Larmor precession

The above applet offers a representation of the spin state of a spin 1/2 particle, using either the Bloch sphere (left) or a direct representation of the complex coefficients \(a_+(t)\) and \(a_-(t)\) of the state as written in the basis \(\{|+\rangle_z, |-\rangle_z\}\), i.e. \(|\psi(t)\rangle = a_+(t) |+\rangle_z + a_-(t) |-\rangle_z\) The two coefficients are shown in the complex plane (right) and can be adjusted using the mouse. A magnetic field \(\vec B_0 = B_0 (\cos \alpha)\vec u_z + B_0 (\sin\alpha) \vec u_x\) in the \(xz\) plane is on the system, resulting in Larmor precession. Using the toolbar, you can adjust both the amplitude \(B_0\) and the angle \(\alpha\) with respect to the \(z\) axis of the field.

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Background
With no loss of generality, the global phase of the state \(|\psi\rangle\) can be chosen so that the complex coefficients \(a_+\) and \(a_-\) have opposite phases. Considering the normalization condition \(|a_+|^2+|a_-|^2=1\), we can thus define two angles \(\theta\in[0, \pi]\) and \(\varphi\in[0, 2\pi[\) so that \(a_+ = e^{-i\varphi/2} \cos\frac{\theta}{2}\)  and  \(a_- = e^{i\varphi/2} \sin\frac{\theta}{2}\). It can be easily shown that such a state happens to be the eigen state of the observable \(\hat{\vec S}\cdot \vec u\), where \(\hat{\vec S}\) is the spin operator and \(\vec u = (\sin\theta\cos\varphi, \sin\theta\sin\varphi, \cos\theta)\) is the unit vector defined by spherical coordinates \(\theta\) and \(\varphi\). It can then be easily shown that for such a state \(|\psi\angle\), we have \(\langle \psi|\hat{\vec S}|\psi\rangle = (\hbar/2) \vec u\). The state of the system is thus entirely characterized by the average value of the spin, shown here on the left panel in the so-called Bloch sphere. You can verify by adjusting the \(a_+\) and \(a_-\) coefficients that there is indeed a one-to-one correspondence between a vector in the Bloch sphere and the state of the system.

When a magnetic field \(B_0\) is applied along the \(z\) axis, the Hamiltonian reads \(\hat H = - \hat {\vec \mu} \cdot \vec B_0 = - \gamma B_0 \hat S_z = \omega_0 \hat S_z\), where \(\hat{\vec \mu}\) is the magnetic moment operator, \(\gamma\) is the gyromagnetic ratio and \(\omega_0\) is defined as \(\omega_0 = -\gamma B_0\). As states \(|\pm \rangle_z\) are eigenstates of \(\hat H\) associated with eigenvalues \(\pm\hbar\omega_0\), the time evolution simply reads \(a_\pm(t) = a_\pm(0) \exp(\mp i \omega_0 t/2)\), of \(\varphi(t) = \varphi(0) + \omega t\). This corresponds to a precession of the spin around the \(z\) axis, as illustrated in the above simulation.