For a spin one system the matrix: $S_y=\hbar \begin{bmatrix} 0 & -i & 0 \\ i & 0& -i \\ 0 & i& 0 \end{bmatrix}$
Suppose an arbitrary vector $|\psi \rangle=[a,b,c]$ where $aa^*+bb^*+cc^*=1$.
Prove that $\langle\psi|S_y |\psi\rangle$ has real value.
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Sign up to join this communityFor a spin one system the matrix: $S_y=\hbar \begin{bmatrix} 0 & -i & 0 \\ i & 0& -i \\ 0 & i& 0 \end{bmatrix}$
Suppose an arbitrary vector $|\psi \rangle=[a,b,c]$ where $aa^*+bb^*+cc^*=1$.
Prove that $\langle\psi|S_y |\psi\rangle$ has real value.
HINT: The quantity $\langle \psi | S_y | \psi \rangle$ can easily be calculated in terms of $a$, $b$, $c$, and their complex conjugates. If this quantity is real, then it must be equal to its own complex conjugate.
Suppose $\langle \psi\vert S_y\vert \psi \rangle = a+ib$ with $a$ and $b$ real. Then, since $S_y$ is hermitian: $$ \langle \psi\vert S_y\vert \psi \rangle^* = a-ib = \langle \psi\vert S^\dagger _y\vert \psi \rangle =\langle \psi\vert S_y\vert \psi \rangle =a +ib $$ from which it follows that $-b=b=0$. Thus, $\langle \psi\vert S_y\vert \psi \rangle = a$ is purely real.