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.11P
Interpretation Introduction

Interpretation:

The operator l^z in spherical polar co-ordinates can be expressed as l^z=i/ϕ has to be shown.

Concept introduction:

The vector model of angular momentum calculates the value of orbital angular momentum for an electron in an atom.  The vector model is represented by a cone.  The length of the cone should be equal to {l(l+1)}1/2 and a projection of ml along the z axis.

Expert Solution & Answer
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Answer to Problem 7F.11P

It has been proved that the operator l^z in spherical polar co-ordinates can be expressed as l^z=i/ϕ.

Explanation of Solution

The operator l^z in cartesian co-ordinates is expressed as shown below.

    l^z=i(xyyx)        (1)

The relation between the cartesian co-ordinates and polar-co-ordinates is shown below.s

    x=rsinθcosϕ, y=rsinθsinϕ and z=rcosθ        (2)

The relation between r and x,y and z is shown below.

    r2=x2+y2+z2        (3)

Differentitate equation (3) with respect to x as shown below.

    2rrx=2x2rrx=2rsinθcosϕ(x=rsinθcosϕ)rx=sinθcosϕ

Differentitate equation (3) with respect to y as shown below.

    2rry=2y2rry=2rsinθsinϕ(y=rsinθsinϕ)ry=sinθsinϕ

The relation of z co-ordinate with spherical co-ordinate is calculated as shown below.

    z=rcosθcosθ=zr        (4)

Substitute the value of r from equation (3) in equation (4).

    cosθ=zx2+y2+z2        (5)

Differentiate equation (5) with respect to x.

    cosθx=x(zx2+y2+z2)sinθθx=12z(x2+y2+z2)3/2x2xθx=12sinθz(r2)3/22x(r2=x2+y2+z2)=xsinθzr3        (6)

Substitute the value of x and z from equation (2) to equation (6).

    θx=rsinθcosϕsinθrcosθr3=cosϕcosθr

Differentiate equation (5) with respect to y.

    cosθy=y(zx2+y2+z2)sinθθy=12z(x2+y2+z2)3/2y2yθy=12sinθz(r2)3/22y(r2=x2+y2+z2)=ysinθzr3        (7)

Substitute the value of y and z from equation (2) to equation (7).

    θy=rsinθsinϕsinθrcosθr3=sinϕcosθr

The value of yx is calculated as shown below.

    yx=rsinθsinϕrsinθcosϕ=tanϕ        (8)

Differentiate equation (8) with respect to x.

    x(yx)=tanϕxyx2=sec2ϕϕxrsinθsinϕ(rsinθcosϕ)2=sec2ϕϕxsinϕrsinθcos2ϕ=sec2ϕϕx        (9)

Rearrange and further solve the equation (9) as shown below.

    ϕx=sinϕrsinθcos2ϕsec2ϕ=sinϕrsinθ(cos2ϕsec2ϕ=1)

Differentiate equation (8) with respect to y.

    y(yx)=tanϕy1x=sec2ϕϕy1rsinθcosϕ=sec2ϕϕy        (10)

Rearrange and further solve the equation (10) as shown below.

    sec2ϕϕy=1rsinθcosϕϕy=1rsinθcosϕsec2ϕ(cosϕsec2ϕ=secϕ)=1rsinθsecϕ=cosϕrsinθ(1secϕ=cosϕ)

The value of x is calculated by the following formula.

    x=(rx)r+(θx)θ+(ϕx)ϕ        (11)

Substitute the value of rx, θx and ϕx in equation (11).

    x=(sinθcosϕ)r+(cosϕcosθr)θ+(sinϕrsinθ)ϕ

The value of y is calculated by the following formula.

    y=(ry)r+(θy)θ+(ϕy)ϕ        (12)

Substitute the value of ry, θy and ϕy in equation (12).

    y=(sinθsinϕ)r+(sinϕcosθr)θ(cosϕrsinθ)ϕ

Substitute the value of x, y, x and y in equation (1).

    l^z=i(rsinθcosϕ((sinθsinϕ)r+(sinϕcosθr)θ(cosϕrsinθ)ϕ)rsinθsinϕ((sinθcosϕ)r+(cosϕcosθr)θ+(sinϕrsinθ)ϕ))=i(((rsin2θcosϕsinϕ)r+(sinθcosϕsinϕcosθ)θ(cos2ϕ)ϕ)((rsin2θsinϕcosϕ)r+(sinθsinϕcosϕcosθ)θ(sin2ϕ)ϕ))=i((rsin2θcosϕsinϕ)r+(sinθcosϕsinϕcosθ)θ(cos2ϕ)ϕ(rsin2θsinϕcosϕ)r(sinθcosϕsinϕcosθ)θ(sin2ϕ)ϕ)=i((cos2ϕ)ϕ+(sin2ϕ)ϕ)

Further solving the above equation using identity (cos2ϕ)+(sin2ϕ)=1.

    l^z=i(ϕ)

Hence, it has been proved that the value of operator l^z in spherical polar co-ordinates can be expressed as l^z=i/ϕ.

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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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