π.
2. The tilt angle is 45 degrees, so sin(45°) = 1/√2 ≈ 0.7071067811865476 and cos(45°) = 1/√2 ≈ 0.7071067811865476.
3. The spin vector at t = 0.5π is (1/√2 cos(-0.5π), 1/√2 sin(-0.5π), 1/√2) = (0, -0.7071067811865475, 0.7071067811865476).
4. The magnetic energy is E = -B × S_z = -1 × 0.7071067811865476 = -0.7071067811865476.
5. The torque is S × B = (0, -0.7071067811865475, 0.7071067811865476) × (0, 0, 1) = (-0.7071067811865475, 0, 0).
6. The spin norm squared is 1, and the signed precession frequency is -1.

Experiment

Open Larmor Magnetic-Moment Precession to observe the dynamics interactively. Use these exact labelled controls to verify the classical precession properties.

1. Set Field B to 1, Tilt degrees to 45, Initial azimuth degrees to 0, and Time over pi to 0.5. Verify that Spin x is 0.0, Spin y is -0.7071, and Spin z is 0.7071.
2. Change Field B to -2. Verify that the rotation reverses direction, the signed precession frequency becomes 2, the final Spin x is -0.7071, and the final Spin y is 0.0. The magnetic energy becomes 1.4142.
3. Set Field B to 1, Tilt degrees to 45, Initial azimuth degrees to 90, and Time over pi to 0.5. Verify that Spin x is 0.7071, Spin y is 0.0, and the unwrapped phase is 0.0.
4. Set Field B to 0. Verify that the spin remains stationary at its initial position, with a precession frequency of 0 and magnetic energy of 0.
5. Set Field B to 1, Tilt degrees to 0, Initial azimuth degrees to 0, and Time over pi to 0.5. Notice that despite a formal precession frequency of -1, the spin is stationary pointing straight up with Spin z = 1.0.

Limitations

This simulation models classical undamped Larmor precession of a single magnetic moment in a uniform, static magnetic field. It does not account for radiation damping, population relaxation (T1/T2), or interactions with driving fields (like in a quantum Rabi experiment). The numerical results are computed using ordinary double-precision floating-point arithmetic.

References

The classical equation of motion for a magnetic moment in a uniform magnetic field and its resulting precession is derived and discussed in The Feynman Lectures on Physics, Vol. II, Chapter 35.