# Time evolution of spin

See attached file for equations and answer the following:

(a) Calculate the matrix representing in the { | + >, | - > } basis, the operator H, the Hamiltonian of the system.

(b) Calculate the eigenvalues and the eigenvalues of H

(c) The system at the time t = 0 is in the state | _ >. What values can be found if the energy is measured, and with what probabilities?

(d) Calculate the state vector at time t. At this instant, S, is measured; what is the mean value of the reuslts that can be obtained? Give a geometrical interpretation.

https://brainmass.com/physics/schrodinger/time-evolution-spin-571766

#### Solution Preview

The magnetic field is:

(1.1)

The spin operator is a vector sum as well:

(1.2)

The magnetic moment of the particle is:

(1.3)

Where is the mass of the electron, e is the charge and c is the speed of light. In other words we can write the magnetic moment as

(1.4)

Where is a constant.

The energy (Hamiltonian) of the interaction between the field and the spin is

(1.5)

The matrix representation of the spin operators in the basis of are:

(1.6)

Thus the matrix representation of the Hamiltonian is:

(1.7)

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The eigenvalues of the matrix are:

(1.8)

Thus, the eigenvalues (energy levels) of the Hamiltonian are

(1.9)

Where the ...

#### Solution Summary

This step-by-step solution demonstrates how to use vector mechanics to calculate the time evolution and expectation value of the spin.