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 7C.15P

(a)

Interpretation Introduction

Interpretation:

The commutator for [ˆH,ˆpx] has to be calculated using (i) V(x)=V0 and (ii) V(x)=12ktx2.

Concept introduction:

The commutators are the operators obtained from the operators which commute with each other.  The commutator of the operators like position and momentum operators is given by the formula shown below.

    [ˆx,ˆpx]=(ˆxˆpxˆpxˆx)

(a)

Expert Solution
Check Mark

Answer to Problem 7C.15P

The commutator for [ˆH,ˆpx] using V(x)=V0 and V(x)=12ktx2 is zero and xikt respectively.

Explanation of Solution

The commutator of the operators is given by the formula shown below.

    [ˆH,ˆpx]=(ˆHˆpxˆpxˆH)        (1)

The value of operator ˆH is given as shown below.

    ˆH=ˆp2x2m+ˆV(x)

Substitute the values of operators in the equation (1) for V(x)=V0 as shown below.    [ˆH,ˆpx]ψ=(ˆHˆpxˆpxˆH)ψ=((ˆp2x2m+V0)ˆpxˆpx(ˆp2x2m+V0))ψ

The above is equation is simplified by substituting the value of momentum operator (iddx) as shown below.

    [ˆH,ˆpx]ψ=((ˆp2x2m+V0)ˆpxˆpx(ˆp2x2m+V0))ψ=((12m(iddx)2+V0)(iddx)ψ(iddx)(12m(iddx)2+V0)ψ)=((12m(2d2dx2)+V0)(idψdx)(iddx)(12m(2d2ψdx2)+V0ψ))=(12m(i3d3ψdx3)+V0(idψdx))(12m(i3d3ψdx3)+V0(idψdx))

The above equation on solving gives the value of zero.

Therefore, the value of [ˆH,ˆpx] for V(x)=V0 is zero.

Substitute the values of operators in the equation (1) for V(x)=12ktx2 as shown below.    [ˆH,ˆpx]ψ=(ˆHˆpxˆpxˆH)ψ=((ˆp2x2m+12ktx2)ˆpxˆpx(ˆp2x2m+12ktx2))ψ

The above is equation is simplified by substituting the value of momentum operator (iddx) as shown below.

    [ˆH,ˆpx]ψ=((ˆp2x2m+12ktx2)ˆpxˆpx(ˆp2x2m+12ktx2))ψ=((12m(iddx)2+12ktx2)(iddx)ψ(iddx)(12m(iddx)2+12ktx2)ψ)=((12m(2d2dx2)+12ktx2)(idψdx)(iddx)(12m(2d2ψdx2)+12ktx2ψ))=(12m(i3d3ψdx3)+12ktx2(idψdx))(12m(i3d3ψdx3)+12ktx2(idψdx)i12ktdx2dxψ)

The above equation can be solved as shown below.

=(12m(i3d3ψdx3)+12ktx2(idψdx))(12m(i3d3ψdx3)+12ktx2(idψdx)xiktψ)=(xikt)ψ

Therefore, the value of [ˆH,ˆpx] for V(x)=12ktx2 is xikt.

(b)

Interpretation Introduction

Interpretation:

The commutator for [ˆH,ˆx] has to be calculated using (i) V(x)=V0 and (ii) V(x)=12ktx2.

Concept introduction:

The commutators are the operator obtained from the operators which commute with each other.  The commutator of the operators like position and momentum operators is given by the formula shown below.

    [ˆx,ˆpx]=(ˆxˆpxˆpxˆx)

(b)

Expert Solution
Check Mark

Answer to Problem 7C.15P

The commutator for [ˆH,ˆx] using V(x)=V0 and V(x)=12ktx2 is same and is equals to 12m(2d2xdx2).

Explanation of Solution

The commutator of the operators is given by the formula shown below.

    [ˆH,ˆx]=(ˆHˆxˆxˆH)        (2)

The value of operator ˆH is given as shown below.

    ˆH=ˆp2x2m+ˆV(x)

Substitute the values of operators in the equation (2) for V(x)=V0 as shown below.    [ˆH,ˆx]ψ=(ˆHˆxˆxˆH)ψ=((ˆp2x2m+V0)ˆxˆx(ˆp2x2m+V0))ψ

The above is equation is simplified by substituting the value of momentum operator (iddx) as shown below.

    [ˆH,ˆx]ψ=((ˆp2x2m+V0)ˆxˆx(ˆp2x2m+V0))ψ=((12m(iddx)2+V0)xψx(12m(iddx)2+V0)ψ)=((12m(x2d2ψdx2)+12m(2d2xdx2)ψ+V0xψ)x(12m(2d2ψdx2)+V0ψ))=12m(2d2xdx2)ψ

Therefore, the value of [ˆH,ˆx] for V(x)=V0 is 12m(2d2xdx2).

Substitute the values of operators in the equation (2) for V(x)=12ktx2 as shown below.    [ˆH,ˆx]ψ=(ˆHˆxˆxˆH)ψ=((ˆp2x2m+12ktx2)ˆxˆx(ˆp2x2m+12ktx2))ψ

The above is equation is simplified by substituting the value of momentum operator (iddx) as shown below.

    [ˆH,ˆx]ψ=((ˆp2x2m+12ktx2)ˆxˆx(ˆp2x2m+12ktx2))ψ=((12m(iddx)2+12ktx2)xψx(12m(iddx)2+12ktx2)ψ)=((12m(2d2dx2)+12ktx2)xψx(12m(2d2ψdx2)+12ktx2ψ))=(12m(x2d2ψdx2)+12m(2d2xdx2)ψ+12ktx3ψ)(12m(x2d2ψdx2)+12ktx3ψ)

The above equation can be solved as shown below.

  =(12m(x2d2ψdx2)+12m(2d2xdx2)ψ+12ktx3ψ)(12m(x2d2ψdx2)+12ktx3ψ)=12m(2d2xdx2)ψ

Therefore, the value of [ˆH,ˆx] for V(x)=12ktx2 is 12m(2d2xdx2).

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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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