Problem $1[4]$

For $=1,$ the angular$-$momentum components $$Lx$,$Ly and $$Lz are represented by the

matrices

Lx $=$

$$

$$

$0/20$

$/20/2$

$0/20$

$$

$$

Ly $=$

$$

$$

$0$i$/20$

i$/20$i$/2$

$0$ i$/20$

$$

$$

Lz $=$

$$

$$

$00$

$000$

$00$

$$

$,$

$($i$)$ Compute L$2=$ L$2$

x $+$ L$2$

y $+$ L$2$

z and show that it is a multiple of the identity matrix.

$($ii$)$ Compute the eigenvalues of Lx and Ly$.$

$($iii$)$ Show that

$$

$$

$1$

$0$

$1$

$$

$$ is an eigenvector of Ly $.$ If a system is in this state, what are the

probabilities for finding the values $,0,$ and $,$ respectively, in a single measurement

of Lz $?$