[Click here for a PDF of this post with nicer formatting]
Problems from angular momentum chapter of [1].
Q: S_y eigenvectors
Find the eigenvectors of \sigma_y , and then find the probability that a measurement of S_y will be \Hbar/2 when the state is initially
\begin{equation}\label{eqn:someSpinProblems:20} \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \end{equation}
A:
The eigenvalues should be \pm 1 , which is easily checked
\begin{equation}\label{eqn:someSpinProblems:40} \begin{aligned} 0 &= \Abs{ \sigma_y – \lambda } \\ &= \begin{vmatrix} -\lambda & -i \\ i & -\lambda \end{vmatrix} \\ &= \lambda^2 – 1. \end{aligned} \end{equation}
For \ket{+} = (a,b)^\T we must have
\begin{equation}\label{eqn:someSpinProblems:60} -1 a – i b = 0, \end{equation}
so
\begin{equation}\label{eqn:someSpinProblems:80} \ket{+} \propto \begin{bmatrix} -i \\ 1 \end{bmatrix}, \end{equation}
or
\begin{equation}\label{eqn:someSpinProblems:100}
\ket{+} =
\inv{\sqrt{2}}
\begin{bmatrix}
1 \\
i
\end{bmatrix}.
\end{equation}
For \ket{-} we must have
\begin{equation}\label{eqn:someSpinProblems:120} a – i b = 0, \end{equation}
so
\begin{equation}\label{eqn:someSpinProblems:140} \ket{+} \propto \begin{bmatrix} i \\ 1 \end{bmatrix}, \end{equation}
or
\begin{equation}\label{eqn:someSpinProblems:160}
\ket{+} =
\inv{\sqrt{2}}
\begin{bmatrix}
1 \\
-i
\end{bmatrix}.
\end{equation}
The normalized eigenvectors are
\begin{equation}\label{eqn:someSpinProblems:180} \boxed{ \ket{\pm} = \inv{\sqrt{2}} \begin{bmatrix} 1 \\ \pm i \end{bmatrix}. } \end{equation}
For the probability question we are interested in
\begin{equation}\label{eqn:someSpinProblems:200} \begin{aligned} \Abs{\bra{S_y; +} \begin{bmatrix} \alpha \\ \beta \end{bmatrix} }^2 &= \inv{2} \Abs{ \begin{bmatrix} 1 & -i \end{bmatrix} \begin{bmatrix} \alpha \\ \beta \end{bmatrix} }^2 \\ &= \inv{2} \lr{ \Abs{\alpha}^2 + \Abs{\beta}^2 } \\ &= \inv{2}. \end{aligned} \end{equation}
There is a 50 % chance of finding the particle in the \ket{S_x;+} state, independent of the initial state.
Q: Magnetic Hamiltonian eigenvectors
Using Pauli matrices, find the eigenvectors for the magnetic spin interaction Hamiltonian
\begin{equation}\label{eqn:someSpinProblems:220} H = – \inv{\Hbar} 2 \mu \BS \cdot \BB. \end{equation}
A:
\begin{equation}\label{eqn:someSpinProblems:240} \begin{aligned} H &= – \mu \Bsigma \cdot \BB \\ &= – \mu \lr{ B_x \begin{bmatrix} 0 & 1 \\ 1 & 0 \\ \end{bmatrix} + B_y \begin{bmatrix} 0 & -i \\ i & 0 \\ \end{bmatrix} + B_z \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix} } \\ &= – \mu \begin{bmatrix} B_z & B_x – i B_y \\ B_x + i B_y & -B_z \end{bmatrix}. \end{aligned} \end{equation}
The characteristic equation is
\begin{equation}\label{eqn:someSpinProblems:260}
\begin{aligned}
0
&=
\begin{vmatrix}
-\mu B_z -\lambda & -\mu(B_x – i B_y) \\
-\mu(B_x + i B_y) & \mu B_z – \lambda
\end{vmatrix} \\
&=
-\lr{ (\mu B_z)^2 – \lambda^2 }
– \mu^2\lr{ B_x^2 – (iB_y)^2 } \\
&=
\lambda^2 – \mu^2 \BB^2.
\end{aligned}
\end{equation}
That is
\begin{equation}\label{eqn:someSpinProblems:360}
\boxed{
\lambda = \pm \mu B.
}
\end{equation}
Now for the eigenvectors. We are looking for \ket{\pm} = (a,b)^\T such that
\begin{equation}\label{eqn:someSpinProblems:300} 0 = (-\mu B_z \mp \mu B) a -\mu(B_x – i B_y) b \end{equation}
or
\begin{equation}\label{eqn:someSpinProblems:320} \ket{\pm} \propto \begin{bmatrix} B_x – i B_y \\ B_z \pm B \end{bmatrix}. \end{equation}
This squares to
\begin{equation}\label{eqn:someSpinProblems:340} B_x^2 + B_y^2 + B_z^2 + B^2 \pm 2 B B_z = 2 B( B \pm B_z ), \end{equation}
so the normalized eigenkets are
\begin{equation}\label{eqn:someSpinProblems:380}
\boxed{
\ket{\pm}
=
\inv{\sqrt{2 B( B \pm B_z )}}
\begin{bmatrix}
B_x – i B_y \\
B_z \pm B
\end{bmatrix}.
}
\end{equation}
References
[1] Jun John Sakurai and Jim J Napolitano. Modern quantum mechanics. Pearson Higher Ed, 2014.