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A spin-1/2 is placed in a magnetic field with both x and z-components so that its

Hamiltonian is [itex]H=-b_x \sigma^x-b_z\sigma^z[/itex], where [itex]\sigma^x[/itex] and [itex]\sigma^z[/itex] are the Pauli matrices. The real constants [itex]b_x[/itex] and [itex]b_z[/itex] have units of energy, and account for both the magnetic field components and coupling constants between the spin and the magnetic field.

Consider that the spin-component along the z-axis of the spin-1/2 is known to be [itex]+\hbar /2[/itex] at [itex]t = 0[/itex].

What is the probability that the spin component along the z-axis at time [itex]t ≥ 0[/itex] will be measured to be [itex]-\hbar /2[/itex]?

Solution

Time-evolution of the initial state yields [itex]\vert \psi(t)\rangle=e^{-i \hat H t/ \hbar}\vert \uparrow_z \rangle[/itex], hence the probability of measuring [itex]-\hbar /2[/itex] is [itex]\vert \langle \downarrow_z \vert \psi(t) \rangle \vert ^2=0[/itex].

Is this correct or am I missing something important here?

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# Spin-1/2 in a magnetic field

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