Atkins' Physical Chemistry
Atkins' Physical Chemistry
11th Edition
ISBN: 9780198769866
Author: ATKINS, P. W. (peter William), De Paula, Julio, Keeler, JAMES
Publisher: Oxford University Press
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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^yfl^yl^xf

(a)

Expert Solution
Check Mark

Answer to Problem 7F.9P

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

    l^xl^yf=2(yfx+zy2fzxyx2fz2z22fyx+zx2fyz)

Explanation of Solution

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

    l^xl^yf=2(yzzy)(zfxxfz)

Multiply the parentheses as shown below.

    l^xl^yf=2(yz(zfxxfz)zy(zfxxfz))=2((yfx+zy2fzxyx2fz2)(z22fyxzx2fyz))=2(yfx+zy2fzxyx2fz2z22fyx+zx2fyz)

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

    l^xl^yf=2(yfx+zy2fzxyx2fz2z22fyx+zx2fyz)

(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^yfl^yl^xf

(b)

Expert Solution
Check Mark

Answer to Problem 7F.9P

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

    l^yl^xf=2(zy2fxzz22fxyxy2fz2+zx2fzy+xfy)

Explanation of Solution

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

    l^yl^xf=2(zxxz)(yfzzfy)

Multiply the parentheses as shown below.

    l^yl^xf=2(zxxz)(yfzzfy)=2(zx(yfzzfy)xz(yfzzfy))=2((zy2fxzz22fxy)(xy2fz2zx2fzyxfy))=2(zy2fxzz22fxyxy2fz2+zx2fzy+xfy)

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

    l^yl^xf=2(zy2fxzz22fxyxy2fz2+zx2fzy+xfy)

(c)

Interpretation Introduction

Interpretation:

The value of l^xl^yfl^yl^xf is il^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^yfl^yl^xf

(c)

Expert Solution
Check Mark

Answer to Problem 7F.9P

It has been proved that the value of l^xl^yfl^yl^xf is il^zf.

Explanation of Solution

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

    l^xl^yfl^yl^xf=((2(yfx+zy2fzxyx2fz2z22fyx+zx2fyz))(2(zy2fxzz22fxyxy2fz2+zx2fzy+xfy)))=2((yfx+zy2fzxyx2fz2z22fyx+zx2fyz)(zy2fxzz22fxyxy2fz2+zx2fzy+xfy))=2(yfx+zy2fzxyx2fz2z22fyx+zx2fyzzy2fxz+z22fxy+xy2fz2zx2fzyxfy)=2(yfxxfy)

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

    l^xl^yfl^yl^xf=i(i(yfxxfy))=i(i(xyyx))f(i(xyyx)=l^z)=il^zf

Hence, it has been proved that the value of l^xl^yfl^yl^xf is il^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^yfl^yl^xf

(d)

Expert Solution
Check Mark

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=i(yzzy)                                                                                                 (1)

The y component of linear momentum operator is shown below.

    l^yf=i(zxxz)                                                                                                 (2)

The z component of linear momentum operator is shown below.

    l^zf=i(xyyx)                                                                                                 (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 xyz.  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 yzx.  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 zxy.  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^yfl^yl^xf

(e)

Expert Solution
Check Mark

Answer to Problem 7F.9P

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

    [l^y,l^z]=il^x

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

    [l^z,l^x]=il^y

Explanation of Solution

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

    [l^x,l^y]=il^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]=il^x

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

    [l^z,l^x]=il^y

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

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Chapter 7 Solutions

Atkins' Physical Chemistry

Ch. 7 - Prob. 7D.1STCh. 7 - Prob. 7E.1STCh. 7 - Prob. 7E.2STCh. 7 - Prob. 7F.1STCh. 7 - Prob. 7A.1DQCh. 7 - Prob. 7A.2DQCh. 7 - Prob. 7A.3DQCh. 7 - Prob. 7A.4DQCh. 7 - Prob. 7A.1AECh. 7 - Prob. 7A.1BECh. 7 - Prob. 7A.2AECh. 7 - Prob. 7A.2BECh. 7 - Prob. 7A.3AECh. 7 - Prob. 7A.3BECh. 7 - Prob. 7A.4AECh. 7 - Prob. 7A.4BECh. 7 - Prob. 7A.5AECh. 7 - Prob. 7A.5BECh. 7 - Prob. 7A.6AECh. 7 - Prob. 7A.6BECh. 7 - Prob. 7A.7AECh. 7 - Prob. 7A.7BECh. 7 - Prob. 7A.8AECh. 7 - Prob. 7A.8BECh. 7 - Prob. 7A.9AECh. 7 - Prob. 7A.9BECh. 7 - Prob. 7A.10AECh. 7 - Prob. 7A.10BECh. 7 - Prob. 7A.11AECh. 7 - Prob. 7A.11BECh. 7 - Prob. 7A.12AECh. 7 - Prob. 7A.12BECh. 7 - Prob. 7A.13AECh. 7 - Prob. 7A.13BECh. 7 - Prob. 7A.1PCh. 7 - Prob. 7A.2PCh. 7 - Prob. 7A.3PCh. 7 - Prob. 7A.4PCh. 7 - Prob. 7A.5PCh. 7 - Prob. 7A.6PCh. 7 - Prob. 7A.7PCh. 7 - Prob. 7A.8PCh. 7 - Prob. 7A.9PCh. 7 - Prob. 7A.10PCh. 7 - Prob. 7B.1DQCh. 7 - Prob. 7B.2DQCh. 7 - Prob. 7B.3DQCh. 7 - Prob. 7B.1AECh. 7 - Prob. 7B.1BECh. 7 - Prob. 7B.2AECh. 7 - Prob. 7B.2BECh. 7 - Prob. 7B.3AECh. 7 - Prob. 7B.3BECh. 7 - Prob. 7B.4AECh. 7 - Prob. 7B.4BECh. 7 - Prob. 7B.5AECh. 7 - Prob. 7B.5BECh. 7 - Prob. 7B.6AECh. 7 - Prob. 7B.6BECh. 7 - Prob. 7B.7AECh. 7 - Prob. 7B.7BECh. 7 - Prob. 7B.8AECh. 7 - Prob. 7B.8BECh. 7 - Prob. 7B.1PCh. 7 - Prob. 7B.2PCh. 7 - Prob. 7B.3PCh. 7 - Prob. 7B.4PCh. 7 - Prob. 7B.5PCh. 7 - Prob. 7B.7PCh. 7 - Prob. 7B.8PCh. 7 - Prob. 7B.9PCh. 7 - Prob. 7B.11PCh. 7 - Prob. 7C.1DQCh. 7 - Prob. 7C.2DQCh. 7 - Prob. 7C.3DQCh. 7 - Prob. 7C.1AECh. 7 - Prob. 7C.1BECh. 7 - Prob. 7C.2AECh. 7 - Prob. 7C.2BECh. 7 - Prob. 7C.3AECh. 7 - Prob. 7C.3BECh. 7 - Prob. 7C.4AECh. 7 - Prob. 7C.4BECh. 7 - Prob. 7C.5AECh. 7 - Prob. 7C.5BECh. 7 - Prob. 7C.6AECh. 7 - Prob. 7C.6BECh. 7 - Prob. 7C.7AECh. 7 - Prob. 7C.7BECh. 7 - Prob. 7C.8AECh. 7 - Prob. 7C.8BECh. 7 - Prob. 7C.9AECh. 7 - Prob. 7C.9BECh. 7 - Prob. 7C.10AECh. 7 - Prob. 7C.10BECh. 7 - Prob. 7C.1PCh. 7 - Prob. 7C.2PCh. 7 - Prob. 7C.3PCh. 7 - Prob. 7C.4PCh. 7 - Prob. 7C.5PCh. 7 - Prob. 7C.6PCh. 7 - Prob. 7C.7PCh. 7 - Prob. 7C.8PCh. 7 - Prob. 7C.9PCh. 7 - Prob. 7C.11PCh. 7 - Prob. 7C.12PCh. 7 - Prob. 7C.13PCh. 7 - Prob. 7C.14PCh. 7 - Prob. 7C.15PCh. 7 - Prob. 7D.1DQCh. 7 - Prob. 7D.2DQCh. 7 - Prob. 7D.3DQCh. 7 - Prob. 7D.1AECh. 7 - Prob. 7D.1BECh. 7 - Prob. 7D.2AECh. 7 - Prob. 7D.2BECh. 7 - Prob. 7D.3AECh. 7 - Prob. 7D.3BECh. 7 - Prob. 7D.4AECh. 7 - Prob. 7D.4BECh. 7 - Prob. 7D.5AECh. 7 - Prob. 7D.5BECh. 7 - Prob. 7D.6AECh. 7 - Prob. 7D.6BECh. 7 - Prob. 7D.7AECh. 7 - Prob. 7D.7BECh. 7 - Prob. 7D.8AECh. 7 - Prob. 7D.8BECh. 7 - Prob. 7D.9AECh. 7 - Prob. 7D.9BECh. 7 - Prob. 7D.10AECh. 7 - Prob. 7D.10BECh. 7 - Prob. 7D.11AECh. 7 - Prob. 7D.11BECh. 7 - Prob. 7D.12AECh. 7 - Prob. 7D.12BECh. 7 - Prob. 7D.13AECh. 7 - Prob. 7D.13BECh. 7 - Prob. 7D.14AECh. 7 - Prob. 7D.14BECh. 7 - Prob. 7D.15AECh. 7 - Prob. 7D.15BECh. 7 - Prob. 7D.1PCh. 7 - Prob. 7D.2PCh. 7 - Prob. 7D.3PCh. 7 - Prob. 7D.4PCh. 7 - Prob. 7D.5PCh. 7 - Prob. 7D.6PCh. 7 - Prob. 7D.7PCh. 7 - Prob. 7D.8PCh. 7 - Prob. 7D.9PCh. 7 - Prob. 7D.11PCh. 7 - Prob. 7D.12PCh. 7 - Prob. 7D.14PCh. 7 - Prob. 7E.1DQCh. 7 - Prob. 7E.2DQCh. 7 - Prob. 7E.3DQCh. 7 - Prob. 7E.1AECh. 7 - Prob. 7E.1BECh. 7 - Prob. 7E.2AECh. 7 - Prob. 7E.2BECh. 7 - Prob. 7E.3AECh. 7 - Prob. 7E.3BECh. 7 - Prob. 7E.4AECh. 7 - Prob. 7E.4BECh. 7 - Prob. 7E.5AECh. 7 - Prob. 7E.5BECh. 7 - Prob. 7E.6AECh. 7 - Prob. 7E.6BECh. 7 - Prob. 7E.7AECh. 7 - Prob. 7E.7BECh. 7 - Prob. 7E.8AECh. 7 - Prob. 7E.8BECh. 7 - Prob. 7E.9AECh. 7 - Prob. 7E.9BECh. 7 - Prob. 7E.1PCh. 7 - Prob. 7E.2PCh. 7 - Prob. 7E.3PCh. 7 - Prob. 7E.4PCh. 7 - Prob. 7E.5PCh. 7 - Prob. 7E.6PCh. 7 - Prob. 7E.7PCh. 7 - Prob. 7E.8PCh. 7 - Prob. 7E.9PCh. 7 - Prob. 7E.12PCh. 7 - Prob. 7E.15PCh. 7 - Prob. 7E.16PCh. 7 - Prob. 7E.17PCh. 7 - Prob. 7F.1DQCh. 7 - Prob. 7F.2DQCh. 7 - Prob. 7F.3DQCh. 7 - Prob. 7F.1AECh. 7 - Prob. 7F.1BECh. 7 - Prob. 7F.2AECh. 7 - Prob. 7F.2BECh. 7 - Prob. 7F.3AECh. 7 - Prob. 7F.3BECh. 7 - Prob. 7F.4AECh. 7 - Prob. 7F.4BECh. 7 - Prob. 7F.5AECh. 7 - Prob. 7F.5BECh. 7 - Prob. 7F.6AECh. 7 - Prob. 7F.6BECh. 7 - Prob. 7F.7AECh. 7 - Prob. 7F.7BECh. 7 - Prob. 7F.8AECh. 7 - Prob. 7F.8BECh. 7 - Prob. 7F.9AECh. 7 - Prob. 7F.9BECh. 7 - Prob. 7F.10AECh. 7 - Prob. 7F.10BECh. 7 - Prob. 7F.11AECh. 7 - Prob. 7F.11BECh. 7 - Prob. 7F.12AECh. 7 - Prob. 7F.12BECh. 7 - Prob. 7F.13AECh. 7 - Prob. 7F.13BECh. 7 - Prob. 7F.14AECh. 7 - Prob. 7F.14BECh. 7 - Prob. 7F.1PCh. 7 - Prob. 7F.4PCh. 7 - Prob. 7F.6PCh. 7 - Prob. 7F.7PCh. 7 - Prob. 7F.8PCh. 7 - Prob. 7F.9PCh. 7 - Prob. 7F.10PCh. 7 - Prob. 7F.11PCh. 7 - Prob. 7.3IACh. 7 - Prob. 7.4IACh. 7 - Prob. 7.5IACh. 7 - Prob. 7.6IA
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