Concept explainers
(a)
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]=(ˆx⋅ˆpx−ˆpx⋅ˆx)
(a)
Answer to Problem 7C.15P
The commutator for [ˆH,ˆpx] using V(x)=V0 and V(x)=12ktx2 is zero and xiℏkt respectively.
Explanation of Solution
The commutator of the operators is given by the formula shown below.
[ˆH,ˆpx]=(ˆH⋅ˆpx−ˆpx⋅ˆH) (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=(ˆH⋅ˆpx−ˆpx⋅ˆH)ψ=((ˆp2x2m+V0)ˆpx−ˆpx(ˆp2x2m+V0))ψ
The above is equation is simplified by substituting the value of momentum operator (−iℏddx) as shown below.
[ˆH,ˆpx]ψ=((ˆp2x2m+V0)ˆpx−ˆpx(ˆp2x2m+V0))ψ=((12m(−iℏddx)2+V0)(−iℏddx)ψ−(−iℏddx)(12m(−iℏddx)2+V0)ψ)=((12m(−ℏ2d2dx2)+V0)(−iℏdψdx)−(−iℏddx)(12m(−ℏ2d2ψdx2)+V0ψ))=(12m(iℏ3d3ψdx3)+V0(−iℏdψdx))−(12m(iℏ3d3ψdx3)+V0(−iℏdψdx))
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=(ˆH⋅ˆpx−ˆpx⋅ˆH)ψ=((ˆp2x2m+12ktx2)ˆpx−ˆpx(ˆp2x2m+12ktx2))ψ
The above is equation is simplified by substituting the value of momentum operator (−iℏddx) as shown below.
[ˆH,ˆpx]ψ=((ˆp2x2m+12ktx2)ˆpx−ˆpx(ˆp2x2m+12ktx2))ψ=((12m(−iℏddx)2+12ktx2)(−iℏddx)ψ−(−iℏddx)(12m(−iℏddx)2+12ktx2)ψ)=((12m(−ℏ2d2dx2)+12ktx2)(−iℏdψdx)−(−iℏddx)(12m(−ℏ2d2ψdx2)+12ktx2ψ))=(12m(iℏ3d3ψdx3)+12ktx2(−iℏdψdx))−(12m(iℏ3d3ψdx3)+12ktx2(−iℏdψdx)−iℏ12ktdx2dxψ)
The above equation can be solved as shown below.
=(12m(iℏ3d3ψdx3)+12ktx2(−iℏdψdx))−(12m(iℏ3d3ψdx3)+12ktx2(−iℏdψdx)−xiℏktψ)=(xiℏkt)ψ
Therefore, the value of [ˆH,ˆpx] for V(x)=12ktx2 is xiℏkt.
(b)
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]=(ˆx⋅ˆpx−ˆpx⋅ˆx)
(b)
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(−ℏ2d2xdx2).
Explanation of Solution
The commutator of the operators is given by the formula shown below.
[ˆH,ˆx]=(ˆH⋅ˆx−ˆx⋅ˆH) (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=(ˆH⋅ˆx−ˆx⋅ˆH)ψ=((ˆp2x2m+V0)ˆx−ˆx(ˆp2x2m+V0))ψ
The above is equation is simplified by substituting the value of momentum operator (−iℏddx) as shown below.
[ˆH,ˆx]ψ=((ˆp2x2m+V0)ˆx−ˆx(ˆp2x2m+V0))ψ=((12m(−iℏddx)2+V0)xψ−x(12m(−iℏddx)2+V0)ψ)=((12m(−xℏ2d2ψdx2)+12m(−ℏ2d2xdx2)ψ+V0xψ)−x(12m(−ℏ2d2ψdx2)+V0ψ))=12m(−ℏ2d2xdx2)ψ
Therefore, the value of [ˆH,ˆx] for V(x)=V0 is 12m(−ℏ2d2xdx2).
Substitute the values of operators in the equation (2) for V(x)=12ktx2 as shown below. [ˆH,ˆx]ψ=(ˆH⋅ˆx−ˆx⋅ˆH)ψ=((ˆp2x2m+12ktx2)ˆx−ˆx(ˆp2x2m+12ktx2))ψ
The above is equation is simplified by substituting the value of momentum operator (−iℏddx) as shown below.
[ˆH,ˆx]ψ=((ˆp2x2m+12ktx2)ˆx−ˆx(ˆp2x2m+12ktx2))ψ=((12m(−iℏddx)2+12ktx2)xψ−x(12m(−iℏddx)2+12ktx2)ψ)=((12m(−ℏ2d2dx2)+12ktx2)xψ−x(12m(−ℏ2d2ψdx2)+12ktx2ψ))=(12m(−xℏ2d2ψdx2)+12m(−ℏ2d2xdx2)ψ+12ktx3ψ)−(12m(−xℏ2d2ψdx2)+12ktx3ψ)
The above equation can be solved as shown below.
=(12m(−xℏ2d2ψdx2)+12m(−ℏ2d2xdx2)ψ+12ktx3ψ)−(12m(−xℏ2d2ψdx2)+12ktx3ψ)=12m(−ℏ2d2xdx2)ψ
Therefore, the value of [ˆH,ˆx] for V(x)=12ktx2 is 12m(−ℏ2d2xdx2).
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Chapter 7 Solutions
Atkins' Physical Chemistry
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