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Physics LibreTexts

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

μ=e2meL,

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

μ=ge2meS,

where g is called the gyromagnetic ratio. Classically, we would expect g=1. In fact,

g=2(1+α2π+)=2.0023192,

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

H=μB.

Assuming that the previous expression also holds good in quantum mechanics, the Hamiltonian of an electron in a z-directed magnetic field of magnitude B takes the form

H=ΩSz,

where Ω=geB2me.
Here, for the sake of simplicity, we are neglecting the electron’s translational degrees of freedom.

Schrödinger’s equation can be written [see Equation ([etimed])]

iχt=Hχ,

where the spin state of the electron is characterized by the spinor χ. Adopting the Pauli representation, we obtain

χ=(c+(t)c(t)),

where |c+|2+|c|2=1. Here, |c+|2 is the probability of observing the spin-up state, and |c|2 the probability of observing the spin-down state. It follows from Equations ([e10.46]), ([e10.53]), ([e10.60]), ([e10.62]), and ([e10.63]) that i(˙c+˙c)=Ω2(1,00,1)(c+c),
where ˙ d/dt. Hence,

˙c±=iΩ2c±

Let

c+(0)=cos(α/2),c(0)=sin(α/2).

The significance of the angle α will become apparent presently. Solving Equation ([e10.65]), subject to the initial conditions ([e10.66]) and ([e10.67]), we obtain

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=χAχ.

Hence, the expectation value of Sz is Sz=2(c+,c)(1,00,1)(c+c),
which reduces to

Sz=2cosα

with the help of Equations ([e10.68]) and ([e10.69]). Likewise, the expectation value of Sx is Sx=2(c+,c)(0,11,0)(c+c),
which reduces to

Sx=2sinαcos(Ωt).

Finally, the expectation value of Sy is

Sy=2sinαsin(Ωt).

According to Equations ([e10.72]), ([e10.74]), and ([e10.75]), the expectation value of the spin angular momentum vector subtends a constant angle α with the z-axis, and precesses about this axis at the frequency ΩeBme.
This behavior is actually equivalent to that predicted by classical physics. Note, however, that a measurement of Sx, Sy, or Sz will always yield either +/2 or /2. It is the relative probabilities of obtaining these two results that varies as the expectation value of a given component of the spin varies.

Contributors and Attributions

  • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)


This page titled 9.5: Spin Precession is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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