Q. 32 Prove that the normalization con... [FREE SOLUTION]

Chapter 41: Q. 32 (page 1208)

Prove that the normalization constant of the $1 s$ radial wave function of the hydrogen atom is localid="1650358688008" ${(蟺 {a_{B}}^{3})}^{- 1 / 2}$ , as given in Equations 41.7.

Short Answer

Expert verified

The normalization constant of the $1 s$ radial wave function of the hydrogen atom is $A_{1 s} =$ ${(蟺 {a_{B}}^{3})}^{- 1 / 2}$

Step by step solution

Given information

We have to prove that the normalization constant of the $1 s$ radial wave function of the hydrogen atom is ${(蟺 {a_{B}}^{3})}^{- 1 / 2}$ .

Simplification

The normalization condition for the three-dimensional hydrogen atom is

${鈭�}_{0}^{鈭�} P_{r} (r) d r = 4 蟺 {鈭�}_{0}^{鈭�} r^{2} {|R_{n l} (r)|}^{2} d r = 1$ ( $P_{r}$ = probability density , $R_{n l}$ = radial wave function , $a_{B}$ = Bohr radius)

From Equation 41.7, the $1 s$ radial wave function can be written $R_{1 s} (r) = A_{1 s} e^{- r / a_{B}}$ . Thus,

$1 = 4 蟺 {A^{2}}_{1 s} {鈭�}_{0}^{鈭�} r^{2} e^{- 2 r / a_{B}} d r = 4 蟺 {A^{2}}_{1 s} \frac{2!}{{(2 / a_{B})}^{3}} = 蟺 {a^{3}}_{B} {A^{2}}_{1 s}$

$鈬� A_{1 s} = {(蟺 {a^{3}}_{B})}^{- 1 / 2}$

Hence proved.

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Prove that the normalization constant of the $1 s$ radial wave function of the hydrogen atom is localid="1650358688008" ${(蟺 {a_{B}}^{3})}^{- 1 / 2}$ , as given in Equations 41.7.

Short Answer

Step by step solution

Given information

Simplification

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91影视

Prove that the normalization constant of the 1sradial wave function of the hydrogen atom is localid="1650358688008" 蟺aB3-1/2, as given in Equations 41.7.

Short Answer

Step by step solution

Given information

Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Rotational Dynamics

Work Energy and Power

Waves Physics

Energy Physics

Wave Optics

Electricity

Study anywhere. Anytime. Across all devices.

Prove that the normalization constant of the $1 s$ radial wave function of the hydrogen atom is localid="1650358688008" ${(蟺 {a_{B}}^{3})}^{- 1 / 2}$ , as given in Equations 41.7.