PHYSICAL CHEMISTRY. VOL.1+2 (LL)(11TH)
PHYSICAL CHEMISTRY. VOL.1+2 (LL)(11TH)
11th Edition
ISBN: 9780198826910
Author: ATKINS
Publisher: Oxford University Press
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Question
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Chapter 7, Problem 7B.8P

(a)

Interpretation Introduction

Interpretation:

The probability of the finding of the particle between x=0 and x=L/2, y=0 and y=L/2 has to be calculated.

Concept introduction:

The wave function represents the exact position of the electron in an atom.  For a wave function to be acceptable, it must be normalized.  The wave function is represented by ψ in the quantum mechanics.

(a)

Expert Solution
Check Mark

Answer to Problem 7B.8P

The probability of finding the particle between x=0 and x=L/2, y=0 and y=L/2 is L34.

Explanation of Solution

The probability of finding the particle between x=0 and x=L/2, y=0 and y=L/2 is calculated by the following formula.

    Probability=y=0y=L/2x=0x=L/2ψ2dxdy                                                                      (1)

The given wave function is shown below.

    ψ=(2/L)sin(πx/L)sin(πy/L)

Substitute the value of wavefunction in equation (1) and separate the integrals as shown below.

    Probability=y=0y=L/2x=0x=L/2((2/L)sin(πx/L)sin(πy/L))2dxdy=(2/L)2y=0L/2sin2(πx/L)dyx=0L/2sin2(πy/L)dx

Use the identity shown below.

    sin2x=12(1cos2x)

On further solving the above integration,

    Probability=(2/L)2y=0L/2(1cos2(πy/L)2)dyx=0L/2(1cos2(πx/L)2)dx=14(4/L2)y=0L/2(1cos2(πy/L))dyx=0L/21cos2(πx/L)dx=L2[ysin2(πy/L)2(π/L)]y=0L/2[xsin2(πx/L)2(π/L)]x=0L/2=L2([(L2sin2(πL/2L)2(π/L))(0sin2(π×0/L)2(π/L))][(L2sin2(π×L/2L)2(π/L))(0sin2(π×0/L)2(π/L))])

On further solving the above equation,

    Probability=L2([(L2sin2(πL/2L)2(π/L))(0sin2(π×0/L)2(π/L))][(L2sin2(π×L/2L)2(π/L))(0sin2(π×0/L)2(π/L))])=L2([(L2sinπ2(π/L))(0sin02(π/L))][(L2sinπ2(π/L))(0sin02(π/L))])=L2([(L20)(00)][(L20)(00)])(sinnπ=0;n=0,1,2,3....)=L2(L2×L2)

On further solving the above integration,

    Probability=L34

Hence, the probability of finding the particle between x=0 and x=L/2, y=0 and y=L/2 is L34.

(b)

Interpretation Introduction

Interpretation:

The probability of finding the particle between x=L/4 and x=3L/4, y=L/4 and y=3L/4 is to be calculated.

Concept introduction:

The wave function represents the exact position of the electron in an atom.  For a wave function to be acceptable, it must be normalized.  The wave function is represented by ψ in the quantum mechanics.

(b)

Expert Solution
Check Mark

Answer to Problem 7B.8P

The probability of finding the particle between x=L/4 and x=3L/4, y=L/4 and y=3L/4 is L2(L2+Lπ)2.

Explanation of Solution

The probability of finding the particle between x=L/4 and x=3L/4, y=L/4 and y=3L/4 is calculated by the following formula.

    Probability=y=L/4y=3L/4x=L/4x=3L/4ψ2dxdy                                                                    (2)

The given wave function is shown below.

    ψ=(2/L)sin(πx/L)sin(πy/L)

Substitute the value of wavefunction in equation (2) and separate the integrals as shown below.

    Probability=y=L/4y=3L/4x=L/4x=3L/4((2/L)sin(πx/L)sin(πy/L))2dxdy=(2/L)2y=L/43L/4sin2(πx/L)dyx=03L/4sin2(πy/L)dx

Use the identity shown below.

    sin2x=12(1cos2x)

On further solving the above integration,

    Probability=(2/L)2y=L/43L/4(1cos2(πy/L)2)dyx=L/43L/4(1cos2(πx/L)2)dx=14(4/L2)y=L/43L/4(1cos2(πy/L))dyx=L/43L/41cos2(πx/L)dx=L2[ysin2(πy/L)2(π/L)]y=L/43L/4[xsin2(πx/L)2(π/L)]x=L/43L/4=L2([(3L4sin2(3πL/4L)2(π/L))(L4sin2(π×L/4L)2(π/L))][(3L4sin2(3π×L/4L)2(π/L))(L4sin2(π×L/4L)2(π/L))])

On further solving the above equation,

    Probability=L2([(3L4sin(3π/2)2(π/L))(L4sin(π/2)2(π/L))][(3L4sin(3π/2)2(π/L))(L4sin(π/2)2(π/L))])=L2([(3L412(π/L))(L412(π/L))][(L212(π/L))(012(π/L))])(sin(3π/2)=1andsin(π/2)=1)=L2([(3L4+L2π)(L4L2π)][(3L4+L2π)(L4L2π)])=L2((3L4+L2πL4+L2π)(3L4+L2πL4+L2π))

On further solving the above integration,

    Probability=L2((3L4+L2πL4+L2π)(3L4+L2πL4+L2π))=L2((2L4+2L2π)(2L4+2L2π))=L2((L2+Lπ)(L2+Lπ))=L2(L2+Lπ)2

Hence, the probability of finding the particle between x=L/4 and x=3L/4, y=L/4 and y=3L/4 is L2(L2+Lπ)2.

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Chapter 7 Solutions

PHYSICAL CHEMISTRY. VOL.1+2 (LL)(11TH)

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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