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Time Evolution of Spin in a Magnetic Field
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[QUOTE="JoseGG, post: 5874852, member: 635046"] [h2]Homework Statement [/h2] The Hamiltonian of a spin 1/2 particle is given by: $$H=g\overrightarrow { S }\cdot \overrightarrow { B } $$ where ##\overrightarrow { S }=\hbar \overrightarrow{\sigma }/2## is the spin operator and ##\overrightarrow { B }## is an external magnetic field. 1. Determine ##\dot { \overrightarrow{ S } } ## as a function of S⃗ and B⃗ . 2. Consider now the particular case in which B⃗ = ##\hat{z}##B is oriented along ##\hat{z}##. Calculate the eigenstates and eigenvalues of ##\dot{S_y}##. 3. For t = 0 the system is in one of the eigenstates of ##\dot{S_y}## . Calculate the time evolution of the spin state and of the expectation value of the energy. [B] 2. Homework Equations [/B] 1. So we are working with Heisenberg, equaiton of motion. They ask what is the rate of change of Spin operator with time. We are dealing with, $$ \frac { dS }{ dt } =\frac { 1 }{ i\hbar } \left[ \overrightarrow { S } ,H \right] $$ 2. ? 3. I think the timeevolution operator on the state $$e^{-iHt/\hbar}\left |s \right> $$ [h2]The Attempt at a Solution[/h2] 1. I interpet the ##\overrightarrow { S}## in the Heisenberg equation of motion as, a vector of [Sx,Sy,Sz], I don't know how to work with the Heisenberg equation to find the dirrevative. [B][/B] 2. Use the found ##\dot{\overrightarrow {S}}## vector and pick ##\dot{S_y}##, solve it as an eigenvalue problem with, spin up, with an eigen value of one. I am not able to move forward without the first task. Any help would be nice. It is possible my attempt are not correct. [/QUOTE]
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Time Evolution of Spin in a Magnetic Field
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