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Physics 481 Recitation Session 1: Constructing Spin Operators

TA: Jeff Schueler January 20th 2016

1 Introduction

Welcome to the physics 481 recitation session! The goal of these sessions is to provide students a settingwhere they can work in groups to discuss how to work through problems. Since this is the first timerecitation sessions are being offered in upper division courses at UHM, the format of these recitationsessions is subject to change. Your feedback is also much appreciated, as it will give me a better idea asto how to best use our time. Please bring your textbook with you to all recitation sessions, as itmay come in handy!

For today, our topic of interest is the construction of spin operators. According to Dr. Vahsen, youhave seen the construction ofS2, Sx, Sy, Sz, and ladder operators S for spin-1/2 particles in lecture, sowe will begin with the construction of the S2 andSz operators for spin 1 particles. The actual problemsbegin in section 2 (at the bottom half of page 2), so you may start those whenever you feelready. Below is a more detailed outline of how to construct S2 and Sz matrices than what is provided in

Griffiths.

1.1 A guide for constructing spin matrices fors= 1/2

Given a spin state |s,ms, we take the following relations to be true for all allowed values ofs andms:

S2 |s,ms= s(s+ 1)2 |s,ms Sz |s,ms= ms|s,ms

S |s,ms=

s(s+ 1) m(m 1)|s,ms 1 (1)

With these three relationships at our disposal, we can construct matrix operators for S2, Sx, Sy, Sz,and S for any value ofs! Lets start with a brief summary of the construction of the spin-1/2 matrices,then you guys will construct the operators for spin-1, and if time allows, spin-3/2.

1.2 Spin-1/2S2 matrix

The matrix elements for the S2 operator are given bys,m

S2 s,m. When computing these matrixelements for spin-1/2 particles, we note thats= 1/2 always, and the allowed values for ms are +1/2 and1/2. Given this, we see that there are four possible combinations ofs and ms, leading to four martrixelements, i.e., a 2 2 matrix. Computing the individual matrix elements in the most general case will give

s,m

S2s,m= s(s+ 1)2s,ms, m=s(s+ 1)2ssmm

The Kronecker delta symbols are a result of the s,m|s, m states being orthonormal. The symbols alsotell us that s and s must always be equal and m and m must also always be equal, which tells us thatthe only nonzero elements of the S2 matrix for any value of spin s are the diagonal elements. For s = 1/2,we find

1/2,m

S21/2,m= 1(1 + 1)21/2,m1/2,m (2)= 22mm (3)

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(2) tells us that we can write out ourS2 matrix comprised of the four S2 matrix elements as

S2 = 22

1/2,m = 1/2|1/2,m= 1/2 1/2,m = 1/2|1/2,m= 1/21/2,m =1/2|1/2,m= 1/2 1/2,m =1/2|1/2,m= 1/2

,

where it is to be understood that s = s = 1/2 for each of the four matrix elements. (3) tells us that all ofthe components withm =m will vanish, and all of the components with m =m will equal 1 (due to the

orthonormality of the spin states), thus

S2 = 22

1 00 1

. (4)

1.3 Sz for s= 1/2

Following the procedure of the S2 operator, we can construct the matrix elements for Sz as follows:

s,m

Sz s,m= ms,ms, m=mssmm

For s= 1/2, we know that the only allowed values for m are 1/2, so:

1/2, 1/2| Sz |1/2, 1/2= mmm =

21/2, 1/2|Sz |1/2, 1/2= mmm = 0

1/2, 1/2| Sz |1/2, 1/2= mmm = 0

1/2,1/2|Sz |1/2, 1/2= mmm =

2.

Substituting these four elements into Sz gives us

Sz =

21 0

0 1 . (5)

The construction of the remaining spin-1/2 matrices are described in sufficient detail in Griffiths pages 173and 174, so I will not go through those details here.

2 Construction of Spin 1 Operators

In the space below, work together with your group to answer the following questions. I will walk aroundto check on your progress. Please collaborate with and bounce ideas off of each other! For all questions inthis section, we will be assuming s= 1.

1. What are the allowed values ofms? (Hint: There are 3)

2. What should be the dimensions of all of the spin-1 matrices?

3. Using the fact that Sz |s,ms= ms|s,ms, construct the matrix operator for Sz.

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(More space for work)

4. Using the fact that S2 |s,ms= s(s+ 1)2 |s,ms, construct the matrix operator forS

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5. Write out the spinors for the three spin-1 states |1, 1, |1, 0, and |1, 1.

6. (Optional, as this is already assigned for homework) Construct the matrices for Sx and Sy. Hint:Proceeding as is done in Griffiths pages 173-174 will be most helpful here: First construct the S+ andS matrices by letting them act on the three spinors you wrote out in question 5 (equation (1) on thefirst page gives you the eigenstates and eigenvalues ofS acting on a given spin ket). Once S+ andSare constructed, you know (or can easily check) that Sx= (S++S)/2 and Sy = (S+ S)/2i.

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3 Construction of Spin 3/2 Operators

1 What are the allowed values ofms? (Hint: There are 4)

2 Write out the spinors (and all of the kets in the form of|s,ms) for the four spin-3/2 states.

3 What should be the dimensions of all of the spin-3/2 matrices?

4 Using the fact thatSz |s,ms= ms|s,ms, construct the matrix operator for Sz.

5 Using the fact thatS2 |s,ms= s(s+ 1)2 |s,ms, construct the matrix operator for S

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6 Construct the matrices for Sx and Sy (You may need to attach a separate sheet of work for this one. Iftime allows, we will go over a general outline of solving this problem as a class.)

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