The spin matrices ( S x , S y , S z ) for arbitrary spin s .

Question

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Answer 1

Question

Chapter 4, Problem 4.62P

To determine

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

Expert Solution & Answer

Answer to Problem 4.62P

The spin matrices (Sx,Sy,Sz) for arbitrary spin s are Sx=ℏ2(0bs00...00bs0bs−10...000bs−10bs−2...0000bs−20...00.....................0000...0b−s+10000...b−s+10), Sy=ℏ2i(0bs00...00−bs0bs−10...000−bs−10bs−2...0000−bs−20...00.....................0000...0b−s+10000...−b−s+10), and Sz=ℏ(s00...00s−10...000s−2...0...............000...−s).

Explanation of Solution

From Equation 4.135, the quantization of Sz is

Sz|s m〉=ℏm|s m〉

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〉=ℏm〈n|m〉=ℏmδn m

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

Sz=ℏ(s00...00s−10...000s−2...0...............000...−s)

From Equation 4.136, the quantization of S± is

S±|s m〉=ℏs(s+1)−m(m±1)|s(m±1)〉=ℏ(s∓m)(s±m+1)|s(m±1)〉

(S+)nm=〈n|S+|m〉=ℏ(s−m)(s+m+1)〈n|m+1〉=ℏbm+1δn(m+1)=ℏbnδn(m+1)

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

S+=ℏ(0bs00...000bs−10...0000bs−2...0..................0000...b−s+10000...0)

Similarly, for S−

(S−)nm=〈n|S−|m〉=ℏ(s+m)(s−m+1)〈n|m−1〉=ℏbmδn(m−1)

Therefore,

S−=ℏ(000..00bs00..000bs−10..00.................000..b−s+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=ℏ2(0bs00...00bs0bs−10...000bs−10bs−2...0000bs−20...00.....................0000...0b−s+10000...b−s+10)

And,

Sy=ℏ2i(0bs00...00−bs0bs−10...000−bs−10bs−2...0000−bs−20...00.....................0000...0b−s+10000...−b−s+10)

Conclusion:

Thus, the spin matrices (Sx,Sy,Sz) for arbitrary spin s are Sx=ℏ2(0bs00...00bs0bs−10...000bs−10bs−2...0000bs−20...00.....................0000...0b−s+10000...b−s+10), Sy=ℏ2i(0bs00...00−bs0bs−10...000−bs−10bs−2...0000−bs−20...00.....................0000...0b−s+10000...−b−s+10), and Sz=ℏ(s00...00s−10...000s−2...0...............000...−s).

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Answer 2

Question

Chapter 4, Problem 4.62P

To determine

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

Expert Solution & Answer

Answer to Problem 4.62P

The spin matrices (Sx,Sy,Sz) for arbitrary spin s are Sx=ℏ2(0bs00...00bs0bs−10...000bs−10bs−2...0000bs−20...00.....................0000...0b−s+10000...b−s+10), Sy=ℏ2i(0bs00...00−bs0bs−10...000−bs−10bs−2...0000−bs−20...00.....................0000...0b−s+10000...−b−s+10), and Sz=ℏ(s00...00s−10...000s−2...0...............000...−s).

Explanation of Solution

From Equation 4.135, the quantization of Sz is

Sz|s m〉=ℏm|s m〉

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〉=ℏm〈n|m〉=ℏmδn m

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

Sz=ℏ(s00...00s−10...000s−2...0...............000...−s)

From Equation 4.136, the quantization of S± is

S±|s m〉=ℏs(s+1)−m(m±1)|s(m±1)〉=ℏ(s∓m)(s±m+1)|s(m±1)〉

(S+)nm=〈n|S+|m〉=ℏ(s−m)(s+m+1)〈n|m+1〉=ℏbm+1δn(m+1)=ℏbnδn(m+1)

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

S+=ℏ(0bs00...000bs−10...0000bs−2...0..................0000...b−s+10000...0)

Similarly, for S−

(S−)nm=〈n|S−|m〉=ℏ(s+m)(s−m+1)〈n|m−1〉=ℏbmδn(m−1)

Therefore,

S−=ℏ(000..00bs00..000bs−10..00.................000..b−s+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=ℏ2(0bs00...00bs0bs−10...000bs−10bs−2...0000bs−20...00.....................0000...0b−s+10000...b−s+10)

And,

Sy=ℏ2i(0bs00...00−bs0bs−10...000−bs−10bs−2...0000−bs−20...00.....................0000...0b−s+10000...−b−s+10)

Conclusion:

Thus, the spin matrices (Sx,Sy,Sz) for arbitrary spin s are Sx=ℏ2(0bs00...00bs0bs−10...000bs−10bs−2...0000bs−20...00.....................0000...0b−s+10000...b−s+10), Sy=ℏ2i(0bs00...00−bs0bs−10...000−bs−10bs−2...0000−bs−20...00.....................0000...0b−s+10000...−b−s+10), and Sz=ℏ(s00...00s−10...000s−2...0...............000...−s).

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Answer 3

Question

Chapter 4, Problem 4.62P

To determine

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

Expert Solution & Answer

Answer to Problem 4.62P

The spin matrices (Sx,Sy,Sz) for arbitrary spin s are Sx=ℏ2(0bs00...00bs0bs−10...000bs−10bs−2...0000bs−20...00.....................0000...0b−s+10000...b−s+10), Sy=ℏ2i(0bs00...00−bs0bs−10...000−bs−10bs−2...0000−bs−20...00.....................0000...0b−s+10000...−b−s+10), and Sz=ℏ(s00...00s−10...000s−2...0...............000...−s).

Explanation of Solution

From Equation 4.135, the quantization of Sz is

Sz|s m〉=ℏm|s m〉

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〉=ℏm〈n|m〉=ℏmδn m

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

Sz=ℏ(s00...00s−10...000s−2...0...............000...−s)

From Equation 4.136, the quantization of S± is

S±|s m〉=ℏs(s+1)−m(m±1)|s(m±1)〉=ℏ(s∓m)(s±m+1)|s(m±1)〉

(S+)nm=〈n|S+|m〉=ℏ(s−m)(s+m+1)〈n|m+1〉=ℏbm+1δn(m+1)=ℏbnδn(m+1)

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

S+=ℏ(0bs00...000bs−10...0000bs−2...0..................0000...b−s+10000...0)

Similarly, for S−

(S−)nm=〈n|S−|m〉=ℏ(s+m)(s−m+1)〈n|m−1〉=ℏbmδn(m−1)

Therefore,

S−=ℏ(000..00bs00..000bs−10..00.................000..b−s+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=ℏ2(0bs00...00bs0bs−10...000bs−10bs−2...0000bs−20...00.....................0000...0b−s+10000...b−s+10)

And,

Sy=ℏ2i(0bs00...00−bs0bs−10...000−bs−10bs−2...0000−bs−20...00.....................0000...0b−s+10000...−b−s+10)

Conclusion:

Thus, the spin matrices (Sx,Sy,Sz) for arbitrary spin s are Sx=ℏ2(0bs00...00bs0bs−10...000bs−10bs−2...0000bs−20...00.....................0000...0b−s+10000...b−s+10), Sy=ℏ2i(0bs00...00−bs0bs−10...000−bs−10bs−2...0000−bs−20...00.....................0000...0b−s+10000...−b−s+10), and Sz=ℏ(s00...00s−10...000s−2...0...............000...−s).

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Answer 4

Question

Chapter 4, Problem 4.62P

To determine

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

Expert Solution & Answer

Answer to Problem 4.62P

The spin matrices (Sx,Sy,Sz) for arbitrary spin s are Sx=ℏ2(0bs00...00bs0bs−10...000bs−10bs−2...0000bs−20...00.....................0000...0b−s+10000...b−s+10), Sy=ℏ2i(0bs00...00−bs0bs−10...000−bs−10bs−2...0000−bs−20...00.....................0000...0b−s+10000...−b−s+10), and Sz=ℏ(s00...00s−10...000s−2...0...............000...−s).

Explanation of Solution

From Equation 4.135, the quantization of Sz is

Sz|s m〉=ℏm|s m〉

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〉=ℏm〈n|m〉=ℏmδn m

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

Sz=ℏ(s00...00s−10...000s−2...0...............000...−s)

From Equation 4.136, the quantization of S± is

S±|s m〉=ℏs(s+1)−m(m±1)|s(m±1)〉=ℏ(s∓m)(s±m+1)|s(m±1)〉

(S+)nm=〈n|S+|m〉=ℏ(s−m)(s+m+1)〈n|m+1〉=ℏbm+1δn(m+1)=ℏbnδn(m+1)

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

S+=ℏ(0bs00...000bs−10...0000bs−2...0..................0000...b−s+10000...0)

Similarly, for S−

(S−)nm=〈n|S−|m〉=ℏ(s+m)(s−m+1)〈n|m−1〉=ℏbmδn(m−1)

Therefore,

S−=ℏ(000..00bs00..000bs−10..00.................000..b−s+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=ℏ2(0bs00...00bs0bs−10...000bs−10bs−2...0000bs−20...00.....................0000...0b−s+10000...b−s+10)

And,

Sy=ℏ2i(0bs00...00−bs0bs−10...000−bs−10bs−2...0000−bs−20...00.....................0000...0b−s+10000...−b−s+10)

Conclusion:

Thus, the spin matrices (Sx,Sy,Sz) for arbitrary spin s are Sx=ℏ2(0bs00...00bs0bs−10...000bs−10bs−2...0000bs−20...00.....................0000...0b−s+10000...b−s+10), Sy=ℏ2i(0bs00...00−bs0bs−10...000−bs−10bs−2...0000−bs−20...00.....................0000...0b−s+10000...−b−s+10), and Sz=ℏ(s00...00s−10...000s−2...0...............000...−s).

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Answer 5

Chapter 4, Problem 4.62P

To determine

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

Expert Solution & Answer

Answer to Problem 4.62P

The spin matrices (Sx,Sy,Sz) for arbitrary spin s are Sx=ℏ2(0bs00...00bs0bs−10...000bs−10bs−2...0000bs−20...00.....................0000...0b−s+10000...b−s+10), Sy=ℏ2i(0bs00...00−bs0bs−10...000−bs−10bs−2...0000−bs−20...00.....................0000...0b−s+10000...−b−s+10), and Sz=ℏ(s00...00s−10...000s−2...0...............000...−s).

Explanation of Solution

From Equation 4.135, the quantization of Sz is

Sz|s m〉=ℏm|s m〉

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〉=ℏm〈n|m〉=ℏmδn m

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

Sz=ℏ(s00...00s−10...000s−2...0...............000...−s)

From Equation 4.136, the quantization of S± is

S±|s m〉=ℏs(s+1)−m(m±1)|s(m±1)〉=ℏ(s∓m)(s±m+1)|s(m±1)〉

(S+)nm=〈n|S+|m〉=ℏ(s−m)(s+m+1)〈n|m+1〉=ℏbm+1δn(m+1)=ℏbnδn(m+1)

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

S+=ℏ(0bs00...000bs−10...0000bs−2...0..................0000...b−s+10000...0)

Similarly, for S−

(S−)nm=〈n|S−|m〉=ℏ(s+m)(s−m+1)〈n|m−1〉=ℏbmδn(m−1)

Therefore,

S−=ℏ(000..00bs00..000bs−10..00.................000..b−s+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=ℏ2(0bs00...00bs0bs−10...000bs−10bs−2...0000bs−20...00.....................0000...0b−s+10000...b−s+10)

And,

Sy=ℏ2i(0bs00...00−bs0bs−10...000−bs−10bs−2...0000−bs−20...00.....................0000...0b−s+10000...−b−s+10)

Conclusion:

Thus, the spin matrices (Sx,Sy,Sz) for arbitrary spin s are Sx=ℏ2(0bs00...00bs0bs−10...000bs−10bs−2...0000bs−20...00.....................0000...0b−s+10000...b−s+10), Sy=ℏ2i(0bs00...00−bs0bs−10...000−bs−10bs−2...0000−bs−20...00.....................0000...0b−s+10000...−b−s+10), and Sz=ℏ(s00...00s−10...000s−2...0...............000...−s).

Answer 6

Chapter 4, Problem 4.62P

To determine

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

Expert Solution & Answer

Answer to Problem 4.62P

The spin matrices (Sx,Sy,Sz) for arbitrary spin s are Sx=ℏ2(0bs00...00bs0bs−10...000bs−10bs−2...0000bs−20...00.....................0000...0b−s+10000...b−s+10), Sy=ℏ2i(0bs00...00−bs0bs−10...000−bs−10bs−2...0000−bs−20...00.....................0000...0b−s+10000...−b−s+10), and Sz=ℏ(s00...00s−10...000s−2...0...............000...−s).

Explanation of Solution

From Equation 4.135, the quantization of Sz is

Sz|s m〉=ℏm|s m〉

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〉=ℏm〈n|m〉=ℏmδn m

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

Sz=ℏ(s00...00s−10...000s−2...0...............000...−s)

From Equation 4.136, the quantization of S± is

S±|s m〉=ℏs(s+1)−m(m±1)|s(m±1)〉=ℏ(s∓m)(s±m+1)|s(m±1)〉

(S+)nm=〈n|S+|m〉=ℏ(s−m)(s+m+1)〈n|m+1〉=ℏbm+1δn(m+1)=ℏbnδn(m+1)

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

S+=ℏ(0bs00...000bs−10...0000bs−2...0..................0000...b−s+10000...0)

Similarly, for S−

(S−)nm=〈n|S−|m〉=ℏ(s+m)(s−m+1)〈n|m−1〉=ℏbmδn(m−1)

Therefore,

S−=ℏ(000..00bs00..000bs−10..00.................000..b−s+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=ℏ2(0bs00...00bs0bs−10...000bs−10bs−2...0000bs−20...00.....................0000...0b−s+10000...b−s+10)

And,

Sy=ℏ2i(0bs00...00−bs0bs−10...000−bs−10bs−2...0000−bs−20...00.....................0000...0b−s+10000...−b−s+10)

Conclusion:

Thus, the spin matrices (Sx,Sy,Sz) for arbitrary spin s are Sx=ℏ2(0bs00...00bs0bs−10...000bs−10bs−2...0000bs−20...00.....................0000...0b−s+10000...b−s+10), Sy=ℏ2i(0bs00...00−bs0bs−10...000−bs−10bs−2...0000−bs−20...00.....................0000...0b−s+10000...−b−s+10), and Sz=ℏ(s00...00s−10...000s−2...0...............000...−s).

Answer 7

Chapter 4, Problem 4.62P

To determine

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

Answer 8

Expert Solution & Answer

Answer to Problem 4.62P

The spin matrices (Sx,Sy,Sz) for arbitrary spin s are Sx=ℏ2(0bs00...00bs0bs−10...000bs−10bs−2...0000bs−20...00.....................0000...0b−s+10000...b−s+10), Sy=ℏ2i(0bs00...00−bs0bs−10...000−bs−10bs−2...0000−bs−20...00.....................0000...0b−s+10000...−b−s+10), and Sz=ℏ(s00...00s−10...000s−2...0...............000...−s).

Answer 9

Expert Solution & Answer

Answer 10

Expert Solution & Answer

Answer 11

Explanation of Solution

From Equation 4.135, the quantization of Sz is

Sz|s m〉=ℏm|s m〉

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〉=ℏm〈n|m〉=ℏmδn m

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

Sz=ℏ(s00...00s−10...000s−2...0...............000...−s)

From Equation 4.136, the quantization of S± is

S±|s m〉=ℏs(s+1)−m(m±1)|s(m±1)〉=ℏ(s∓m)(s±m+1)|s(m±1)〉

(S+)nm=〈n|S+|m〉=ℏ(s−m)(s+m+1)〈n|m+1〉=ℏbm+1δn(m+1)=ℏbnδn(m+1)

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

S+=ℏ(0bs00...000bs−10...0000bs−2...0..................0000...b−s+10000...0)

Similarly, for S−

(S−)nm=〈n|S−|m〉=ℏ(s+m)(s−m+1)〈n|m−1〉=ℏbmδn(m−1)

Therefore,

S−=ℏ(000..00bs00..000bs−10..00.................000..b−s+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=ℏ2(0bs00...00bs0bs−10...000bs−10bs−2...0000bs−20...00.....................0000...0b−s+10000...b−s+10)

And,

Sy=ℏ2i(0bs00...00−bs0bs−10...000−bs−10bs−2...0000−bs−20...00.....................0000...0b−s+10000...−b−s+10)

Conclusion:

Thus, the spin matrices (Sx,Sy,Sz) for arbitrary spin s are Sx=ℏ2(0bs00...00bs0bs−10...000bs−10bs−2...0000bs−20...00.....................0000...0b−s+10000...b−s+10), Sy=ℏ2i(0bs00...00−bs0bs−10...000−bs−10bs−2...0000−bs−20...00.....................0000...0b−s+10000...−b−s+10), and Sz=ℏ(s00...00s−10...000s−2...0...............000...−s).

Answer 12

Ch. 4.1 - Prob. 4.1P Ch. 4.1 - Prob. 4.3P Ch. 4.1 - Prob. 4.4P Ch. 4.1 - Prob. 4.5P Ch. 4.1 - Prob. 4.6P Ch. 4.1 - Prob. 4.7P Ch. 4.1 - Prob. 4.8P Ch. 4.1 - Prob. 4.9P Ch. 4.1 - Prob. 4.10P Ch. 4.1 - Prob. 4.11P

The spin matrices ( S x , S y , S z ) for arbitrary spin s .

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

Answer to Problem 4.62P

Explanation of Solution

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