Question

Given that

友一2

where $\sigma_x = \begin{pmatrix} 0 &1 \\ 1&0 \end{pmatrix}$ $\sigma_y = \begin{pmatrix} 0 &-i \\ i&0 \end{pmatrix}$ , $\sigma_z = \begin{pmatrix} 1 &0 \\ 0&-1 \end{pmatrix}$

a) Calculate the uncertainty relations for the following pairs of operators: $(S_x,S_y),(S_x,S_z),(S_y,S_z)$

b) Assume an unknown state $\chi = a\chi_++b\chi_-$ with. Now we measure $S_y$ on that state. What results of that measurement can we expect and how likely is it to measure them?

c) What are the results of a measurement of $S_y^2$ ? and how likely is it to measure them?

Answer 1

S^vector eigen value is hwithstroke^2/4 Eigen state 1 + > as 1 rightarrow P (hwithstroke^2/4) = (< + 1 middot (a1 + > + b1- >)^2/ Vertical-bar a Vertical-bar ^2 + Vertical-bar b Vertical-bar ^2 because Vertical-bar a Vertical-bar ^2 + Vertical-bar b Vertical-bar ^2 = 1 = Vertical-bar a Vertical-bar ^2 P (hwithstroke^2/4) = (< -1 middot (a1 + > + b1- >)^2/ Vertical-bar a Vertical-bar ^2 + Vertical-bar b Vertical-bar ^2 = Vertical-bar b Vertical-bar ^2 Total Probability of measuring S^vector = Vertical-bar a Vertical-bar ^2 + Vertical-bar b Vertical-bar ^2 = 1 most general spinor (a b) < sx > = hwithstroke/2 lx = hwithstroke/2 (a* b*) (0 1 1 0) (a b) = hwithstroke/2 (a* b*) (b a) = hwithstroke/2 (a* b + b* a) = hwithstroke/2 2 Re (ab*) < sx > = hwithstroke Re (ab*) \r\n < sy > = hwithstroke/2 (a* b*) (0 1 - 1 0) (a b) = hwithstroke/2 (a* b*) (-1 1 b a) = hwithstroke/2 (-1 a* b + 1 ab*) = 1 hwithstroke/2 (ab* - a*b) = -hwithstroke Im (ab*) < sz > = hwithstroke/2 (a* b*) (1 0 0 1) (a b) = hwithstroke/2 (a* b*) (a b) = hwithstroke/2 (a* a - b* b) = hwithstroke/2