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Quantum Theory I: Homework 4 for Phys 321, Fall 2007, Assignments of Physics

The solution to homework 4 for the quantum theory i course in physics 321, which was due on september 17, 2007. The homework covers measurement operators for a spin-1/2 particle, specifically the projection operators sy associated with sg y measurements. The tasks include determining the matrix representation of |+ˆy⟩ ⟨+ˆy| and |−ˆy⟩ ⟨−ˆy| in the {|+ˆz⟩ , |−ˆz⟩} basis, finding the probability of obtaining measurement outcomes ±ℏ/2 when the initial state is |+ˆx⟩, and determining the matrix representation of ˆsy and the expectation value ⟨sy⟩ for an ensemble of particles in the state |+ˆx⟩.

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Uploaded on 08/19/2009

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bg1
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

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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 yˆ measurement. The projection operators associated with the measurement outcomes are

Sy = +

! |+yˆ〉 〈+yˆ|

Sy = −

! |−yˆ〉 〈−yˆ|

a) Determine the matrix representation of |+yˆ〉 〈+yˆ| in the {|+ˆz〉 , |−zˆ〉} basis. b) Determine the matrix representation of |−yˆ〉 〈−yˆ| in the {|+ˆz〉 , |−zˆ〉} basis. c) Suppose that the spin-1/2 particle is initially in the state |+xˆ〉. Using row and col- umn vector representations of bra and ket vectors and the projection operations above, determine Pr (Sy = ±ℏ/2). d) Determine the matrix representation of Sˆy in the {|+ˆz〉 , |−ˆz〉} basis and use this to determine the expectation value 〈Sy〉 for an ensemble of particles in the state |+xˆ〉.

2 Quantum Theory I: Problems, problem I.13.

3 Quantum Theory I: Problems, problem I.14.