A spin-1/2 particle with a magnetic moment is known to be in the state |Ψ(t=0)〉 = |+〉.

a) If the observable S𝓍 is measured at t=0, the possible results are ħ/2 and -ħ/2 with an equal probability of measuring either.

b) The system evolves in a uniform magnetic field 𝐁 = B₀ŷ. What is the state of the system at t=T?

Since the magnetic field is oriented along the y axis, the energy eigenstates will be associated with that direction. The eigenstates are |±𝓎〉.

The Hamiltonian for this system is 
    H = - ω₀ S𝓎, with ω₀ = g q/2mₑ B₀ ≈ e/mₑ B₀. 

The eigenvalue equations in the energy basis are therefore

    Ĥ|E±〉 = ω₀ ±ħ/2 |E±〉 = ω₀ ±ħ/2 |±〉 = E± |±〉

The initial state is prepared to |+〉, which means, in the y basis,

    |Ψ(t=0)〉 = 1/√2 |+𝓎〉 - 1/√2 |-𝓎〉

This Hamiltonian is time independent, so the time evolution is given by multiplying each eigenstate with the time-pdependent phase factor, with ι the imaginary unit.

    |Ψ(t)〉 = exp(-ι E₊ t/ħ)/√2 |+𝓎〉 - exp(-ι E₋ t/ħ)/√2 |-𝓎〉



If the time-evolution of the general state is known in the energy basis (aligned with the spin y basis), the time-evolved state at some time t in the z basis can be predicted by projecting the z basis onto the state to determine the coefficients in the z basis. In mathematical terms,

    〈+|Ψ(t)〉|+〉 = 
    〈-|Ψ(t)〉|-〉 = (