Larmor precession

A spin- $12$ particle in a constant magnetic field $𝐁$ experiences a Hamiltonian¹

ˆ 𝐻 = - 𝛾 𝐁 \cdot ˆ 𝐒

and given an initial state (measured along the direction of $𝐁$ , which we will call the $𝑧$ -axis) of $| 𝜒 (0) ⟩ = c o s (𝛼 / 2) | ↑ ⟩ + s i n (𝛼 / 2) | ↓ ⟩$ , the state evolves as

| 𝜒 (𝑡) ⟩ = c o s (𝛼 / 2) 𝑒 𝑖 𝜔 𝑡 / 2 | ↑ ⟩ + s i n (𝛼 / 2) 𝑒 - 𝑖 𝜔 𝑡 / 2 | ↓ ⟩

where $𝜔 = 𝛾 ‖ 𝐁 ‖$ is the Larmor (angular) frequency. This corresponds to $⟨ 𝐒 ⟩$ tilted at angle $𝛼$ to the $𝑧$ -axis precessing with angular frequency $𝜔$ .

#state/tidy | #lang/en | #SemBr

2018. Introduction to quantum mechanics, §4.4.2, pp. 172–173 ↩

Larmor precession

Footnotes