Phys 321
Fall 2007
Quantum Theory I: Homework 4
Due: 17 September 2007
1 Measurement Operators
Consider a spin-1/2 particle subjected to an SG ˆymeasurement. The projection operators
associated with the measurement outcomes are
Sy= +~
2
!|+ˆyi h+ˆy|
Sy=−~
2
!|−ˆyi h− ˆy|
a) Determine the matrix representation of |+ˆyi h+ˆy|in the {|+ˆzi,|−ˆzi} basis.
b) Determine the matrix representation of |−ˆyi h− ˆy|in the {|+ˆzi,|−ˆzi} basis.
c) Suppose that the spin-1/2 particle is initially in the state |+ˆxi.Using row and col-
umn vector representations of bra and ket vectors and the projection operations above,
determine Pr (Sy=±~/2).
d) Determine the matrix representation of ˆ
Syin the {|+ˆzi,|−ˆzi} basis and use this to
determine the expectation value hSyifor an ensemble of particles in the state |+ˆxi.
2Quantum Theory I: Problems, problem I.13.
3Quantum Theory I: Problems, problem I.14.
1