9.5: Spin Precession
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According to classical physics, a small current loop possesses a magnetic moment of magnitude μ=IA, where I is the current circulating around the loop, and A the area of the loop. The direction of the magnetic moment is conventionally taken to be normal to the plane of the loop, in the sense given by a standard right-hand circulation rule. Consider a small current loop consisting of an electron in uniform circular motion. It is easily demonstrated that the electron’s orbital angular momentum L is related to the magnetic moment μ of the loop via
where e is the magnitude of the electron charge, and me the electron mass.
The previous expression suggests that there may be a similar relationship between magnetic moment and spin angular momentum. We can write
here α=e2/(2ϵ0hc)≃1/137 is the so-called fine-structure constant. The fact that the gyromagnetic ratio is (almost) twice that expected from classical physics is only explicable using relativistic quantum mechanics . Furthermore, the small corrections to the relativistic result g=2 come from quantum field theory .
The energy of a classical magnetic moment μ in a uniform magnetic field B is
Schrödinger’s equation can be written [see Equation ([etimed])]
c+(0)=cos(α/2),c−(0)=sin(α/2).
c+(t)=cos(α/2)exp(−iΩt/2),c−(t)=sin(α/2)exp(+iΩt/2).
We can most easily visualize the effect of the time dependence in the previous expressions for c± by calculating the expectation values of the three Cartesian components of the electron’s spin angular momentum. By analogy with Equation ([e3.55]), the expectation value of a general spin operator A is simply ⟨A⟩=χ†Aχ.
Contributors and Attributions
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)