Question

Problem (4) Let us consider the following operators on a Hilbert space $V^3$: $L_x = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & 1 & 0 \ 1 & 0 & 1 \ 0 & 1 & 0 \end{pmatrix}$, $L_y = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & -i & 0 \ i & 0 & -i \ 0 & i & 0 \end{pmatrix}$, $L_z = \begin{pmatrix} 1 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & -1 \end{pmatrix}$. (a) Find the eigenvalues $l_z$ and normalized eigenvectors $|l_z\rangle$ of $L_z$. (b) Take the state in which $l_z = 1$. In this state, what are $\langle L_x \rangle$, $\langle L_x^2 \rangle$, and $\Delta L_x$. (c) Find the eigenvalues and the normalized eigenvectors of $L_x$ in $l_z$ basis. (d) If the particle is in the state with $l_z = -1$, and $L_x$ is measured, what are the possible outcomes and their probabilities?

          Problem (4)
Let us consider the following operators on a Hilbert space $V^3$:
$L_x = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & 1 & 0 \ 1 & 0 & 1 \ 0 & 1 & 0 \end{pmatrix}$, $L_y = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & -i & 0 \ i & 0 & -i \ 0 & i & 0 \end{pmatrix}$, $L_z = \begin{pmatrix} 1 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & -1 \end{pmatrix}$.
(a) Find the eigenvalues $l_z$ and normalized eigenvectors $|l_z\rangle$ of $L_z$.
(b) Take the state in which $l_z = 1$. In this state, what are $\langle L_x \rangle$, $\langle L_x^2 \rangle$, and $\Delta L_x$.
(c) Find the eigenvalues and the normalized eigenvectors of $L_x$ in $l_z$ basis.
(d) If the particle is in the state with $l_z = -1$, and $L_x$ is measured, what are the
possible outcomes and their probabilities?

Problem (4)
Let us consider the following operators on a Hilbert space V^3:
Lx = (1)/(√(2))
< p m a t r i x >, Ly = (1)/(√(2))
< p m a t r i x >, Lz =
< p m a t r i x >.
(a) Find the eigenvalues lz and normalized eigenvectors |lz⟩ of Lz.
(b) Take the state in which lz = 1. In this state, what are ⟨ Lx ⟩, ⟨ Lx^2 ⟩, and Δ Lx.
(c) Find the eigenvalues and the normalized eigenvectors of Lx in lz basis.
(d) If the particle is in the state with lz = -1, and Lx is measured, what are the
possible outcomes and their probabilities?

Added by Kurt K.

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