Physics for Scientists and Engineers
Physics for Scientists and Engineers
6th Edition
ISBN: 9781429281843
Author: Tipler
Publisher: MAC HIGHER
Question
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Chapter 36, Problem 39P

(a)

To determine

The value of ψ(r) at r=a0 .

(a)

Expert Solution
Check Mark

Answer to Problem 39P

The value of ψ(r) at r=a0 is 142eπ( a 0 )3/2 .

Explanation of Solution

Given:

The radius of Bohr bolt is r=a0 .

Formula used:

The expression for the spherical wave function for n=2,l=0,ml=0 is given by,

  ψ200=C200(2Zra0)eZr/2a0

The expression for the constant term C200 is given by,

  C200=142π(Z a 0 )3/2

The new expression for the spherical wave function is given as,

  ψ200=C200(2 Zr a 0 )eZr/2 a 0=14 2π( Z a 0 )3/2(2 Zr a 0 )eZr/2 a 0

Calculation:

The atomic number of hydrogen atom is 1 .

The spherical wave function for n=2,l=0,ml=0 is calculated as,

  ψ200=14 2π( Z a 0 )3/2(2 Zr a 0 )eZr/2 a 0=14 2π( 1 a 0 )3/2(2 1( a 0 ) a 0 )e( 1)( a 0 )/2 a 0=14 2eπ ( a 0 ) 3/2

Conclusion:

Therefore, the value of ψ(r) at r=a0 is 142eπ( a 0 )3/2 .

(b)

To determine

The value of ψ2(r) at r=a0 .

(b)

Expert Solution
Check Mark

Answer to Problem 39P

The value of ψ2(r) at r=a0 is 0.00366a03 .

Explanation of Solution

Given:

The radius of Bohr bolt is r=a0 .

Formula used:

The expression for the spherical wave function for n=2,l=0,ml=0 is given by,

  ψ200=C200(2Zra0)eZr/2a0

The expression for the constant term C200 is given by,

  C200=142π(Z a 0 )3/2

The new expression for the spherical wave function is given as,

  ψ200=C200(2 Zr a 0 )eZr/2 a 0=14 2π( Z a 0 )3/2(2 Zr a 0 )eZr/2 a 0

Calculation:

The atomic number of hydrogen atom is 1 .

The spherical wave function for n=2,l=0,ml=0 is calculated as,

  ψ200=14 2π( Z a 0 )3/2(2 Zr a 0 )eZr/2 a 0=14 2π( 1 a 0 )3/2(2 1( a 0 ) a 0 )e( 1)( a 0 )/2 a 0ψ2002=( 1 4 2eπ ( a 0 ) 3/2 )2ψ2002=0.00366a03

Conclusion:

Therefore, the value of ψ2(r) at r=a0 is 0.00366a03 .

(c)

To determine

The value of radial probability density P(r) at r=a0 .

(c)

Expert Solution
Check Mark

Answer to Problem 39P

The value of radial probability density P(r) at r=a0 is 0.0460a0 .

Explanation of Solution

Given:

The radius of Bohr bolt is r=a0 .

Formula used:

The expression for the radial probability density of finding a particle at a position r is given by,

  P(r)=4πr2|ψ(r)|2

Calculation:

The radial probability density is calculated as,

  P(r)=4πr2|ψ(r)|2=4π( a 0)2( 0.00366 a 0 3 )=0.0460a0

Conclusion:

Therefore, the value of radial probability density P(r) at r=a0 is 0.0460a0 .

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