Introduction To Quantum Mechanics
Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
Question
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Chapter 4, Problem 4.62P
To determine

The spin matrices (Sx,Sy,Sz) for arbitrary spin s.

Expert Solution & Answer
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Answer to Problem 4.62P

The spin matrices (Sx,Sy,Sz) for arbitrary spin s are Sx=2(0bs00...00bs0bs10...000bs10bs2...0000bs20...00.....................0000...0bs+10000...bs+10), Sy=2i(0bs00...00bs0bs10...000bs10bs2...0000bs20...00.....................0000...0bs+10000...bs+10), and Sz=(s00...00s10...000s2...0...............000...s).

Explanation of Solution

From Equation 4.135, the quantization of Sz is

Sz|sm=m|sm

Identifying the states by the value of m (which is s to +s) as s is fixed.

The matrix element of Sz are

Snm=n|Sz|m=mn|m=mδnm

Sz is a diagonal matrix, with element m. Where, from m=s to m=s.

Sz=(s00...00s10...000s2...0...............000...s)

From Equation 4.136, the quantization of S± is

S±|sm=s(s+1)m(m±1)|s(m±1)=(sm)(s±m+1)|s(m±1)

(S+)nm=n|S+|m=(sm)(s+m+1)n|m+1=bm+1δn(m+1)=bnδn(m+1)

All non-zero elements have row index (n) one greater than the column index (m), so

S+=(0bs00...000bs10...0000bs2...0..................0000...bs+10000...0)

Similarly, for S

(S)nm=n|S|m=(s+m)(sm+1)n|m1=bmδn(m1)

Therefore,

S=(000..00bs00..000bs10..00.................000..bs+10)

Write the expression to fine the spin matrix Sy

  Sy=12i(S+S)

Write the expression to fine the spin matrix Sx

  Sx=12(S++S)

Therefore,

Sx=2(0bs00...00bs0bs10...000bs10bs2...0000bs20...00.....................0000...0bs+10000...bs+10)

And,

Sy=2i(0bs00...00bs0bs10...000bs10bs2...0000bs20...00.....................0000...0bs+10000...bs+10)

Conclusion:

Thus, the spin matrices (Sx,Sy,Sz) for arbitrary spin s are Sx=2(0bs00...00bs0bs10...000bs10bs2...0000bs20...00.....................0000...0bs+10000...bs+10), Sy=2i(0bs00...00bs0bs10...000bs10bs2...0000bs20...00.....................0000...0bs+10000...bs+10), and Sz=(s00...00s10...000s2...0...............000...s).

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