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# Quantum Probability of Eigenvalue Measurement

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A particle of spin-one and magnetic moment  is in a uniform ~B field of strength B. At t = 0, the component of the spin along an axis is at angle  with the ~B field direction is measured to be 1hbar. What is the probability that a measurement at time t(> 0 will yield the eigenvalues mhbar (with m = +1; 0, or -1)?

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#### Solution Preview

A particle of spin 1 and magnetic moment is in a uniform magnetic field . At the component of the spin along an axis at an angle with respect to is measured to be What is the probability that a measurement at time will yield the eigenvalues (with m equal to 1, 0, or -1)?

We set up a coordinate system in which points in the z-direction and the magnetic moment is initially in the xz-plane. The particle experiences a torque

where S is the spin angular momentum of the particle. The magnitude of the torque is given by

where is the angular frequency of precession of the magnetic moment of the particle about the z-axis. Thus we have

Now the spin angular momentum of the particle at time is ...

#### Solution Summary

We calculate the probability of measuring each of three possible eigenvalues (z-component of spin) of a spin-one particle precessing in a magnetic field.

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