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.11P

(a)

Interpretation Introduction

Interpretation:

The expectation values x, x2, px, and px2 have to be determined.

Concept introduction:

The expectation value is the average value of a large number of measurements of an observable.  The expectation value of a measurable parameter is calculated by the formula shown below.

  Ω=ψ*(Ω)^ψdτ

Where,

  • ψ is the wave function.
  • Ω is an observable.

(a)

Expert Solution
Check Mark

Answer to Problem 7C.11P

The value of x is 0_.  The value of x2 is 14a.  The value of px is 0_.  The value of px2 is a2.

Explanation of Solution

The value of ψ is given below.

    ψ(x)=(2aπ)1/4eax2        (1)

Where,

  • a is a constant.

The expectation value of x for a particle is given by the expression.

    x=ψ*(x)^ψdx        (2)

Substitute the value of ψ in equation (2).

    x=((2aπ)1/4eax2)*(x)^((2aπ)1/4eax2)dx=(2aπ)1/2(e2ax2)xdx        (3)

As the integral in equation (3) is odd, therefore, the value of the integrand will be zero.

    x=(2aπ)1/2(e2ax2)xdx=0_

Therefore, the expectation value of x is 0_.

The expectation value of x2 for a particle is given by the expression.

    x2=ψ*(x2)^ψdx        (4)

Substitute the value of ψ in equation (4).

    x2=((2aπ)1/4eax2)*(x2)^((2aπ)1/4eax2)dx=(2aπ)1/2(e2ax2)x2dx=(2aπ)1/2×12(π(2a)3)1/2(x2ekx2dx=12(πk3)1/2)=(2aπ)1/2×12a×(2π4a)1/2        (5)

Equation (5) can be solved further to determine the value of x2 as shown below.

    x2=(2aπ)1/2×12a×(2π4a)1/2=(2aπ)1/2×18a×(2πa)1/2=14a

Therefore, the value of x2 is 14a.

The momentum operator, px^ is given by the expression.

    px^=iddx        (6)

Where,

  • is a constant.

The expectation value of px^ for a particle is given by the expression.

    px=ψ*(px^)ψdx        (7)

Substitute the values of ψ and equation (6) in equation (7).

    px=((2aπ)1/4eax2)*(iddx)((2aπ)1/4eax2)dx=(2aπ)1/2×(hi)(eax2)(ddx)(eax2)dx=(2aπ)1/2×(hi)(eax2)(2ax)(eax2)dx=(2a)×(2aπ)1/2×(hi)x(e2ax2)dx        (8)

As the integral in equation (8) is odd, therefore, the value of the integrand will be zero.

    px=(2a)×(2aπ)1/2×(hi)x(e2ax2)dx=0_

Therefore, the expectation value of px is 0_.

The momentum operator, px2^ is given by the expression.

    px2^=2d2dx2        (9)

The expectation value of px2^ for a particle is given by the expression.

    px2=0Lψ*(px2^)ψdx        (10)

Substitute the value of ψ and equation (9) in equation (10).

    px2=((2aπ)1/4eax2)(2d2dx2)((2aπ)1/4eax2)dx=(2aπ)1/2×(2)×eax2ddx(eax2(2ax))dx=(2aπ)1/2×2×(2a)eax2(eax2+(2ax2eax2))dx=(2aπ)1/2×2×(2a)(eax2eax2dx2ax2eax2eax2dx)        (11)

The equation (11) can be solved further as shown below.

  px2=(2aπ)1/2×(2)×(2a)(eax2eax2dx2ax2e2ax2dx)=(2aπ)1/2×2×(2a)((π2a)1/22a(12(π(2a)3)1/2))(x2ekx2dx=12(πk3)1/2)=2a2((2aπ)1/2×(π2a)1/2(2a×(2aπ)1/2×18a×(2πa)1/2))=2a2(112)

The above equation can be solved further as shown below.

    px2=a2

Therefore, the value of px2 is a2.

(b)

Interpretation Introduction

Interpretation:

The value of Δpx and Δx have to be determined.

Concept introduction:

According to the Heisenberg uncertainty principle, it is impossible to specify simultaneously, with arbitrary precision, both the linear momentum and the position of a particle.  According to it if the position is known, the momentum cannot be predicted simultaneously. Δx represents the uncertainty in position and Δpx represents the uncertainty in momentum.

(b)

Expert Solution
Check Mark

Answer to Problem 7C.11P

The value of Δx is 12a1/2.  The value of Δpx is a1/2.

Explanation of Solution

The expression to determine Δx value is given below.

    Δx={x2x2}1/2        (12)

The value of x is 0.

The value of x2 is 14a.

Substitute the values of x and x2 in equation (12).

    Δx={14a0}1/2=12a1/2

Therefore, the value of Δx is 12a1/2.

The expression to determine Δpx value is given below.

    Δpx={px2px2}1/2        (13)

The value of px is 0.

The value of px2 is a2.

Substitute the values of px and px2 in equation (13).

    Δpx={a20}1/2=a1/2

Therefore, the value of Δpx is a1/2.

(c)

Interpretation Introduction

Interpretation:

Whether the value for the product, ΔpxΔx is consistent with the predictions from the uncertainty principle or not has to be determined.

Concept introduction:

According to the Heisenberg uncertainty principle, it is impossible to specify simultaneously, with arbitrary precision, both the linear momentum and the position of a particle.  According to it if the position is known, the momentum cannot be predicted simultaneously.

(c)

Expert Solution
Check Mark

Answer to Problem 7C.11P

The value for the product, ΔpxΔx is consistent with the predictions from the uncertainty principle.

Explanation of Solution

The value of Δx is 12a1/2.

The value of Δpx is a1/2.

Therefore, the value of ΔpxΔx is determined below.

    ΔpxΔx=12a1/2×a1/2=2

This is in accordance with the uncertainty principle according to which the expression given below holds true.

    ΔpqΔq2

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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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