- #1
gasar8
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Homework Statement
Two particles with spin [itex]S_1={1 \over 2}[/itex] and [itex]S_2={1 \over 2}[/itex] are at t=0 in a state with [itex]S=0[/itex].
a) Find wave function at t=0 in [itex]S_{1z},[/itex], [itex]S_{2z}[/itex] basis.
b) Second particle is in a magnetic field [itex]B = (\sin\theta,0,\cos\theta)[/itex], the Hamiltonian is [itex]H=\lambda \vec{S_2} \cdot \vec{B}.[/itex] Find the probability that we find particles at time t in [itex]S=1[/itex] state.
The Attempt at a Solution
a) [itex] |10\rangle ={1 \over \sqrt{2}} \big(|{1 \over 2}-{1 \over 2}\rangle + |-{1 \over 2}{1 \over 2}\rangle\big)[/itex]
b) If I write Hamiltonian with Pauli matrices, I get:
[tex]\lambda \vec{S_2} \cdot \vec{B}=\frac{\lambda \hbar B}{2}
\left(
\begin{array}{cc}
\cos\theta & \sin\theta\\
\sin\theta & -\cos\theta
\end{array}
\right)
[/tex]
Then I wrote Schrödingers equation and got eigenvalues [itex]E=\pm \frac{\lambda \hbar B}{2}[/itex] and eigenvectors ([itex]\cot\theta \pm {1 \over \sin\theta},1)[/itex].
I should be right to this point, right? But now, I have some problems. Is the wave function really: [tex]|\psi,0\rangle= |\uparrow\rangle+ \bigg(\cot\theta \pm {1 \over \sin\theta} \bigg) |\downarrow\rangle.[/tex]
How do I normalize it? And after that, how do I find its time evolution? I tried to put Hamiltonian on both states, so [itex]H|\uparrow\rangle[/itex] and [itex]H|\downarrow\rangle,[/itex] but is it right? Because H is a matrix and [itex]|\uparrow\rangle=|{1 \over 2}{1 \over 2}\rangle[/itex] is vector, so I get another vector?