Explanation Given Information: 2 s and 2 p state Formula Used: The expression for the uncertainty in r is as follows: Δ r = r 2 ¯ − r ¯ 2 Calculation: For 2 s state: n = 2 and l = 0 the radial solution for the 2 s state is as follows. R 2 s = 2 ( 2 a 0 ) 3 2 ( 1 − r 2 a 0 ) e − r 2 a 0 For 2 p state: n = 2 and l = 1 the radial solution for the 2 s state is as follows. R 2 p = 2 ( 2 a 0 ) 3 2 ( r 3 a 0 ) e − r 2 a 0 Here, a 0 is the Bohr radius and its value is a 0 = 0.0529 n m For 2 s state The expression for the expectation value of r is as follows. r ¯ = ∫ 0 ∞ r P ( r ) d r Here P ( r ) is the probability per unit radial distance and the value of P ( r ) is P ( r ) = r 2 R 2 ( r ) Substitute 2 ( 2 a 0 ) 3 2 ( 1 − r 2 a 0 ) e − r 2 a 0 for R ( r ) in equation P ( r ) = r 2 R 2 ( r ) P ( r ) = r 2 ( 2 ( 2 a 0 ) 3 2 ( 1 − r 2 a 0 ) e − r 2 a 0 ) 2 = r 2 2 a 0 3 ( 1 − r a 0 + r 2 4 a 0 2 ) e − r a 0 = 1 2 a 0 3 ( r 2 − r 3 a 0 + r 4 4 a 0 2 ) e − r a 0 Substitute 1 2 a 0 3 ( r 2 − r 3 a 0 + r 4 4 a 0 2 ) e − r a 0 for P ( r ) in equation r ¯ = ∫ 0 ∞ r P ( r ) d r and solve for r ¯ . r ¯ = ∫ 0 ∞ ( r ) 1 2 a 0 3 ( r 2 − r 3 a 0 + r 4 4 a 0 2 ) e − r a 0 d r = 1 2 a 0 3 ∫ 0 ∞ ( r 3 − r 4 a 0 + r 5 4 a 0 2 ) e − r a 0 d r = 1 2 a 0 3 ( 3 ! ( 1 / a 0 ) 4 − 1 a 0 ( 4 ! ( 1 / a 0 ) 5 ) + 1 4 a 0 ( 5 ! ( 1 / a 0 ) 6 ) ) = 6 a 0 The expression for the expectation value of r is as follows. r 2 ¯ = ∫ 0 ∞ r 2 P ( r ) d r Substitute 1 2 a 0 3 ( r 2 − r 3 a 0 + r 4 4 a 0 2 ) e − r a 0 for P ( r ) in equation r 2 ¯ = ∫ 0 ∞ r 2 P ( r ) d r and solve for r 2 ¯ . r 2 ¯ = ∫ 0 ∞ ( r 2 ) 1 2 a 0 3 ( r 2 − r 3 a 0 + r 4 4 a 0 2 ) e − r a 0 d r = 1 2 a 0 3 ∫ 0 ∞ ( r 4 − r 5 a 0 + r 6 4 a 0 2 ) e − r a 0 d r = 1 2 a 0 3 ( 4 ! ( 1 / a 0 ) 5 − 1 a 0 ( 5 ! ( 1 / a 0 ) 6 ) + 1 4 a 0 ( 6 ! ( 1 / a 0 ) 7 ) ) = 42 a 0 2 Calculate the uncertainty in r for 2 s state as follows: The expression for the uncertainty in r is as follows: Δ r = r 2 ¯ − r ¯ 2 Substitute 42 a 0 2 for r 2 ¯ and 6 a 0 for r ¯ in equation Δ r = r 2 ¯ − r ¯ 2 and solve for Δ r

2nd Edition

ISBN: 9780805303087

Author: Randy Harris

Publisher: Addison Wesley

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Question

Chapter 7, Problem 62E

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To Calculate:

The uncertainties in r for the 2s and 2p states.

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Chapter 7, Problem 61E

Chapter 7, Problem 63E

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Chapter 7 Solutions

Modern Physics

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Knowledge Booster

The uncertainties in r for the 2 s and 2 p states.

To Calculate: The uncertainties in r for the 2s and 2p states.

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Chapter 7 Solutions

To Calculate:

The uncertainties in r for the 2s and 2p states.