The value of l ^ x l ^ y f has to be predicted. Concept introduction: The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics , commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below. [ l ^ x , l ^ y ] f = l ^ x l ^ y f − l ^ y l ^ x f

Question

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Answer 1

Question

Chapter 7, Problem 7F.9P

(a)

Interpretation Introduction

Interpretation:

The value of l^xl^yf has to be predicted.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(a)

Expert Solution

Answer to Problem 7F.9P

The value of l^xl^yf is shown below.

l^xl^yf=−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z)

Explanation of Solution

The value of l^xl^yf is shown below.

l^xl^yf=−ℏ2(y∂∂z−z∂∂y)(z∂f∂x−x∂f∂z)

Multiply the parentheses as shown below.

l^xl^yf=−ℏ2(y∂∂z(z∂f∂x−x∂f∂z)−z∂∂y(z∂f∂x−x∂f∂z))=−ℏ2((y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2)−(z2∂2f∂y∂x−zx∂2f∂y∂z))=−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z)

Hence, the value of l^xl^yf is shown below.

l^xl^yf=−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z)

(b)

Interpretation Introduction

Interpretation:

The value of l^yl^xf has to be predicted.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(b)

Expert Solution

Answer to Problem 7F.9P

The value of l^xl^yf is shown below.

l^yl^xf=−ℏ2(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y)

Explanation of Solution

The value of l^yl^xf is shown below.

l^yl^xf=−ℏ2(z∂∂x−x∂∂z)(y∂f∂z−z∂f∂y)

Multiply the parentheses as shown below.

l^yl^xf=−ℏ2(z∂∂x−x∂∂z)(y∂f∂z−z∂f∂y)=−ℏ2(z∂∂x(y∂f∂z−z∂f∂y)−x∂∂z(y∂f∂z−z∂f∂y))=−ℏ2((zy∂2f∂x∂z−z2∂2f∂x∂y)−(xy∂2f∂z2−zx∂2f∂z∂y−x∂f∂y))=−ℏ2(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y)

Hence, the value of l^xl^yf is shown below.

l^yl^xf=−ℏ2(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y)

(c)

Interpretation Introduction

Interpretation:

The value of l^xl^yf−l^yl^xf is ℏil^zf has to be proved.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(c)

Expert Solution

Answer to Problem 7F.9P

It has been proved that the value of l^xl^yf−l^yl^xf is ℏil^zf.

Explanation of Solution

The value of l^xl^yf−l^yl^xf is calculated by substituting the value of l^xl^yf and l^xl^yf as shown below.

l^xl^yf−l^yl^xf=((−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z))−(−ℏ2(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y)))=−ℏ2((y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z)−(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y))=−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z−zy∂2f∂x∂z+z2∂2f∂x∂y+xy∂2f∂z2−zx∂2f∂z∂y−x∂f∂y)=−ℏ2(y∂f∂x−x∂f∂y)

Further solve the above equation, by multiplying and dividing by i as shown below.

l^xl^yf−l^yl^xf=−ℏi(ℏi(y∂f∂x−x∂f∂y))=ℏi(ℏi(x∂∂y−y∂∂x))f (∵ℏi(x∂∂y−y∂∂x)=l^z)=ℏil^zf

Hence, it has been proved that the value of l^xl^yf−l^yl^xf is ℏil^zf.

(d)

Interpretation Introduction

Interpretation:

The cyclic permutation order for linear momentum operators has to be proved.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(d)

Expert Solution

Answer to Problem 7F.9P

It has been proved that linear momentum operators follow the cyclic permutation order.

Explanation of Solution

The x component of linear momentum operator is shown below.

l^xf=ℏi(y∂∂z−z∂∂y) (1)

The y component of linear momentum operator is shown below.

l^yf=ℏi(z∂∂x−x∂∂z) (2)

The z component of linear momentum operator is shown below.

l^zf=ℏi(x∂∂y−y∂∂x) (3)

From equation (1), it is clear that the x component of linear momentum operator contains a variable function of angular momentum along y and z direction. This makes the permutation x→y→z. Similarly, from equation (2), the y component of linear momentum operator contains a variable function of angular momentum along x and z direction. This makes the permutation y→z→x. From equation (3), the z component of linear momentum operator contains a variable function of angular momentum along y and x direction. This makes the permutation z→x→y. Therefore, the angular momentum operator along x, y and z direction follows a cyclic permutation.

(e)

Interpretation Introduction

Interpretation:

The commutations of other two angular momentum operators, by following the cyclic order permutation have to be predicted.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(e)

Expert Solution

Answer to Problem 7F.9P

The value of the commutation of l^y and l^z is shown below.

[l^y, l^z]=ℏil^x

The value of the commutation of l^z and l^x is shown below.

[l^z, l^x]=ℏil^y

Explanation of Solution

The value of the commutation of l^x and l^y is shown below.

[l^x, l^y]=ℏil^z

By following the cyclic order permutation, the value of the commutation of l^y and l^z is shown below.

[l^y, l^z]=ℏil^x

Similarly, the value of the commutation of l^z and l^x is shown below.

[l^z, l^x]=ℏil^y

This proves that the commutations of other two angular momentum operators follow cyclic permutation order.

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Answer 2

Question

Chapter 7, Problem 7F.9P

(a)

Interpretation Introduction

Interpretation:

The value of l^xl^yf has to be predicted.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(a)

Expert Solution

Answer to Problem 7F.9P

The value of l^xl^yf is shown below.

l^xl^yf=−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z)

Explanation of Solution

The value of l^xl^yf is shown below.

l^xl^yf=−ℏ2(y∂∂z−z∂∂y)(z∂f∂x−x∂f∂z)

Multiply the parentheses as shown below.

l^xl^yf=−ℏ2(y∂∂z(z∂f∂x−x∂f∂z)−z∂∂y(z∂f∂x−x∂f∂z))=−ℏ2((y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2)−(z2∂2f∂y∂x−zx∂2f∂y∂z))=−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z)

Hence, the value of l^xl^yf is shown below.

l^xl^yf=−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z)

(b)

Interpretation Introduction

Interpretation:

The value of l^yl^xf has to be predicted.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(b)

Expert Solution

Answer to Problem 7F.9P

The value of l^xl^yf is shown below.

l^yl^xf=−ℏ2(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y)

Explanation of Solution

The value of l^yl^xf is shown below.

l^yl^xf=−ℏ2(z∂∂x−x∂∂z)(y∂f∂z−z∂f∂y)

Multiply the parentheses as shown below.

l^yl^xf=−ℏ2(z∂∂x−x∂∂z)(y∂f∂z−z∂f∂y)=−ℏ2(z∂∂x(y∂f∂z−z∂f∂y)−x∂∂z(y∂f∂z−z∂f∂y))=−ℏ2((zy∂2f∂x∂z−z2∂2f∂x∂y)−(xy∂2f∂z2−zx∂2f∂z∂y−x∂f∂y))=−ℏ2(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y)

Hence, the value of l^xl^yf is shown below.

l^yl^xf=−ℏ2(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y)

(c)

Interpretation Introduction

Interpretation:

The value of l^xl^yf−l^yl^xf is ℏil^zf has to be proved.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(c)

Expert Solution

Answer to Problem 7F.9P

It has been proved that the value of l^xl^yf−l^yl^xf is ℏil^zf.

Explanation of Solution

The value of l^xl^yf−l^yl^xf is calculated by substituting the value of l^xl^yf and l^xl^yf as shown below.

l^xl^yf−l^yl^xf=((−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z))−(−ℏ2(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y)))=−ℏ2((y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z)−(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y))=−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z−zy∂2f∂x∂z+z2∂2f∂x∂y+xy∂2f∂z2−zx∂2f∂z∂y−x∂f∂y)=−ℏ2(y∂f∂x−x∂f∂y)

Further solve the above equation, by multiplying and dividing by i as shown below.

l^xl^yf−l^yl^xf=−ℏi(ℏi(y∂f∂x−x∂f∂y))=ℏi(ℏi(x∂∂y−y∂∂x))f (∵ℏi(x∂∂y−y∂∂x)=l^z)=ℏil^zf

Hence, it has been proved that the value of l^xl^yf−l^yl^xf is ℏil^zf.

(d)

Interpretation Introduction

Interpretation:

The cyclic permutation order for linear momentum operators has to be proved.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(d)

Expert Solution

Answer to Problem 7F.9P

It has been proved that linear momentum operators follow the cyclic permutation order.

Explanation of Solution

The x component of linear momentum operator is shown below.

l^xf=ℏi(y∂∂z−z∂∂y) (1)

The y component of linear momentum operator is shown below.

l^yf=ℏi(z∂∂x−x∂∂z) (2)

The z component of linear momentum operator is shown below.

l^zf=ℏi(x∂∂y−y∂∂x) (3)

From equation (1), it is clear that the x component of linear momentum operator contains a variable function of angular momentum along y and z direction. This makes the permutation x→y→z. Similarly, from equation (2), the y component of linear momentum operator contains a variable function of angular momentum along x and z direction. This makes the permutation y→z→x. From equation (3), the z component of linear momentum operator contains a variable function of angular momentum along y and x direction. This makes the permutation z→x→y. Therefore, the angular momentum operator along x, y and z direction follows a cyclic permutation.

(e)

Interpretation Introduction

Interpretation:

The commutations of other two angular momentum operators, by following the cyclic order permutation have to be predicted.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(e)

Expert Solution

Answer to Problem 7F.9P

The value of the commutation of l^y and l^z is shown below.

[l^y, l^z]=ℏil^x

The value of the commutation of l^z and l^x is shown below.

[l^z, l^x]=ℏil^y

Explanation of Solution

The value of the commutation of l^x and l^y is shown below.

[l^x, l^y]=ℏil^z

By following the cyclic order permutation, the value of the commutation of l^y and l^z is shown below.

[l^y, l^z]=ℏil^x

Similarly, the value of the commutation of l^z and l^x is shown below.

[l^z, l^x]=ℏil^y

This proves that the commutations of other two angular momentum operators follow cyclic permutation order.

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Answer 3

Question

Chapter 7, Problem 7F.9P

(a)

Interpretation Introduction

Interpretation:

The value of l^xl^yf has to be predicted.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(a)

Expert Solution

Answer to Problem 7F.9P

The value of l^xl^yf is shown below.

l^xl^yf=−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z)

Explanation of Solution

The value of l^xl^yf is shown below.

l^xl^yf=−ℏ2(y∂∂z−z∂∂y)(z∂f∂x−x∂f∂z)

Multiply the parentheses as shown below.

l^xl^yf=−ℏ2(y∂∂z(z∂f∂x−x∂f∂z)−z∂∂y(z∂f∂x−x∂f∂z))=−ℏ2((y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2)−(z2∂2f∂y∂x−zx∂2f∂y∂z))=−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z)

Hence, the value of l^xl^yf is shown below.

l^xl^yf=−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z)

(b)

Interpretation Introduction

Interpretation:

The value of l^yl^xf has to be predicted.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(b)

Expert Solution

Answer to Problem 7F.9P

The value of l^xl^yf is shown below.

l^yl^xf=−ℏ2(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y)

Explanation of Solution

The value of l^yl^xf is shown below.

l^yl^xf=−ℏ2(z∂∂x−x∂∂z)(y∂f∂z−z∂f∂y)

Multiply the parentheses as shown below.

l^yl^xf=−ℏ2(z∂∂x−x∂∂z)(y∂f∂z−z∂f∂y)=−ℏ2(z∂∂x(y∂f∂z−z∂f∂y)−x∂∂z(y∂f∂z−z∂f∂y))=−ℏ2((zy∂2f∂x∂z−z2∂2f∂x∂y)−(xy∂2f∂z2−zx∂2f∂z∂y−x∂f∂y))=−ℏ2(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y)

Hence, the value of l^xl^yf is shown below.

l^yl^xf=−ℏ2(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y)

(c)

Interpretation Introduction

Interpretation:

The value of l^xl^yf−l^yl^xf is ℏil^zf has to be proved.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(c)

Expert Solution

Answer to Problem 7F.9P

It has been proved that the value of l^xl^yf−l^yl^xf is ℏil^zf.

Explanation of Solution

The value of l^xl^yf−l^yl^xf is calculated by substituting the value of l^xl^yf and l^xl^yf as shown below.

l^xl^yf−l^yl^xf=((−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z))−(−ℏ2(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y)))=−ℏ2((y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z)−(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y))=−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z−zy∂2f∂x∂z+z2∂2f∂x∂y+xy∂2f∂z2−zx∂2f∂z∂y−x∂f∂y)=−ℏ2(y∂f∂x−x∂f∂y)

Further solve the above equation, by multiplying and dividing by i as shown below.

l^xl^yf−l^yl^xf=−ℏi(ℏi(y∂f∂x−x∂f∂y))=ℏi(ℏi(x∂∂y−y∂∂x))f (∵ℏi(x∂∂y−y∂∂x)=l^z)=ℏil^zf

Hence, it has been proved that the value of l^xl^yf−l^yl^xf is ℏil^zf.

(d)

Interpretation Introduction

Interpretation:

The cyclic permutation order for linear momentum operators has to be proved.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(d)

Expert Solution

Answer to Problem 7F.9P

It has been proved that linear momentum operators follow the cyclic permutation order.

Explanation of Solution

The x component of linear momentum operator is shown below.

l^xf=ℏi(y∂∂z−z∂∂y) (1)

The y component of linear momentum operator is shown below.

l^yf=ℏi(z∂∂x−x∂∂z) (2)

The z component of linear momentum operator is shown below.

l^zf=ℏi(x∂∂y−y∂∂x) (3)

From equation (1), it is clear that the x component of linear momentum operator contains a variable function of angular momentum along y and z direction. This makes the permutation x→y→z. Similarly, from equation (2), the y component of linear momentum operator contains a variable function of angular momentum along x and z direction. This makes the permutation y→z→x. From equation (3), the z component of linear momentum operator contains a variable function of angular momentum along y and x direction. This makes the permutation z→x→y. Therefore, the angular momentum operator along x, y and z direction follows a cyclic permutation.

(e)

Interpretation Introduction

Interpretation:

The commutations of other two angular momentum operators, by following the cyclic order permutation have to be predicted.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(e)

Expert Solution

Answer to Problem 7F.9P

The value of the commutation of l^y and l^z is shown below.

[l^y, l^z]=ℏil^x

The value of the commutation of l^z and l^x is shown below.

[l^z, l^x]=ℏil^y

Explanation of Solution

The value of the commutation of l^x and l^y is shown below.

[l^x, l^y]=ℏil^z

By following the cyclic order permutation, the value of the commutation of l^y and l^z is shown below.

[l^y, l^z]=ℏil^x

Similarly, the value of the commutation of l^z and l^x is shown below.

[l^z, l^x]=ℏil^y

This proves that the commutations of other two angular momentum operators follow cyclic permutation order.

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Answer 4

Question

Chapter 7, Problem 7F.9P

(a)

Interpretation Introduction

Interpretation:

The value of l^xl^yf has to be predicted.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(a)

Expert Solution

Answer to Problem 7F.9P

The value of l^xl^yf is shown below.

l^xl^yf=−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z)

Explanation of Solution

The value of l^xl^yf is shown below.

l^xl^yf=−ℏ2(y∂∂z−z∂∂y)(z∂f∂x−x∂f∂z)

Multiply the parentheses as shown below.

l^xl^yf=−ℏ2(y∂∂z(z∂f∂x−x∂f∂z)−z∂∂y(z∂f∂x−x∂f∂z))=−ℏ2((y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2)−(z2∂2f∂y∂x−zx∂2f∂y∂z))=−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z)

Hence, the value of l^xl^yf is shown below.

l^xl^yf=−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z)

(b)

Interpretation Introduction

Interpretation:

The value of l^yl^xf has to be predicted.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(b)

Expert Solution

Answer to Problem 7F.9P

The value of l^xl^yf is shown below.

l^yl^xf=−ℏ2(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y)

Explanation of Solution

The value of l^yl^xf is shown below.

l^yl^xf=−ℏ2(z∂∂x−x∂∂z)(y∂f∂z−z∂f∂y)

Multiply the parentheses as shown below.

l^yl^xf=−ℏ2(z∂∂x−x∂∂z)(y∂f∂z−z∂f∂y)=−ℏ2(z∂∂x(y∂f∂z−z∂f∂y)−x∂∂z(y∂f∂z−z∂f∂y))=−ℏ2((zy∂2f∂x∂z−z2∂2f∂x∂y)−(xy∂2f∂z2−zx∂2f∂z∂y−x∂f∂y))=−ℏ2(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y)

Hence, the value of l^xl^yf is shown below.

l^yl^xf=−ℏ2(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y)

(c)

Interpretation Introduction

Interpretation:

The value of l^xl^yf−l^yl^xf is ℏil^zf has to be proved.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(c)

Expert Solution

Answer to Problem 7F.9P

It has been proved that the value of l^xl^yf−l^yl^xf is ℏil^zf.

Explanation of Solution

The value of l^xl^yf−l^yl^xf is calculated by substituting the value of l^xl^yf and l^xl^yf as shown below.

l^xl^yf−l^yl^xf=((−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z))−(−ℏ2(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y)))=−ℏ2((y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z)−(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y))=−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z−zy∂2f∂x∂z+z2∂2f∂x∂y+xy∂2f∂z2−zx∂2f∂z∂y−x∂f∂y)=−ℏ2(y∂f∂x−x∂f∂y)

Further solve the above equation, by multiplying and dividing by i as shown below.

l^xl^yf−l^yl^xf=−ℏi(ℏi(y∂f∂x−x∂f∂y))=ℏi(ℏi(x∂∂y−y∂∂x))f (∵ℏi(x∂∂y−y∂∂x)=l^z)=ℏil^zf

Hence, it has been proved that the value of l^xl^yf−l^yl^xf is ℏil^zf.

(d)

Interpretation Introduction

Interpretation:

The cyclic permutation order for linear momentum operators has to be proved.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(d)

Expert Solution

Answer to Problem 7F.9P

It has been proved that linear momentum operators follow the cyclic permutation order.

Explanation of Solution

The x component of linear momentum operator is shown below.

l^xf=ℏi(y∂∂z−z∂∂y) (1)

The y component of linear momentum operator is shown below.

l^yf=ℏi(z∂∂x−x∂∂z) (2)

The z component of linear momentum operator is shown below.

l^zf=ℏi(x∂∂y−y∂∂x) (3)

From equation (1), it is clear that the x component of linear momentum operator contains a variable function of angular momentum along y and z direction. This makes the permutation x→y→z. Similarly, from equation (2), the y component of linear momentum operator contains a variable function of angular momentum along x and z direction. This makes the permutation y→z→x. From equation (3), the z component of linear momentum operator contains a variable function of angular momentum along y and x direction. This makes the permutation z→x→y. Therefore, the angular momentum operator along x, y and z direction follows a cyclic permutation.

(e)

Interpretation Introduction

Interpretation:

The commutations of other two angular momentum operators, by following the cyclic order permutation have to be predicted.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(e)

Expert Solution

Answer to Problem 7F.9P

The value of the commutation of l^y and l^z is shown below.

[l^y, l^z]=ℏil^x

The value of the commutation of l^z and l^x is shown below.

[l^z, l^x]=ℏil^y

Explanation of Solution

The value of the commutation of l^x and l^y is shown below.

[l^x, l^y]=ℏil^z

By following the cyclic order permutation, the value of the commutation of l^y and l^z is shown below.

[l^y, l^z]=ℏil^x

Similarly, the value of the commutation of l^z and l^x is shown below.

[l^z, l^x]=ℏil^y

This proves that the commutations of other two angular momentum operators follow cyclic permutation order.

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Answer 5

Chapter 7, Problem 7F.9P

(a)

Interpretation Introduction

Interpretation:

The value of l^xl^yf has to be predicted.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(a)

Expert Solution

Answer to Problem 7F.9P

The value of l^xl^yf is shown below.

l^xl^yf=−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z)

Explanation of Solution

The value of l^xl^yf is shown below.

l^xl^yf=−ℏ2(y∂∂z−z∂∂y)(z∂f∂x−x∂f∂z)

Multiply the parentheses as shown below.

l^xl^yf=−ℏ2(y∂∂z(z∂f∂x−x∂f∂z)−z∂∂y(z∂f∂x−x∂f∂z))=−ℏ2((y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2)−(z2∂2f∂y∂x−zx∂2f∂y∂z))=−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z)

Hence, the value of l^xl^yf is shown below.

l^xl^yf=−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z)

(b)

Interpretation Introduction

Interpretation:

The value of l^yl^xf has to be predicted.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(b)

Expert Solution

Answer to Problem 7F.9P

The value of l^xl^yf is shown below.

l^yl^xf=−ℏ2(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y)

Explanation of Solution

The value of l^yl^xf is shown below.

l^yl^xf=−ℏ2(z∂∂x−x∂∂z)(y∂f∂z−z∂f∂y)

Multiply the parentheses as shown below.

l^yl^xf=−ℏ2(z∂∂x−x∂∂z)(y∂f∂z−z∂f∂y)=−ℏ2(z∂∂x(y∂f∂z−z∂f∂y)−x∂∂z(y∂f∂z−z∂f∂y))=−ℏ2((zy∂2f∂x∂z−z2∂2f∂x∂y)−(xy∂2f∂z2−zx∂2f∂z∂y−x∂f∂y))=−ℏ2(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y)

Hence, the value of l^xl^yf is shown below.

l^yl^xf=−ℏ2(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y)

(c)

Interpretation Introduction

Interpretation:

The value of l^xl^yf−l^yl^xf is ℏil^zf has to be proved.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(c)

Expert Solution

Answer to Problem 7F.9P

It has been proved that the value of l^xl^yf−l^yl^xf is ℏil^zf.

Explanation of Solution

The value of l^xl^yf−l^yl^xf is calculated by substituting the value of l^xl^yf and l^xl^yf as shown below.

l^xl^yf−l^yl^xf=((−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z))−(−ℏ2(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y)))=−ℏ2((y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z)−(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y))=−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z−zy∂2f∂x∂z+z2∂2f∂x∂y+xy∂2f∂z2−zx∂2f∂z∂y−x∂f∂y)=−ℏ2(y∂f∂x−x∂f∂y)

Further solve the above equation, by multiplying and dividing by i as shown below.

l^xl^yf−l^yl^xf=−ℏi(ℏi(y∂f∂x−x∂f∂y))=ℏi(ℏi(x∂∂y−y∂∂x))f (∵ℏi(x∂∂y−y∂∂x)=l^z)=ℏil^zf

Hence, it has been proved that the value of l^xl^yf−l^yl^xf is ℏil^zf.

(d)

Interpretation Introduction

Interpretation:

The cyclic permutation order for linear momentum operators has to be proved.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(d)

Expert Solution

Answer to Problem 7F.9P

It has been proved that linear momentum operators follow the cyclic permutation order.

Explanation of Solution

The x component of linear momentum operator is shown below.

l^xf=ℏi(y∂∂z−z∂∂y) (1)

The y component of linear momentum operator is shown below.

l^yf=ℏi(z∂∂x−x∂∂z) (2)

The z component of linear momentum operator is shown below.

l^zf=ℏi(x∂∂y−y∂∂x) (3)

From equation (1), it is clear that the x component of linear momentum operator contains a variable function of angular momentum along y and z direction. This makes the permutation x→y→z. Similarly, from equation (2), the y component of linear momentum operator contains a variable function of angular momentum along x and z direction. This makes the permutation y→z→x. From equation (3), the z component of linear momentum operator contains a variable function of angular momentum along y and x direction. This makes the permutation z→x→y. Therefore, the angular momentum operator along x, y and z direction follows a cyclic permutation.

(e)

Interpretation Introduction

Interpretation:

The commutations of other two angular momentum operators, by following the cyclic order permutation have to be predicted.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(e)

Expert Solution

Answer to Problem 7F.9P

The value of the commutation of l^y and l^z is shown below.

[l^y, l^z]=ℏil^x

The value of the commutation of l^z and l^x is shown below.

[l^z, l^x]=ℏil^y

Explanation of Solution

The value of the commutation of l^x and l^y is shown below.

[l^x, l^y]=ℏil^z

By following the cyclic order permutation, the value of the commutation of l^y and l^z is shown below.

[l^y, l^z]=ℏil^x

Similarly, the value of the commutation of l^z and l^x is shown below.

[l^z, l^x]=ℏil^y

This proves that the commutations of other two angular momentum operators follow cyclic permutation order.

Answer 6

Chapter 7, Problem 7F.9P

(a)

Interpretation Introduction

Interpretation:

The value of l^xl^yf has to be predicted.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(a)

Expert Solution

Answer to Problem 7F.9P

The value of l^xl^yf is shown below.

l^xl^yf=−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z)

Explanation of Solution

The value of l^xl^yf is shown below.

l^xl^yf=−ℏ2(y∂∂z−z∂∂y)(z∂f∂x−x∂f∂z)

Multiply the parentheses as shown below.

l^xl^yf=−ℏ2(y∂∂z(z∂f∂x−x∂f∂z)−z∂∂y(z∂f∂x−x∂f∂z))=−ℏ2((y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2)−(z2∂2f∂y∂x−zx∂2f∂y∂z))=−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z)

Hence, the value of l^xl^yf is shown below.

l^xl^yf=−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z)

Answer 7

Chapter 7, Problem 7F.9P

(a)

Interpretation Introduction

Interpretation:

The value of l^xl^yf has to be predicted.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

Answer 8

(a)

Expert Solution

Answer to Problem 7F.9P

The value of l^xl^yf is shown below.

l^xl^yf=−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z)

Answer 9

(a)

Expert Solution

Answer 10

(a)

Expert Solution

Answer 11

Expert Solution

Answer 12

Explanation of Solution

The value of l^xl^yf is shown below.

l^xl^yf=−ℏ2(y∂∂z−z∂∂y)(z∂f∂x−x∂f∂z)

Multiply the parentheses as shown below.

l^xl^yf=−ℏ2(y∂∂z(z∂f∂x−x∂f∂z)−z∂∂y(z∂f∂x−x∂f∂z))=−ℏ2((y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2)−(z2∂2f∂y∂x−zx∂2f∂y∂z))=−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z)

Hence, the value of l^xl^yf is shown below.

l^xl^yf=−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z)

Answer 13

(b)

Interpretation Introduction

Interpretation:

The value of l^yl^xf has to be predicted.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(b)

Expert Solution

Answer to Problem 7F.9P

The value of l^xl^yf is shown below.

l^yl^xf=−ℏ2(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y)

Explanation of Solution

The value of l^yl^xf is shown below.

l^yl^xf=−ℏ2(z∂∂x−x∂∂z)(y∂f∂z−z∂f∂y)

Multiply the parentheses as shown below.

l^yl^xf=−ℏ2(z∂∂x−x∂∂z)(y∂f∂z−z∂f∂y)=−ℏ2(z∂∂x(y∂f∂z−z∂f∂y)−x∂∂z(y∂f∂z−z∂f∂y))=−ℏ2((zy∂2f∂x∂z−z2∂2f∂x∂y)−(xy∂2f∂z2−zx∂2f∂z∂y−x∂f∂y))=−ℏ2(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y)

Hence, the value of l^xl^yf is shown below.

l^yl^xf=−ℏ2(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y)

Answer 14

(b)

Interpretation Introduction

Interpretation:

The value of l^yl^xf has to be predicted.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

Answer 15

(b)

Expert Solution

Answer to Problem 7F.9P

The value of l^xl^yf is shown below.

l^yl^xf=−ℏ2(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y)

Answer 16

(b)

Expert Solution

Answer 17

(b)

Expert Solution

Answer 18

Expert Solution

Answer 19

Explanation of Solution

The value of l^yl^xf is shown below.

l^yl^xf=−ℏ2(z∂∂x−x∂∂z)(y∂f∂z−z∂f∂y)

Multiply the parentheses as shown below.

l^yl^xf=−ℏ2(z∂∂x−x∂∂z)(y∂f∂z−z∂f∂y)=−ℏ2(z∂∂x(y∂f∂z−z∂f∂y)−x∂∂z(y∂f∂z−z∂f∂y))=−ℏ2((zy∂2f∂x∂z−z2∂2f∂x∂y)−(xy∂2f∂z2−zx∂2f∂z∂y−x∂f∂y))=−ℏ2(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y)

Hence, the value of l^xl^yf is shown below.

l^yl^xf=−ℏ2(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y)

Answer 20

(c)

Interpretation Introduction

Interpretation:

The value of l^xl^yf−l^yl^xf is ℏil^zf has to be proved.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(c)

Expert Solution

Answer to Problem 7F.9P

It has been proved that the value of l^xl^yf−l^yl^xf is ℏil^zf.

Explanation of Solution

The value of l^xl^yf−l^yl^xf is calculated by substituting the value of l^xl^yf and l^xl^yf as shown below.

l^xl^yf−l^yl^xf=((−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z))−(−ℏ2(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y)))=−ℏ2((y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z)−(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y))=−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z−zy∂2f∂x∂z+z2∂2f∂x∂y+xy∂2f∂z2−zx∂2f∂z∂y−x∂f∂y)=−ℏ2(y∂f∂x−x∂f∂y)

Further solve the above equation, by multiplying and dividing by i as shown below.

l^xl^yf−l^yl^xf=−ℏi(ℏi(y∂f∂x−x∂f∂y))=ℏi(ℏi(x∂∂y−y∂∂x))f (∵ℏi(x∂∂y−y∂∂x)=l^z)=ℏil^zf

Hence, it has been proved that the value of l^xl^yf−l^yl^xf is ℏil^zf.

Answer 21

(c)

Interpretation Introduction

Interpretation:

The value of l^xl^yf−l^yl^xf is ℏil^zf has to be proved.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

Answer 22

(c)

Expert Solution

Answer to Problem 7F.9P

It has been proved that the value of l^xl^yf−l^yl^xf is ℏil^zf.

Answer 23

(c)

Expert Solution

Answer 24

(c)

Expert Solution

Answer 25

Expert Solution

Answer 26

Explanation of Solution

The value of l^xl^yf−l^yl^xf is calculated by substituting the value of l^xl^yf and l^xl^yf as shown below.

l^xl^yf−l^yl^xf=((−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z))−(−ℏ2(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y)))=−ℏ2((y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z)−(zy∂2f∂x∂z−z2∂2f∂x∂y−xy∂2f∂z2+zx∂2f∂z∂y+x∂f∂y))=−ℏ2(y∂f∂x+zy∂2f∂z∂x−yx∂2f∂z2−z2∂2f∂y∂x+zx∂2f∂y∂z−zy∂2f∂x∂z+z2∂2f∂x∂y+xy∂2f∂z2−zx∂2f∂z∂y−x∂f∂y)=−ℏ2(y∂f∂x−x∂f∂y)

Further solve the above equation, by multiplying and dividing by i as shown below.

l^xl^yf−l^yl^xf=−ℏi(ℏi(y∂f∂x−x∂f∂y))=ℏi(ℏi(x∂∂y−y∂∂x))f (∵ℏi(x∂∂y−y∂∂x)=l^z)=ℏil^zf

Hence, it has been proved that the value of l^xl^yf−l^yl^xf is ℏil^zf.

Answer 27

(d)

Interpretation Introduction

Interpretation:

The cyclic permutation order for linear momentum operators has to be proved.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(d)

Expert Solution

Answer to Problem 7F.9P

It has been proved that linear momentum operators follow the cyclic permutation order.

Explanation of Solution

The x component of linear momentum operator is shown below.

l^xf=ℏi(y∂∂z−z∂∂y) (1)

The y component of linear momentum operator is shown below.

l^yf=ℏi(z∂∂x−x∂∂z) (2)

The z component of linear momentum operator is shown below.

l^zf=ℏi(x∂∂y−y∂∂x) (3)

From equation (1), it is clear that the x component of linear momentum operator contains a variable function of angular momentum along y and z direction. This makes the permutation x→y→z. Similarly, from equation (2), the y component of linear momentum operator contains a variable function of angular momentum along x and z direction. This makes the permutation y→z→x. From equation (3), the z component of linear momentum operator contains a variable function of angular momentum along y and x direction. This makes the permutation z→x→y. Therefore, the angular momentum operator along x, y and z direction follows a cyclic permutation.

Answer 28

(d)

Interpretation Introduction

Interpretation:

The cyclic permutation order for linear momentum operators has to be proved.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

Answer 29

(d)

Expert Solution

Answer to Problem 7F.9P

It has been proved that linear momentum operators follow the cyclic permutation order.

Answer 30

(d)

Expert Solution

Answer 31

(d)

Expert Solution

Answer 32

Expert Solution

Answer 33

Explanation of Solution

The x component of linear momentum operator is shown below.

l^xf=ℏi(y∂∂z−z∂∂y) (1)

The y component of linear momentum operator is shown below.

l^yf=ℏi(z∂∂x−x∂∂z) (2)

The z component of linear momentum operator is shown below.

l^zf=ℏi(x∂∂y−y∂∂x) (3)

From equation (1), it is clear that the x component of linear momentum operator contains a variable function of angular momentum along y and z direction. This makes the permutation x→y→z. Similarly, from equation (2), the y component of linear momentum operator contains a variable function of angular momentum along x and z direction. This makes the permutation y→z→x. From equation (3), the z component of linear momentum operator contains a variable function of angular momentum along y and x direction. This makes the permutation z→x→y. Therefore, the angular momentum operator along x, y and z direction follows a cyclic permutation.

Answer 34

(e)

Interpretation Introduction

Interpretation:

The commutations of other two angular momentum operators, by following the cyclic order permutation have to be predicted.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

(e)

Expert Solution

Answer to Problem 7F.9P

The value of the commutation of l^y and l^z is shown below.

[l^y, l^z]=ℏil^x

The value of the commutation of l^z and l^x is shown below.

[l^z, l^x]=ℏil^y

Explanation of Solution

The value of the commutation of l^x and l^y is shown below.

[l^x, l^y]=ℏil^z

By following the cyclic order permutation, the value of the commutation of l^y and l^z is shown below.

[l^y, l^z]=ℏil^x

Similarly, the value of the commutation of l^z and l^x is shown below.

[l^z, l^x]=ℏil^y

This proves that the commutations of other two angular momentum operators follow cyclic permutation order.

Answer 35

(e)

Interpretation Introduction

Interpretation:

The commutations of other two angular momentum operators, by following the cyclic order permutation have to be predicted.

Concept introduction:

The importance of angular momentum is comparable to the linear momentum in translational motion. In quantum mechanics, commutator of two operators tells the measurement of two operators at a same time. The commutator of angular momentum in two different axes is calculated by the formula shown below.

[l^x, l^y]f=l^xl^yf−l^yl^xf

Answer 36

(e)

Expert Solution

Answer to Problem 7F.9P

The value of the commutation of l^y and l^z is shown below.

[l^y, l^z]=ℏil^x

The value of the commutation of l^z and l^x is shown below.

[l^z, l^x]=ℏil^y

Answer 37

(e)

Expert Solution

Answer 38

(e)

Expert Solution

Answer 39

Expert Solution

Answer 40

Explanation of Solution

The value of the commutation of l^x and l^y is shown below.

[l^x, l^y]=ℏil^z

By following the cyclic order permutation, the value of the commutation of l^y and l^z is shown below.

[l^y, l^z]=ℏil^x

Similarly, the value of the commutation of l^z and l^x is shown below.

[l^z, l^x]=ℏil^y

This proves that the commutations of other two angular momentum operators follow cyclic permutation order.

Answer 41

Ch. 7 - Prob. 7A.1ST Ch. 7 - Prob. 7A.2ST Ch. 7 - Prob. 7A.3ST Ch. 7 - Prob. 7B.1ST Ch. 7 - Prob. 7B.2ST Ch. 7 - Prob. 7B.3ST Ch. 7 - Prob. 7C.1ST Ch. 7 - Prob. 7C.2ST Ch. 7 - Prob. 7C.3ST Ch. 7 - Prob. 7C.4ST

Concept explainers

Answer to Problem 7F.9P

Explanation of Solution

Answer to Problem 7F.9P

Explanation of Solution

Answer to Problem 7F.9P

Explanation of Solution

Answer to Problem 7F.9P

Explanation of Solution

Answer to Problem 7F.9P

Explanation of Solution

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