- #1
logic smogic
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Problem
Consider the spin precession problem in the Heisenberg picture. Using the Hamiltonian
[tex]H=-\omega S_{z}[/tex]
where
[tex]\omega=\frac{eB}{mc}[/tex]
write the Heisenberg equations of motion for the time dependent operators [tex]S_{x}(t)[/tex], [tex]S_{y}(t)[/tex], and [tex]S_{z}(t)[/tex]. Solve them to obtain [tex]\vec{S}[/tex] as a function of t.
Formulae
[tex]\frac{d A_{H}}{dt}=\frac{1}{\imath \hbar}[A_{H}, H][/tex]
[tex]A_{H}=U^{\dagger}A_{S}U[/tex]
[tex]U=e^{\frac{-\imath H t}{\hbar}}[/tex]
Attempt
Well, computing the Heisenberg equations is pretty straitforward:
[tex]\frac{d S_{x}}{dt}=\frac{1}{\imath \hbar}[S_{x}, -\omega S_{z}]
=-\frac{\omega}{\imath \hbar}[S_{x},S_{z}]
=\omega S_{y}[/tex]
[tex]\frac{d S_{y}}{dt}=\frac{1}{\imath \hbar}[S_{y}, -\omega S_{z}]
=-\frac{\omega}{\imath \hbar}[S_{y},S_{z}]
=-\omega S_{x}[/tex]
[tex]\frac{d S_{z}}{dt}=\frac{1}{\imath \hbar}[S_{z}, -\omega S_{z}]
=-\frac{\omega}{\imath \hbar}[S_{z},S_{z}]
=0[/tex]
But when it comes to solving for a function of t, I’m stuck with extra constants. My method here is to differentiate [tex]S_{x}[/tex] twice, and then solve the resulting differential equation.
[tex]\frac{d^{2}S_{x}}{dt^{2}}=-\omega^{2} S_{x}[/tex]
[tex]S_{x} = C_{1} e^{\imath \omega t}+C_{2} e^{-\imath \omega t}[/tex]
What “initial/boundary conditions” do I use to determine the two constants above? Normalization of some sort?
Consider the spin precession problem in the Heisenberg picture. Using the Hamiltonian
[tex]H=-\omega S_{z}[/tex]
where
[tex]\omega=\frac{eB}{mc}[/tex]
write the Heisenberg equations of motion for the time dependent operators [tex]S_{x}(t)[/tex], [tex]S_{y}(t)[/tex], and [tex]S_{z}(t)[/tex]. Solve them to obtain [tex]\vec{S}[/tex] as a function of t.
Formulae
[tex]\frac{d A_{H}}{dt}=\frac{1}{\imath \hbar}[A_{H}, H][/tex]
[tex]A_{H}=U^{\dagger}A_{S}U[/tex]
[tex]U=e^{\frac{-\imath H t}{\hbar}}[/tex]
Attempt
Well, computing the Heisenberg equations is pretty straitforward:
[tex]\frac{d S_{x}}{dt}=\frac{1}{\imath \hbar}[S_{x}, -\omega S_{z}]
=-\frac{\omega}{\imath \hbar}[S_{x},S_{z}]
=\omega S_{y}[/tex]
[tex]\frac{d S_{y}}{dt}=\frac{1}{\imath \hbar}[S_{y}, -\omega S_{z}]
=-\frac{\omega}{\imath \hbar}[S_{y},S_{z}]
=-\omega S_{x}[/tex]
[tex]\frac{d S_{z}}{dt}=\frac{1}{\imath \hbar}[S_{z}, -\omega S_{z}]
=-\frac{\omega}{\imath \hbar}[S_{z},S_{z}]
=0[/tex]
But when it comes to solving for a function of t, I’m stuck with extra constants. My method here is to differentiate [tex]S_{x}[/tex] twice, and then solve the resulting differential equation.
[tex]\frac{d^{2}S_{x}}{dt^{2}}=-\omega^{2} S_{x}[/tex]
[tex]S_{x} = C_{1} e^{\imath \omega t}+C_{2} e^{-\imath \omega t}[/tex]
What “initial/boundary conditions” do I use to determine the two constants above? Normalization of some sort?
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