- #1
Francisco Dahab
- 3
- 0
So I was trying to see what the result for the spin precession would be if the magnetic field pointed in the y-direction instead of z. I feel like either something with what I'm doing is wrong or, I'm just overlooking something because I keep getting complex energy eigenvalues. So what I'm doing is:
Initially, we need to find the Hamiltonian which is just
$$H=-\gamma (B \cdot S)=-\gamma B_0S_y=\frac{-\gamma B_0 \hbar}{2}
\begin{pmatrix}
0 & -i \\
i & 0
\end{pmatrix}
$$
Then I proceed to find the eigenvalues, taking the determinant of this matrix and setting it equal to zero
$$
\begin{pmatrix}
-E & \frac{i \gamma B_0 \hbar}{2} \\
\frac{-i \gamma B_0 \hbar}{2} & -E
\end{pmatrix} \Rightarrow E^2+\left(\frac{\gamma B_0 \hbar}{2}\right)^2=0 \Leftrightarrow E=\pm\frac{i\gamma B_0 \hbar}{2}
$$
which is complex, but what's the meaning of complex energies? And if this isn't wrong why won't the spin precess (because once you apply time evolution you won't have an exponential with an imaginary exponent you'll have an ##e^{- t}##
Initially, we need to find the Hamiltonian which is just
$$H=-\gamma (B \cdot S)=-\gamma B_0S_y=\frac{-\gamma B_0 \hbar}{2}
\begin{pmatrix}
0 & -i \\
i & 0
\end{pmatrix}
$$
Then I proceed to find the eigenvalues, taking the determinant of this matrix and setting it equal to zero
$$
\begin{pmatrix}
-E & \frac{i \gamma B_0 \hbar}{2} \\
\frac{-i \gamma B_0 \hbar}{2} & -E
\end{pmatrix} \Rightarrow E^2+\left(\frac{\gamma B_0 \hbar}{2}\right)^2=0 \Leftrightarrow E=\pm\frac{i\gamma B_0 \hbar}{2}
$$
which is complex, but what's the meaning of complex energies? And if this isn't wrong why won't the spin precess (because once you apply time evolution you won't have an exponential with an imaginary exponent you'll have an ##e^{- t}##
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