Spin 1/2 systemin Sakurai quantum mechanics

BREAD · Nov 27, 2015

Homework Statement

$Printed textbook page titled 'Spin 1/2 Systems, Once Again' with equation (1.4.8), |<+|S_x;+>| = |<-|S_x;+>| = 1/√2, and equation (1.4.9), |S_x;+> = (1/√2)|+> + (1/√2)e^{iδ1}|->, with δ1 real.$

Printed textbook page 27 showing equations 1.4.10 to 1.4.13: the S_x ket |S_x; -> written in terms of |+> and |->, the S_x operator as a sum of outer products with e^(-i delta_1) and e^(i delta_1) terms, and the S_y kets and operator with delta_2.

Homework Equations

This is a passage from Modern Quantum Mechanics by Sakurai ( page 26~27)

The Attempt at a Solution

I wonder how i can get 1.4.8 , 1.4.9 equations . and what do they mean?

blue_leaf77 · Nov 27, 2015

BREAD said:

I wonder how i can get 1.4.8 , 1.4.9 equations . and what do they mean?

They follow from the immediately preceding paragraph. It's mentioned that the probability for a measurement on the ##z## spin from a ##|S_x;+\rangle## state is equal to 1/2 for both up and down ##z## spin. This means the magnitude of the projections from the state ##|S_x;+\rangle## to the eigenstates of ##z## spin, ##|+\rangle## and ##|-\rangle##, must be equal to ##1/\sqrt{2}##.

BREAD · Nov 28, 2015

Then what is e^i*delta

blue_leaf77 · Nov 28, 2015

The expansion coefficient is in general complex, and that complex exponential is set there to account for this fact, neglecting the overall phase factor as this is not influential for a measurement.

Spin 1/2 systemin Sakurai quantum mechanics

Homework Statement

Homework Equations

The Attempt at a Solution

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